Home Mathematics New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
Article Open Access

New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization

  • Tareq Saeed , Muhammad Adil Khan EMAIL logo , Shah Faisal , Hamed H. Alsulami and Mohammed Sh. Alhodaly
Published/Copyright: May 11, 2023
Download Article (PDF)
Become an author with De Gruyter Brill
Author Information Explore this Subject
Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

The Hermite-Hadamard inequality is regarded as one of the most favorable inequalities from the research point of view. Currently, mathematicians are working on extending, improving, and generalizing this inequality. This article presents conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in weighted and unweighted forms by using the idea of majorization and convexity together with generalized conformable fractional integral operators. They not only represent continuous and discrete inequalities in compact form but also produce generalized inequalities connecting various fractional operators such as Hadamard, Katugampola, Riemann-Liouville, conformable, and Rieman integrals into one single form. Also, two new integral identities have been investigated pertaining a differentiable function and three tuples. By using these identities and assuming ∣ f ′ ∣ and ∣ f ′ ∣ q ( q > 1 ) as convex, we deduce bounds concerning the discrepancy of the terms of the main inequalities.

Keywords: Jensen inequality; Mercer inequality; Hermite-Hadamard inequality; Hölder inequality; majorization theory
MSC 2010: 26D15; 26A51; 26A33; 26A42

1 Introduction

In mathematics, inequalities are very helpful, especially when working with the quantities about which we are not sure what exactly they are equal to. Sometimes, rather than writing down the exact solution, one can solve a mathematical problem by approximating the solution [1–3]. Nowadays, inequalities are recognized and taught as some of the most practical areas of mathematics because they have successfully extended their validity to the fields of engineering [4], economics [5], mathematical statistics [6,7], and information theory [8].

Inequalities are deeply connected to convexity theory. Convexity theory can be used to produce numerous concepts concerning mathematical inequalities and their uses in different branches of science [9–13]. Convex functions are one of the fundamental concepts in convexity theory. This concept has led to the discovery of new inequalities, among which the Hermite-Hadamard [14], Jensen’s [15], the Jensen-Mercer [16], Ostrwaski [17], and Fejér [18] inequalities are notable. In recent years, the Hermite-Hadamard inequality, which is considered to be the most attractive inequality in the literature, has drawn a lot of attention. The definition of this inequality is given as follows:

Let f : [ α 1 , α 2 ] → R be a convex function. Then

(1) f α 1 + α 2 2 ≤ 1 α 2 − α 1 ∫ α 1 α 2 f ( u ) d u ≤ f ( α 1 ) + f ( α 2 ) 2 .

If the function being applied is concave, the inequality stated earlier can be obtained in the opposite direction. Researchers have also used several classes of convex functions, including s -convex function [19], coordinate convex function [20], strongly convex function [21], and η -convex function [22] to derive this inequality.

The fractional calculus is another field that has drawn the attention of many researchers working in different fields. The impact of this field seems to have increased significantly over the past years in both pure and applied branches of science and engineering [23–29]. Currently, researchers are using convexity and the concepts from this subject to produce new findings [30–33]. There are various known versions of the fractional integral operators used in fractional calculus, among which the two operators, i.e., Riemann-Lioville [34] and Hadamard [35] integrals have been widely studied due to their extensive applications [34–36]. These operators are given as follows:

Definition 1

[34] (Riemann-Liouville fractional integral operators). Let f : [ α 1 , α 2 ] → R be a function, 0 < ν and Γ ( ⋅ ) represent the gamma function. Then, the following fractional integrals are known as Riemann-Liouville fractional integral operators.

J α 1 + ν f ( ζ ) = 1 Γ ( ν ) ∫ α 1 ζ ( ζ − u ) ν − 1 f ( u ) d u , ζ > α 1 (left-sided integral)

and

J α 2 − ν f ( ζ ) = 1 Γ ( ν ) ∫ ζ α 2 ( u − ζ ) ν − 1 f ( u ) d u , ζ < α 2 (right-sided integral) .

Another, fractional integral operator was introduced by Hadamard [35] and is expressed as follows:

Definition 2

[35] (Hadamard fractional integral operators) Let f : [ α 1 , α 2 ] → R be a function, 0 < ν and Γ ( ⋅ ) represent the gamma function. Then, the following fractional integrals are known as Hadamard fractional integral operators.

(2) H α 1 + ν f ( ζ ) = 1 Γ ( ν ) ∫ α 1 ζ ln ζ u ν − 1 f ( u ) u d u , ζ > α 1 (left-sided integral)

and

H α 2 − ν f ( ζ ) = 1 Γ ( ν ) ∫ ζ α 2 ln u ζ ν − 1 f ( u ) u d u , ζ < α 2 (right-sided integral) .

In [30], Öagülmüş and Sarikaya presented the Hermite-Hadamard inequalities by using Riemann-Liouville fractional integrals along with the convexity of the function. This inequality is expressed in the form of the following theorem.

Theorem 1

Let f : [ α 1 , α 2 ] → R be a convex function with α 2 > α 1 ≥ 0 , ν > 0 and f ∈ L 1 [ α 1 , α 2 ] . Then

(3) f α 1 + α 2 2 ≤ Γ ( ν + 1 ) 2 ( α 2 − α 1 ) ν [ J α 1 + ν f ( α 2 ) + J α 2 − ν f ( α 1 ) ] ≤ f ( α 1 ) + f ( α 2 ) 2 .

In this article, the authors also obtained an integral identity, and then by employing that identity, they obtained bounds for the difference of terms associated to (3). Also, the inequality (3) implies inequality (1) for a specific value of the parameter.

Katugampola, discovered a generalized fractional integral operator that generalizes the Riemann-Liouville and Hadamard fractional integral operators. It is defined as follows:

Definition 3

[37] (Generalized fractional integral operator). Let f : [ α 1 , α 2 ] → R be a function such that f ∈ X c p ( α 1 , α 2 ) ( c ∈ R , 1 ≤ p ≤ ∞ ) , 0 < ν , τ > 0 and Γ ( ⋅ ) represent the gamma function . Then, the following fractional integrals are known as generalized fractional integral operators.

(4) I α 1 + ν τ f ( ζ ) = 1 Γ ( ν ) ∫ α 1 ζ ζ τ − w τ τ ν − 1 w τ − 1 f ( w ) d w , ζ > α 1 (left-sided integral)

and

(5) I α 2 − ν τ f ( ζ ) = 1 Γ ( ν ) ∫ ζ α 2 w τ − ζ τ τ ν − 1 w τ − 1 f ( w ) d w α 2 > ζ (right-sided integral) .

For τ → 1 in Definition 3, we obtain Riemann-Liouville fractional integral operator, and for τ → 0 , we obtain Hadamard fractional integrals. In [38], Jleli et al. introduced the following Hermite-Hadamard type inequality using generalized fractional integrals.

Theorem 2

[38] Let f : [ α 1 , α 2 ] → R be a convex function with 0 < α 1 < α 2 < ∞ , ν > 0 , and τ > 0 . Then

(6) f α 1 + α 2 2 ≤ τ ν Γ ( ν + 1 ) 4 ( α 2 τ − α 1 τ ) ν [ I α 1 + ν τ F ( α 2 ) + I α 2 − ν τ F ( α 1 ) ] ≤ f ( α 1 ) + f ( α 2 ) 2 ,

where F ( z ) = f ( z ) + f ˜ ( z ) and f ˜ ( z ) = f ( α 1 + α 2 − z ) for z ∈ [ α 1 , α 2 ] .

In [39], Khan and Adil Khan introduced new conformable fractional integral operators that are a generalized version of some more significant integral operators. These operators are defined as follows:

Definition 4

[39] (Generalized conformable fractional operators). Let f be a conformable integrable function on [ α 1 , α 2 ] with 0 ≤ α 1 < α 2 < ∞ , ν > 0 , θ ∈ ( 0 , 1 ] , τ ∈ R , and θ + τ ≠ 0 . Then the following fractional integrals are known as generalized conformable fractional integral operators.

(7) K α 1 + ν θ τ f ( ζ ) = 1 Γ ( ν ) ∫ α 1 ζ ζ τ + θ − w τ + θ τ + θ ν − 1 f ( w ) w τ d θ w , ζ > α 1 (left-sided integral)

and

(8) K α 2 − ν θ τ f ( ζ ) = 1 Γ ( ν ) ∫ ζ α 2 w τ + θ − ζ τ + θ τ + θ ν − 1 f ( w ) w τ d θ w , α 2 > ζ (right-sided integral) .

Remark 1

  1. Definition 4 reduces to Katugampola fractional integrals for θ = 1 [37];

  2. If we take τ = 0 and θ → 0 in Definition 4, we obtain Hadamard fractional integrals [35];

  3. By taking τ = 0 and θ = 1 , we obtain the famous Riemann-Liouville fractional integrals [34];

  4. Definition 4 reduces to the conformable fractional integrals and the classical Riemann integrals by choosing ν = 1 , τ = 0 , and ν = θ = 1 , τ = 0 , respectively.

In [40], Khan and Adil Khan also utilized these operators to obtain the Hermite-Hadamard inequality which is expressed as follows:

Theorem 3

[40] Let f : [ α 1 , α 2 ] → R be a convex function such that f is conformal integrable on [ α 1 , α 2 ] with 0 ≤ α 1 < α 2 < ∞ , ν > 0 , and θ ∈ ( 0 , 1 ] , τ ∈ R , and τ + θ > 0 . Then

(9) f α 1 + α 2 2 ≤ ( τ + θ ) ν Γ ( ν + 1 ) 4 ( α 2 τ + θ − α 1 τ + θ ) ν [ I α 1 + ν θ τ F ( α 2 ) + K α 2 − ν θ τ F ( α 1 ) ] ≤ f ( α 1 ) + f ( α 2 ) 2 ,

where F ( z ) = f ( z ) + f ˜ ( z ) and f ˜ ( z ) = f ( α 1 + α 2 − z ) for z ∈ [ α 1 , α 2 ] .

We now describe the concept of majorization, which will be used to express our new findings.

Definition 5

[41] (Majorization). Let θ = ( θ 1 , θ 2 , … , θ ε ) and ϑ = ( ϑ 1 , ϑ 2 , … , ϑ ε ) be two tuples arranged in the order θ [ ε ] ≤ θ [ ε − 1 ] ≤ ⋯ ≤ θ [ 1 ] and ϑ [ ε ] ≤ ϑ [ ε − 1 ] ≤ ⋯ ≤ ϑ [ 1 ] . Then the tuple θ is said to majorize the tuple ϑ (in symbols ϑ ≺ θ ) if:

(10) ∑ ς = 1 k ϑ [ ς ] ≤ ∑ ς = 1 k θ [ ς ] for k = 1 , 2 , … , ε − 1 ,

and

(11) ∑ ς = 1 ε θ ς = ∑ ς = 1 ε ϑ ς .

By using the majorization concept, Neizgoda [42] presented an extended version of the Jensen-Mercer inequality which is stated as follows:

Theorem 4

[42] (Majorized discrete Jensen-Mercer inequality). Let f be a function that is defined to be convex on the interval I of real numbers, and let ( x i ς ) be an n × ε matrix such that x i ς ∈ I for all i = 1 , 2 , … , n , ς = 1 , 2 , … , ε . Let α = ( α 1 , α 2 , … , α ε ) be a tuple with α ς ∈ I for ς = 1 , 2 , … , ε and σ i ≥ 0 for i = 1 , 2 , … , n with ∑ i = 1 n σ i = 1 . Also, suppose that each row of the matrix ( x i ς ) is majorized by α , then

(12) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 ∑ i = 1 n σ i x i ς ≤ ∑ ς = 1 ε f ( α ς ) − ∑ ς = 1 ε − 1 ∑ i = 1 n σ i f ( x i ς ) .

From the overall discussion about the inequalities, we come to know that the researchers have constructed generalized inequalities in discrete or continuous form [43,44] by using integral operators, convexity or both [45–47]. However, it is essential to develop combined inequalities that can act as both continuous and discrete inequalities simultaneously. This requirement can be fulfilled by using the majorization approach discussed earlier. In [48], Faisal et al. utilized Riemann-Liouville fractional integral operators along with the concept of convexity and majorization to establish “conticrete” fractional Hermite-Hadamard-Jensen-Mercer inequalities. The authors have introduced “conticrete” name for these inequalities due to the fact that they represent both discrete and continuous inequalities in a combined form. The authors also obtained integral identities by using which bounds for the major results have been obtained. For some more conticrete inequalities via majorization in terms of Caputo fractional derivative operators and Riemann integrals, see [49,50].

This article is divided into four sections. Section 2 is devoted to the derivation of new conticrete inequalities by using majorization and generalized conformable fractional operators. These results have been expressed in the from of Theorems 5 and 6. Remark 2 demonstrates that our key findings generalize the previously known inequalities in the literature for different fractional operators. They also present the conticerete versions of the Hermite-Hadamard-Jensen-Mercer inequalities for various fractional operators and classical integrals. In Section 3, we derive weighted forms of our main results by first considering decreasing tuples and then taking those tuples, which show the same monotonic behavior. These results have been obtained on the basis of two previously proved lemmas and expressed as Theorems 7 and 8. In Section 4, two new integral identities are investigated by considering three tuples and a differentiable function. These identities are then used to establish bounds for the main results. The conclusion of overall work is presented in the last section.

2 Main results

We present our new results in the context of generalized conformable fractional operators in the following way.

Theorem 5

Let α = ( α 1 , α 2 , … , α ε ) , β = ( β 1 , β 2 , … , β ε ) , and γ = ( γ 1 , γ 2 , … , γ ε ) be three tuples, where β ε > γ ε and α ς , β ς , γ ς ∈ I = [ γ ε , β ε ] , for all ς = 1 , 2 , … , ε . Let ν > 0 and τ ∈ R , θ ∈ ( 0 , 1 ] such that τ + θ > 0 . Let f : I ⊆ [ 0 , ∞ ) → R be a function such that f ∈ L θ ( I ) and F ( z ) = f ( z ) + f ˜ ( z ) , f ˜ ( z ) = f ( β ε + γ ε − z ) . Also, suppose that f is convex function on I with β ≺ α and γ ≺ α , then

(13) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ ( τ + θ ) ν Γ ( ν + 1 ) 4 ( β ε τ + θ − γ ε τ + θ ) ν [ K γ ε + ν θ τ F ( β ε ) + K β ε − ν θ τ F ( γ ε ) ] ≤ f ( β ε ) + f ( γ ε ) 2 ≤ ∑ ς = 1 ε f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς ) .

Proof

By the fact that β ≺ α and γ ≺ α , we have ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς = β ε and ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 γ ς = γ ε . Now, we take start by considering that

(14) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 = f 1 2 ( β ε + γ ε ) = f 1 2 ( η β ε + ( 1 − η ) γ ε + η γ ε + ( 1 − η ) β ε ) .

By using convexity of f in (14), we have

(15) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ 1 2 [ f ( η β ε + ( 1 − η ) γ ε ) + f ( η γ ε + ( 1 − η ) β ε ) ] .

By multiplying both sides of (15) by

∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 Γ ( ν ) ( β ε τ + θ − ( ( 1 − η ) γ ε + η β ε ) τ + θ ) 1 − ν

and integrating with respect to η , we obtain

f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ∫ 0 1 ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 ( β ε τ + θ − ( ( 1 − η ) γ ε + η β ε ) τ + θ ) ν − 1 d η ≤ 1 2 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ∫ 0 1 ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 ( β ε τ + θ − ( ( 1 − η ) γ ε + η β ε ) τ + θ ) ν − 1 × f ( η β ε + ( 1 − η ) γ ε ) d η + 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ∫ 0 1 ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 × ( β ε τ + θ − ( ( 1 − η ) γ ε + η β ε ) τ + θ ) ν − 1 f ( η γ ε + ( 1 − η ) β ε ) d η ]

(16) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν I 1 ≤ 1 2 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν I 2 + 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν I 3 .

Now finding I 1 , I 2 , and I 3 , we have

(17) 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν I 1 = 1 Γ ( ν + 1 ) ( τ + θ ) ν ( β ε τ + θ − γ ε τ + θ ) ν .

Now, by substituting w = ( 1 − η ) γ ε + η β ε , we obtain

(18) 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν I 2 = K γ ε + ν θ τ f ( β ε )

Also, by using the following identity

f ˜ ( ( 1 − η ) γ ε + η β ε ) = f ( ( 1 − η ) β ε + η γ ε )

and by substituting the aforementioned, we obtain I 3 as follows:

(19) 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν I 3 = K γ ε + ν θ τ f ˜ ( β ε )

By using (17), (18), and (19) in (16), we deduce

(20) 1 ( τ + θ ) ν Γ ( ν + 1 ) ( β ε τ + θ − γ ε τ + θ ) ν f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ 1 2 [ K γ ε + ν θ τ F ( β ε ) ] .

Again multiplying (15) by

∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 Γ ( ν ) ( ( ( 1 − η ) γ ε + η β ε ) τ + θ − γ ε τ + θ ) 1 − ν

and then integrating with respect to η and using the same procedure as earlier, we have

(21) 1 ( τ + θ ) ν Γ ( ν + 1 ) ( β ε τ + θ − γ ε τ + θ ) ν f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ 1 2 [ K β ε − ν θ τ F ( γ ε ) ] .

By adding (20) and (21), we obtain

f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ ( τ + θ ) ν Γ ( ν + 1 ) 4 ( β ε τ + θ − γ ε τ + θ ) ν [ K γ ε + ν θ τ F ( β ε ) + K β ε − ν θ τ F ( γ ε ) ] .

Hence, the first part of the inequality (13) is completed.

To prove the second part of the inequality (13), we utilize the convexity of f as follows:

(22) f ( η β ε + ( 1 − η ) γ ε ) ≤ η f ( β ε ) + ( 1 − η ) f ( γ ε )

and

(23) f ( η γ ε + ( 1 − η ) β ε ) ≤ η f ( γ ε ) + ( 1 − η ) f ( β ε ) .

Adding (22) and (23) and then applying Theorem 4 by taking n = 1 and σ 1 = 1 , we have

(24) f ( η β ε + ( 1 − η ) γ ε ) + f ( η γ ε + ( 1 − η ) β ε ) ≤ f ( β ε ) + f ( γ ε ) ≤ 2 ∑ ς = 1 ε f ( α ς ) − ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς )

Multiplying both sides of (24) by

∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 Γ ( ν ) ( β ε τ + θ − ( ( 1 − η ) γ ε + η β ε ) τ + θ ) 1 − ν

and integrating with respect to η , we obtain

(25) ( τ + θ ) ν Γ ( ν + 1 ) 2 ( β ε τ + θ − γ ε τ + θ ) ν K γ ε + ν θ τ F ( β ε ) ≤ 1 2 [ f ( β ε ) + f ( γ ε ) ] ≤ ∑ ς = 1 ε f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς ) .

Again multiplying (24) by

∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 Γ ( ν ) ( ( ( 1 − η ) γ ε + η β ε ) τ + θ − γ ε τ + θ ) 1 − ν

and then integrating with respect to η and using the same procedure as earlier, we have

(26) ( τ + θ ) ν Γ ( ν + 1 ) 2 ( β ε τ + θ − γ ε τ + θ ) ν K β ε − ν θ τ F ( γ ε ) ≤ 1 2 [ f ( β ε ) + f ( γ ε ) ] ≤ ∑ ς = 1 ε f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς ) .

By adding (25) and (26), we obtain the second and third part of the inequality (13).□

Remark 2

  1. By taking ε = 2 , β 1 = α 1 , and γ 1 = α 2 in inequality (13), we obtain inequality (9), which is proved in [40].

  2. By substituting θ = 1 in (13), we acquire the following inequality of the Hermite-Hadamard-Jensen-Mercer type in terms of Katugampola fractional operators:

    (27) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ ( τ + 1 ) ν Γ ( ν + 1 ) 4 ( β ε τ + 1 − γ ε τ + 1 ) ν [ K γ ε + ν 1 τ F ( β ε ) + K β ε − ν 1 τ F ( γ ε ) ] ≤ 1 2 [ f ( β ε ) + f ( γ ε ) ] ≤ ∑ ς = 1 ε f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς ) .

  3. By taking θ = 1 , ε = 2 , β 1 = α 1 , and γ 1 = α 2 in (13), we obtain inequality (6), which is proved in [38].

  4. For τ + θ → 0 in (13) and using L’ Hospital rule and the relation f ∈ [ γ ε , β ε ] , ∫ γ ε β ε f ( w ) d θ w = ∫ γ ε β ε f ( w ) w θ − 1 d w , where ∫ γ ε β ε d θ w represents conformable integration, we obtain the following Hermite-Hadamard-Jensen-Mercer type inequality for Hadamard fractional integral operators:

    (28) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ Γ ( ν + 1 ) 2 ln β ε γ ε ν [ K γ ε + ν 0 + 0 f ( β ε ) + K β ε − ν 0 + 0 f ( γ ε ) ] ≤ 1 2 [ f ( β ε ) + f ( γ ε ) ] ≤ ∑ ς = 1 ε f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς ) .

  5. By inserting τ + θ = 1 , in (13), we obtain the following Hermite-Hadamard-Jensen-Mercer type inequality for Riemann-Liouville fractional integrals [48]:

    f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ Γ ( ν + 1 ) 2 ∑ ς = 1 ε − 1 ( γ ς − β ς ) ν [ K γ ε + ν 1 0 f ( β ε ) + K β ε − ν 1 0 f ( γ ε ) ] .

  6. For τ + θ = 1 , ε = 2 , β 1 = α 1 , γ 1 = α 2 , the inequality (13) produces the inequality (3).

  7. For ν = 1 and τ = 0 in (13), we obtain the Hermite-Hadamard-Jensen-Mercer type inequality in terms of conformable fractional integrals as follows:

    (29) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ 1 2 ( β ε θ − γ ε θ ) ν ∫ γ ε β ε F ( w ) d θ w ≤ 1 2 [ f ( γ ε ) + f ( β ε ) ] ≤ ∑ ς = 1 ε f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς ) .

  8. For θ = ν = 1 , τ = 0 , we deduce the following inequality of the Hermite-Hadamard-Jensen-Mercer type in terms of classical Riemann integrals [49]:

    (30) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) ∫ γ ε β ε f ∑ ς = 1 ε α ς − u d u ≤ ∑ ς = 1 ε f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς ) .

Now, we present another result of Hermite-Hadamard-Jensen-Mercer type by using generalized conformable fractional operators as follows:

Theorem 6

Let all the conditions in the hypotheses of Theorem 5  hold true, then

(31) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ Γ ( ν + 1 ) ( τ + θ ) ν 2 Q τ + θ ν ( β ε , γ ε ) K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 + ν θ τ F ( β ε ) + K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 − ν θ τ F ( γ ε ) ≤ ∑ ς = 1 ε f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς ) ,

where

Q τ + θ ν ( β ε , γ ε ) = β ε τ + θ − ∑ ς = 1 ε α ς − 1 2 ∑ ς = 1 ε − 1 ( β ς + γ ς ) τ + θ ν + ∑ ς = 1 ε α ς − 1 2 ∑ ς = 1 ε − 1 ( β ς + γ ς ) τ + θ − γ ε τ + θ ν .

Proof

For ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς = β ε , ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 γ ς = γ ε , and η ∈ [ 0 , 1 ] , it may be written that

(32) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 = f 1 2 ( β ε + γ ε ) = f 1 2 ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς + ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς .

By using convexity of f in (32), we have

(33) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ 1 2 f ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς + f ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς .

Multiplying both sides of (33) by

∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς τ + θ − 1 2 Γ ( ν ) ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς τ + θ − γ ε τ + θ 1 − ν ,

and integrating with respect to η , we obtain

f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 1 2 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ∫ 0 1 ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς τ + θ − 1 × ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς τ + θ − γ ε τ + θ ν − 1 d η ≤ 1 2 1 2 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ∫ 0 1 ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς τ + θ − γ ε τ + θ ν − 1 × ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς τ + θ − 1 f ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς d η + 1 2 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ∫ 0 1 ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς τ + θ − γ ε τ + θ ν − 1 × ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς τ + θ − 1 f ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς .

(34) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 1 2 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν I 4 ≤ 1 2 1 2 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν I 5 + 1 2 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν I 6 .

Now, finding I 4 , I 5 , and I 6 , we have

(35) 1 2 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν I 4 = 1 Γ ( ν + 1 ) ( τ + θ ) ν ∑ ς = 1 ε α ς − 1 2 ∑ ς = 1 ε − 1 ( β ς + γ ς ) τ + θ − γ ε τ + θ ν .

Now by substituting w = ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς − 2 − η 2 ∑ ς = 1 ε − 1 γ ς , we obtain

(36) 1 2 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν I 5 = K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 − ν θ τ f ( γ ε ) .

Also, by using the identity

f ˜ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς − 2 − η 2 ∑ ς = 1 ε − 1 β ς = f ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς − 2 − η 2 ∑ ς = 1 ε − 1 γ ς

and the aforementioned substitution, we have

(37) 1 2 Γ ( ν ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν I 6 = K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 − ν θ τ f ˜ ( γ ε ) .

By using (35), (36), and (37) in (34), we deduce

(38) 1 Γ ( ν + 1 ) ( τ + θ ) ν ∑ ς = 1 ε α ς − 1 2 ∑ ς = 1 ε − 1 ( β ς + γ ς ) τ + θ − γ ε τ + θ ν f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ 1 2 K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 − ν θ τ F ( γ ε ) .

Similarly, multiplying both sides of (33) by

∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς τ + θ − 1 2 Γ ( ν ) β ε τ + θ − ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς τ + θ 1 − ν ,

and then integrating with respect to η , we finally obtain

(39) 1 Γ ( ν + 1 ) ( τ + θ ) ν β ε τ + θ − ∑ ς = 1 ε α ς − 1 2 ∑ ς = 1 ε − 1 ( β ς + γ ς ) τ + θ ν f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ 1 2 K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 + ν θ τ F ( β ε ) .

By adding (38) and (39), we have

(40) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ Γ ( ν + 1 ) ( τ + θ ) ν 2 Q τ + θ ν ( β ε , γ ε ) K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 + ν θ τ F ( β ε ) + K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 − ν θ τ F ( γ ε ) .

Thus, the first part of the inequality (31) is proved. To prove the second part of the inequality (31), we proceed as follows:

As a consequence of convexity of f , we use Theorem 4 for the values n = 2 , σ 1 = η 2 , and σ 2 = 2 − η 2 , to obtain

(41) f ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς ≤ ∑ ς = 1 ε f ( α ς ) − η 2 ∑ ς = 1 ε − 1 f ( β ς ) + 2 − η 2 ∑ ς = 1 ε − 1 f ( γ ς )

and

(42) f ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς ≤ ∑ ς = 1 ε f ( α ς ) − η 2 ∑ ς = 1 ε − 1 f ( γ ς ) + 2 − η 2 ∑ ς = 1 ε − 1 f ( β ς ) .

By adding (41) and (42), we have

(43) f ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς + f ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς ≤ 2 ∑ ς = 1 ε f ( α ς ) − ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς ) .

By multiplying

∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς τ + θ − 1 2 Γ ( ν ) ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς τ + θ − γ ε τ + θ 1 − ν ,

to both sides of (43), we obtain

(44) K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 − ν θ τ F ( γ ε ) ≤ 1 Γ ( ν + 1 ) ( τ + θ ) ν 2 ∑ ς = 1 ε f ( α ς ) − ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς ) ∑ ς = 1 ε α ς − 1 2 ∑ ς = 1 ε − 1 ( β ς + γ ς ) τ + θ − γ ε τ + θ ν .

Also, by multiplying

∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) 1 − ν ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς τ + θ − 1 2 Γ ( ν ) β ε τ + θ − ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς τ + θ 1 − ν

to both sides of (43), we obtain

(45) K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 + ν θ τ F ( β ε ) ≤ 1 Γ ( ν + 1 ) ( τ + θ ) ν 2 ∑ ς = 1 ε f ( α ς ) − ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς ) β ε τ + θ − ∑ ς = 1 ε α ς − 1 2 ∑ ς = 1 ε − 1 ( β ς + γ ς ) τ + θ ν .

Now, by adding (44) and (45), we obtain the second part of the inequality (31).□

Remark 3

  1. By substituting τ + θ = 1 , in (31), we obtain the following Hermite-Hadamard-Jensen-Mercer type inequality for Riemann-Liouville fractional integrals [48]:

    (46) f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ 2 1 − ν Γ ( ν + 1 ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) ν K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 + ν 1 0 f ( β ε ) + K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 − ν 1 0 f ( γ ε ) ≤ ∑ ς = 1 ε f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 f ( β ς ) + ∑ ς = 1 ε − 1 f ( γ ς ) .

  2. For τ + θ = 1 and ε = 2 , the inequality (31) reduces to the following inequality [30]:

    (47) f α 1 + α 2 − β 1 + γ 1 2 ≤ 2 1 − ν Γ ( ν + 1 ) ( γ 1 − β 1 ) ν K α 1 + α 2 − β 1 + γ 1 2 + ν 1 0 f ( α 1 + α 2 − β 1 ) + K α 1 + α 2 − β 1 + γ 1 2 − ν 1 0 f ( α 1 + α 2 − γ 1 ) ≤ f ( α 1 ) + f ( α 2 ) − 1 2 [ f ( β 1 ) + f ( γ 1 ) ] .

  3. For τ + θ = 1 , ε = 2 , and ν = 1 then the inequality (31) reduces to the inequality (2.2) in [51].

3 Weighted forms of the main results

We generate weighted versions of our first primary result in this section. The similar process can be used to produce weighted versions of the second main outcome. They are acquired by utilizing lemmas that have already been proved. We first recall these lemmas [49].

Lemma 1

Let α = ( α 1 , α 2 , … , α ε ) , r = ( r 1 , r 2 , … , r ε ) be two tuples and ( x i ε ) be a real n × ε matrix such that α ε , x i ε ∈ I , r ς ≥ 0 with r ε ≠ 0 , ε = 1 r ε for all i = 1 , 2 , … , n , ς = 1 , 2 , … , ε , and f be a convex function defined on I . Also, let σ i ≥ 0 for i = 1 , 2 , … , n with ∑ i = 1 n σ i = 1 . If for each i = 1 , 2 , … , n , ( x i 1 , x i 2 , … , x i ε ) is a decreasing tuple and satisfying

∑ ς = 1 k r ς x i ς ≤ ∑ ς = 1 k r ς α ς f o r k = 1 , 2 , … , ε − 1 , ∑ ς = 1 ε r ς α ς = ∑ ς = 1 ε r ς x i ς ,

then

f ∑ ς = 1 ε ε r ς α ς − ∑ ς = 1 ε − 1 ∑ i = 1 n ε σ i r ς x i ς ≤ ∑ ς = 1 ε ε r ς f ( α ς ) − ∑ ς = 1 ε − 1 ∑ i = 1 n ε σ i r ς f ( x i ς ) .

Lemma 2

Let α = ( α 1 , α 2 , … , α ε ) , r = ( r 1 , r 2 , … , r ε ) be two tuples and ( x i ε ) be a real n × ε matrix such that α ε , x i ε ∈ I , r ς ≥ 0 with r ε ≠ 0 , ε = 1 r ε for all i = 1 , 2 , … , n , ς = 1 , 2 , … , ε , and f be a convex function defined on I . Also, let σ i ≥ 0 for i = 1 , 2 , … , n with ∑ i = 1 n σ i = 1 .

If for each i = 1 , 2 , … , n , ( α ς − x i ς ) and x i ς are monotonically in the same sense and

∑ ς = 1 ε r ς α ς = ∑ ς = 1 ε r ς x i ς ,

then

f ∑ ς = 1 ε ε r ς α ς − ∑ ς = 1 ε − 1 ∑ i = 1 n ε σ i r ς x i ς ≤ ∑ ς = 1 ε ε r ς f ( α ς ) − ∑ ς = 1 ε − 1 ∑ i = 1 n ε σ i r ς f ( x i ς ) .

Now, we present weighted form of Theorem 5 on the basis of Lemma 1 as follows:

Theorem 7

Let α = ( α 1 , α 2 , … , α ε ) , β = ( β 1 , β 2 , … , β ε ) , γ = ( γ 1 , γ 2 , … , γ ε ) , and r = ( r 1 , r 2 , … , r ε ) be four tuples, where β ε > γ ε and α ς , β ς , γ ς ∈ I = [ γ ε , β ε ] , r ς ≥ 0 with r ε ≠ 0 for all ς = 1 , 2 , … , ε , ε = 1 r ε . Let ν > 0 and τ ∈ R , θ ∈ ( 0 , 1 ] such that τ + θ > 0 . Let f : I ⊆ [ 0 , ∞ ) → R be a function such that f ∈ L θ ( I ) and F ( z ) = f ( z ) + f ˜ ( z ) , f ˜ ( z ) = f ( β ε + γ ε − z ) . Also, suppose that f is convex function on I , β , and γ are decreasing tuples and

∑ ς = 1 k r ς β ς ≤ ∑ ς = 1 k r ς α ς , ∑ ς = 1 k r ς γ ς ≤ ∑ ς = 1 k r ς α ς for k = 1 , … , ε − 1 ,

∑ ς = 1 ε r ς α ς = ∑ ς = 1 ε r ς β ς , ∑ ς = 1 ε r ς α ς = ∑ ς = 1 ε r ς γ ς ,

then

(48) f ∑ ς = 1 ε ε r ς α ς − ∑ ς = 1 ε − 1 ε r ς β ς + γ ς 2 ≤ ( τ + θ ) ν Γ ( ν + 1 ) 4 ( β ε τ + θ − γ ε τ + θ ) ν [ K γ ε + ν θ τ F ( β ε ) + K β ε − ν θ τ F ( γ ε ) ] ≤ 1 2 [ f ( β ε ) + f ( γ ε ) ] ≤ ∑ ς = 1 ε ε r ς f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 ε r ς f ( β ς ) + ∑ ς = 1 ε − 1 ε r ς f ( γ ς ) .

Proof

From the fact that

∑ ς = 1 ε r ς α ς = ∑ ς = 1 ε r ς β ς and ∑ ς = 1 ε r ς α ς = ∑ ς = 1 ε r ς γ ς ,

we arrive at

∑ ς = 1 ε ε r ς α ς − ∑ ς = 1 ε − 1 ε r ς β ς = β ε and ∑ ς = 1 ε ε r ς α ς − ∑ ς = 1 ε − 1 ε r ς γ ς = γ ε .

Now, for η ∈ [ 0 , 1 ] , we have

(49) f ∑ ς = 1 ε ε r ς α ς − ε ∑ ς = 1 ε − 1 r ς β ς + γ ς 2 = f β ε + γ ε 2 = f 1 2 ( η β ε + ( 1 − η ) γ ε + η γ ε + ( 1 − η ) β ε ) .

By using convexity of f in (49), we have

(50) f ∑ ς = 1 ε ε r ς α ς − ε ∑ ς = 1 ε − 1 r ς β ς + γ ς 2 ≤ 1 2 [ f ( η β ε + ( 1 − η ) γ ε ) + f ( η γ ε + ( 1 − η ) β ε ) ] .

Multiplying both sides of (50) by

∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 Γ ( ν ) ( β ε τ + θ − ( ( 1 − η ) γ ε + η β ε ) τ + θ ) 1 − ν

and integrating with respect to η , we obtain

f ∑ ς = 1 ε ε r ς α ς − ε ∑ ς = 1 ε − 1 r ς β ς + γ ς 2 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν × ∫ 0 1 ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 ( β ε τ + θ − ( ( 1 − η ) γ ε + η β ε ) τ + θ ) ν − 1 d η ≤ 1 2 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν ∫ 0 1 ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 ( β ε τ + θ − ( ( 1 − η ) γ ε + η β ε ) τ + θ ) ν − 1 × f ( η β ε + ( 1 − η ) γ ε ) d η + 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν ∫ 0 1 ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 ( β ε τ + θ − ( ( 1 − η ) γ ε + η β ε ) τ + θ ) ν − 1 f ( η γ ε + ( 1 − η ) β ε ) ] .

(51) f ∑ ς = 1 ε ε r ς α ς − ε ∑ ς = 1 ε − 1 r ς β ς + γ ς 2 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν I 7 ≤ 1 2 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν I 8 + 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν I 9 .

Now, finding I 7 , I 8 , and I 9 in the following way:

(52) 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν I 7 = 1 Γ ( ν + 1 ) ( τ + θ ) ν ( β ε τ + θ − γ ε τ + θ ) ν .

Now, by substituting w = ( 1 − η ) γ ε + η β ε , we obtain

(53) 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν I 8 = K γ ε + ν θ τ f ( β ε )

Also, by using the following identity

f ˜ ( ( 1 − η ) γ ε + η β ε ) = f ( ( 1 − η ) β ε + η γ ε )

and making the aforementioned substitution, we obtain I 9 as follows:

(54) 1 Γ ( ν ) ∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν I 9 = K γ ε + ν θ τ f ˜ ( β ε ) .

By using (52), (53), and (54) in (51), we have

(55) 1 Γ ( ν + 1 ) ( τ + θ ) ν ( β ε τ + θ − γ ε τ + θ ) ν f ∑ ς = 1 ε ε r ς α ς − ε ∑ ς = 1 ε − 1 r ς β ς + γ ς 2 ≤ 1 2 [ K γ ε + ν θ τ F ( β ε ) ] .

Again by multiplying both sides of (50) by

∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 Γ ( ν ) ( ( ( 1 − η ) γ ε + η β ε ) τ + θ − γ ε τ + θ ) 1 − ν

and integrating with respect to η and using the same procedure as earlier, we have

(56) 1 Γ ( ν + 1 ) ( τ + θ ) ν ( β ε τ + θ − γ ε τ + θ ) ν f ∑ ς = 1 ε ε r ς α ς − ε ∑ ς = 1 ε − 1 r ς β ς + γ ς 2 ≤ 1 2 [ K β ε − ν θ τ F ( γ ε ) ] .

By adding (55) and (56), we obtain the first part of the inequality (48).

To prove the second part of the inequality (48), we utilize the convexity of f as follows:

(57) f ( η β ε + ( 1 − η ) γ ε ) ≤ η f ( β ε ) + ( 1 − η ) f ( γ ε )

and

(58) f ( η γ ε + ( 1 − η ) β ε ) ≤ η f ( γ ε ) + ( 1 − η ) f ( β ε ) .

Adding (57) and (58) and then using Lemma 1 for n = 2 , σ 1 = η , and σ 2 = 1 − η , we obtain

(59) f ( η β ε + ( 1 − η ) γ ε ) + f ( η γ ε + ( 1 − η ) β ε ) ≤ f ( β ε ) + f ( γ ε ) ≤ 2 ∑ ς = 1 ε ε r ς f ( α ς ) − ∑ ς = 1 ε − 1 ε r ς f ( β ς ) + ∑ ς = 1 ε − 1 ε r ς f ( γ ς ) .

Multiplying both sides of (59) by

∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 Γ ( ν ) ( β ε τ + θ − ( ( 1 − η ) γ ε + η β ε ) τ + θ ) 1 − ν

and integrating with respect to η , we obtain

(60) ( τ + θ ) ν Γ ( ν + 1 ) 2 ( β ε τ + θ − γ ε τ + θ ) ν K γ ε + ν θ τ F ( β ε ) ≤ 1 2 [ f ( β ε ) + f ( γ ε ) ] ≤ ∑ ς = 1 ε ε r ς f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 ε r ς f ( β ς ) + ∑ ς = 1 ε − 1 ε r ς f ( γ ς ) .

Again by multiplying both sides of (59) by

∑ ς = 1 ε − 1 ( ε r ς γ ς − ε r ς β ς ) ( τ + θ ) 1 − ν ( ( 1 − η ) γ ε + η β ε ) τ + θ − 1 Γ ( ν ) ( ( ( 1 − η ) γ ε + η β ε ) τ + θ − γ ε τ + θ ) 1 − ν

and integrating with respect to η , we have

(61) ( τ + θ ) ν Γ ( ν + 1 ) 2 ( β ε τ + θ − γ ε τ + θ ) ν K β ε − ν θ τ F ( γ ε ) ≤ 1 2 [ f ( β ε ) + f ( γ ε ) ] ≤ ∑ ς = 1 ε ε r ς f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 ε r ς f ( β ς ) + ∑ ς = 1 ε − 1 ε r ς f ( γ ς ) .

By adding (60) and (61), we deduce the second and third part of inequality (48).□

Another, weighted form of Theorem 5 on the basis of Lemma 2 is obtained as follows:

Theorem 8

Let α = ( α 1 , α 2 , … , α ε ) , β = ( β 1 , β 2 , … , β ε ) , γ = ( γ 1 , γ 2 , … , γ ε ) , and r = ( r 1 , r 2 , … , r ε ) be four tuples, where β ε > γ ε and α ς , β ς , γ ς ∈ I = [ γ ε , β ε ] , r ς ≥ 0 with r ε ≠ 0 for all ς = 1 , 2 , … , ε , and ε = 1 r ε . Let ν > 0 and τ ∈ R , θ ∈ ( 0 , 1 ] such that τ + θ > 0 . Let f : I ⊆ [ 0 , ∞ ) → R be a function such that f ∈ L θ ( I ) and F ( z ) = f ( z ) + f ˜ ( z ) , f ˜ ( z ) = f ( β ε + γ ε − z ) . Also, suppose that f is convex function on I and α − β , β , α − γ , and γ are monotonically in the same sense and

∑ ς = 1 ε r ς α ς = ∑ ς = 1 ε r ς β ς , ∑ ς = 1 ε r ς α ς = ∑ ς = 1 ε r ς γ ς ,

then

(62) f ∑ ς = 1 ε ε r ς α ς − ∑ ς = 1 ε − 1 ε r ς β ς + γ ς 2 ≤ ( τ + θ ) ν Γ ( ν + 1 ) 4 ( β ε τ + θ − γ ε τ + θ ) ν [ K γ ε + ν θ τ F ( β ε ) + K β ε − ν θ τ F ( γ ε ) ] ≤ 1 2 [ f ( β ε ) + f ( γ ε ) ] ≤ ∑ ς = 1 ε ε r ς f ( α ς ) − 1 2 ∑ ς = 1 ε − 1 ε r ς f ( β ς ) + ∑ ς = 1 ε − 1 ε r ς f ( γ ς ) .

Proof

By using Lemma 2, we can derive (62) by following the steps given in the proof of Theorem 7.□

Remark 4

Weighted versions of Theorem 6 can also be established in a similar manner by utilizing Lemmas 1 and 2.

4 Derivation of new integral identities and bounds for the main results

First, in this section, we construct two new integral identities connected to the right and left sides of our major inequalities. Then, by using these identities, we bound the absolute difference between the terms on the right and left of the main results.

Lemma 3

Let α = ( α 1 , α 2 , … , α ε ) , β = ( β 1 , β 2 , … , β ε ) , and γ = ( γ 1 , γ 2 , … , γ ε ) be three tuples, where β ε > γ ε and α ς , β ς , γ ς ∈ I = [ γ ε , β ε ] , for all ς = 1 , 2 , … , ε , and ν > 0 , η ∈ [ 0 , 1 ] , τ ∈ R , θ ∈ ( 0 , 1 ] such that τ + θ > 0 . Also, let f : I ⊆ [ 0 , ∞ ) → R be a differentiable function such that f ′ ∈ L θ ( I ) , then

(63) 1 2 [ f ( β ε ) + f ( γ ε ) ] − ( τ + θ ) ν Γ ( ν + 1 ) 4 ( β ε τ + θ − γ ε τ + θ ) ν [ K γ ε + ν θ τ F ( β ε ) + K β ε − ν θ τ F ( γ ε ) ] = ∑ ς = 1 ε − 1 ( γ ς − β ς ) 4 ( β ε τ + θ − γ ε τ + θ ) ν ∫ 0 1 ∇ τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 ( η γ ς + ( 1 − η ) β ς ) d η ,

where

∇ τ + θ ν ( η ) = ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 γ ς − ( 1 − η ) ∑ ς = 1 ε − 1 β ς τ + θ − γ ε τ + θ ν − ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς τ + θ − γ ε τ + θ ν + β ε τ + θ − ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς τ + θ ν − β ε τ + θ − ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 γ ς − ( 1 − η ) ∑ ς = 1 ε − 1 β ς τ + θ ν .

Proof

By using integration by parts, we obtain

(64) K γ ε + ν θ τ F ( β ε ) = ( β ε τ + θ − γ ε τ + θ ) ν ( τ + θ ) ν Γ ( ν + 1 ) F ( γ ε ) + ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) ν Γ ( ν + 1 ) × ∫ 0 1 β ε τ + θ − ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς τ + θ ν × F ′ ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς d η .

Also, we obtain

(65) K β ε − ν θ τ F ( γ ε ) = ( β ε τ + θ − γ ε τ + θ ) ν ( τ + θ ) ν Γ ( ν + 1 ) F ( β ε ) − ∑ ς = 1 ε − 1 ( γ ς − β ς ) ( τ + θ ) ν Γ ( ν + 1 ) × ∫ 0 1 ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς τ + θ − γ ε τ + θ ν × F ′ ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς d η .

By adding (64) and (65), we obtain

(66) f ( β ε ) + f ( γ ε ) 2 − ( τ + θ ) ν Γ ( ν + 1 ) 4 ( β ε τ + θ − γ ε τ + θ ) ν [ K γ ε + ν θ τ F ( β ε ) + K ( β ε ) − ν θ τ F ( γ ε ) ] = ∑ ς = 1 ε − 1 ( γ ς − β ς ) 4 ( β ε τ + θ − γ ε τ + θ ) ν ∫ 0 1 ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς τ + θ − γ ε τ + θ ν − β ε τ + θ − ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς τ + θ ν F ′ ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς d η .

Also, we have

(67) F ′ ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς = f ′ ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς − f ′ ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 γ ς − ( 1 − η ) ∑ ς = 1 ε − 1 β ς .

And

(68) ∫ 0 1 ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς τ + θ − γ ε τ + θ ν F ′ ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς d η = ∫ 0 1 ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 γ ς − ( 1 − η ) ∑ ς = 1 ε − 1 β ς τ + θ − γ ε τ + θ ν f ′ ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 γ ς − ( 1 − η ) ∑ ς = 1 ε − 1 β ς d η − ∫ 0 1 ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς τ + θ − γ ε τ + θ ν f ′ ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 γ ς − ( 1 − η ) ∑ ς = 1 ε − 1 β ς d η .

Also,

(69) ∫ 0 1 β ε τ + θ − ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς τ + θ ν F ′ ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς d η = ∫ 0 1 β ε τ + θ − ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 γ ς − ( 1 − η ) ∑ ς = 1 ε − 1 β ς τ + θ ν f ′ ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 γ ς − ( 1 − η ) ∑ ς = 1 ε − 1 β ς d η − ∫ 0 1 β ε τ + θ − ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 β ς − ( 1 − η ) ∑ ς = 1 ε − 1 γ ς τ + θ ν f ′ ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 γ ς − ( 1 − η ) ∑ ς = 1 ε − 1 β ς d η .

Now by using (68) and (69) in (66), we obtain (63).□

Remark 5

  1. For τ + θ = 1 , Lemma 3 reduces to Lemma 3 given in [48].

  2. For τ + θ = 1 , ν = 1 , ε = 2 , β 1 = α 1 , and γ 1 = α 2 in (63), we obtain the following identity:

    (70) f ( α 1 ) + f ( α 2 ) 2 − 1 α 2 − α 1 ∫ α 1 α 2 f ( u ) d u = α 2 − α 1 2 ∫ 0 1 ( 2 η − 1 ) f ′ ( η α 2 + ( 1 − η ) α 1 ) d η .

    The equality (70) has been proved by Dragomir and Agarwal in [14].

Lemma 4

Let all the conditions in the hypotheses of Lemma 3  hold true, then

(71) Γ ( ν + 1 ) ( τ + θ ) ν 2 Q τ + θ ν ( β ε , γ ε ) K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 + ν θ τ F ( β ε ) + K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 − ν θ τ F ( γ ε ) − f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 = ∑ ς = 1 ε − 1 ( γ ς − β ς ) 4 Q τ + θ ν ( β ε , γ ε ) ∫ 0 1 P τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς − f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς d η ,

where

Q τ + θ ν ( β ε , γ ε ) = β ε τ + θ − ∑ ς = 1 ε α ς − 1 2 ∑ ς = 1 ε − 1 ( β ς + γ ς ) τ + θ ν + ∑ ς = 1 ε α ς − 1 2 ∑ ς = 1 ε − 1 ( β ς + γ ς ) τ + θ − γ ε τ + θ ν

and

P τ + θ ν ( η ) = β ε τ + θ − ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς τ + θ ν + ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς τ + θ − γ ε τ + θ ν .

Proof

We can easily prove (71), by following the same procedure as adopted in the proof of Lemma 3.□

To prove some further results, we first give the following definition.

Definition 6

Let α = ( α 1 , α 2 , … , α ε ) , β = ( β 1 , β 2 , … , β ε ) , and γ = ( γ 1 , γ 2 , … , γ ε ) be three tuples, where β ε > γ ε and α ς , β ς , γ ς ∈ I = [ γ ε , β ε ] ⊆ [ 0 , ∞ ) , for all ς = 1 , 2 , … , ε , ν > 0 , τ + θ > 0 . If ξ , z ∈ I , then we define the operators:

(72) Λ 1 ν ( z , τ + θ ) = ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ∣ z τ + θ − w τ + θ ∣ ν d w − ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ∣ z τ + θ − w τ + θ ∣ ν d w .

(73) Λ 2 ν ( z , τ + θ ) = ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ∣ w τ + θ − z τ + θ ∣ ν d w − ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ∣ w τ + θ − z τ + θ ∣ ν d w .

(74) ϒ 1 ν ( ξ , z , τ + θ ) = ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ∣ ξ − w ∣ ∣ z τ + θ − w τ + θ ∣ ν d w − ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ∣ ξ − w ∣ ∣ z τ + θ − w τ + θ ∣ ν d w .

(75) ϒ 2 ν ( ξ , z , τ + θ ) = ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ∣ ξ − w ∣ ∣ w τ + θ − z τ + θ ∣ ν d w − ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ∣ ξ − w ∣ ∣ w τ + θ − z τ + θ ∣ ν d w .

Theorem 9

Let f be a differentiable function defined on I and α = ( α 1 , α 2 , … , α ε ) , β = ( β 1 , β 2 , … , β ε ) , and γ = ( γ 1 , γ 2 , … , γ ε ) be three tuples, where β ε > γ ε and α ς , β ς , γ ς ∈ I = [ γ ε , β ε ] , for all ς = 1 , 2 , … , ε . Let ν > 0 and τ ∈ R , θ ∈ ( 0 , 1 ] such that τ + θ > 0 . If β ≺ α , γ ≺ α and ∣ f ′ ∣ is convex on I , then

(76) f ( γ ε ) + f ( β ε ) 2 − ( τ + θ ) ν Γ ( ν + 1 ) 4 ( β ε τ + θ − γ ε τ + θ ) ν [ K γ ε + ν θ τ F ( β ε ) + K β ε − ν θ τ F ( γ ε ) ] ≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 ( β ε τ + θ − γ ε τ + θ ) ν 2 M τ + θ ν ( β ε , γ ε ) ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ − N τ + θ ν ( β ε , γ ε ) ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ + ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ ,

where

M τ + θ ν ( β ε , γ ε ) = Λ 1 ν ( β ε , τ + θ ) − Λ 2 ν ( γ ε , τ + θ ) ∑ ς = 1 ε − 1 ( γ ς − β ς )

and

N τ + θ ν ( β ε , γ ε ) = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) 2 [ ϒ 1 ν ( β ε , β ε , τ + θ ) − ϒ 1 ν ( γ ε , β ε , τ + θ ) + ϒ 2 ν ( γ ε , γ ε , τ + θ ) − ϒ 2 ν ( β ε , γ ε , τ + θ ) ] .

Proof

By using Lemma 3, we have

(77) f ( γ ε ) + f ( β ε ) 2 − ( τ + θ ) ν Γ ( ν + 1 ) 4 ( β ε τ + θ − γ ε τ + θ ) ν [ K γ ε + ν θ τ F ( β ε ) + K β ε − ν θ τ F ( γ ε ) ] = ∑ ς = 1 ε − 1 ( γ ς − β ς ) 4 ( β ε τ + θ − γ ε τ + θ ) ν ∫ 0 1 ∇ τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 ( η γ ς + ( 1 − η ) β ς ) d η ≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 ( β ε τ + θ − γ ε τ + θ ) ν ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ f ′ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 ( η γ ς + ( 1 − η ) β ς ) d η .

By taking σ 1 = η , σ 2 = 1 − η , and n = 2 and applying Theorem 4 in (77), we have

(78) ≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 ( β ε τ + θ − γ ε τ + θ ) ν ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ − η ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ − ( 1 − η ) ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ d η = ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 ( β ε τ + θ − γ ε τ + θ ) ν ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ d η − ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ ∫ 0 1 η ∣ ∇ τ + θ ν ( η ) ∣ d η − ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ ∫ 0 1 ( 1 − η ) ∣ ∇ τ + θ ν ( η ) ∣ .

By substituting u = ∑ ς = 1 ε α ς − η ∑ ς = 1 ε − 1 γ ς − ( 1 − η ) ∑ ς = 1 ε − 1 β ς , the function ∇ τ + θ ν ( η ) becomes

(79) ψ ( u ) = ( u τ + θ − γ ε τ + θ ) ν − ( ( ( β ε ) + ( γ ε ) − u ) τ + θ − γ ε τ + θ ) ν + ( β ε τ + θ − ( ( β ε ) + ( γ ε ) − u ) τ + θ ) ν − ( β ε τ + θ − u τ + θ ) ν .

We see that

ψ ( γ ε ) = − 2 ( β ε τ + θ − ( γ ε ) τ + θ ) ν ,

ψ ( β ε ) = 2 ( β ε τ + θ − ( γ ε ) τ + θ ) ν ,

and

ψ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 = 0 .

Therefore, we have

(80) ψ ( u ) ≤ 0 , if γ ε ≤ u ≤ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ψ ( u ) > 0 , if ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 < u ≤ β ε .

Now, we find ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ d η , ∫ 0 1 η ∣ ∇ τ + θ ν ( η ) ∣ d η , and ∫ 0 1 ( 1 − η ) ∣ ∇ τ + θ ν ( η ) ∣ d η as follows:

(81) ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ d η = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) ∫ γ ε β ε ∣ ψ ( u ) ∣ d u = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) − ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ψ ( u ) d u + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ψ ( u ) d u = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) [ L 1 + L 2 + L 3 + L 4 ] .

Here,

(82) L 1 = − ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ( u τ + θ − γ ε τ + θ ) ν d u + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ( u τ + θ − γ ε τ + θ ) ν d u = − Λ 2 ν ( γ ε , τ + θ ) ,

(83) L 2 = ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ( ( β ε + γ ε − u ) τ + θ − γ ε τ + θ ) ν d u − ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 ( β ς + γ ς 2 ) β ε ( ( β ε + γ ε − u ) τ + θ − γ ε τ + θ ) ν d u ,

(84) L 3 = − ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ( β ε τ + θ − ( β ε + γ ε − u ) τ + θ ) ν d u + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 ( β ς + γ ς 2 ) β ε ( β ε τ + θ − ( β ε + γ ε − u ) τ + θ ) ν d u .

By the change of variables v = β ε + γ ε − u , we have

(85) L 2 = − Λ 2 ν ( γ ε , τ + θ ) , L 3 = Λ 1 ν ( β ε , τ + θ ) ,

(86) L 4 = ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ( β ε τ + θ − u τ + θ ) ν d u − ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ( β ε τ + θ − u τ + θ ) ν d u = Λ 1 ν ( β ε , τ + θ ) .

By using (82), (85), and (86) in (81), we obtain

(87) ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ d η = 2 ∑ ς = 1 ε − 1 ( γ ς − β ς ) [ Λ 1 ν ( β ε , τ + θ ) − Λ 2 ν ( γ ε , τ + θ ) ] .

And

(88) ∫ 0 1 η ∣ ∇ τ + θ ν ( η ) ∣ d η = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) 2 ∫ γ ε β ε ( β ε − u ) ∣ ψ ( u ) ∣ d u = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) 2 − ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ( β ε − u ) ψ ( u ) d u + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ( β ε − u ) ψ ( u ) d u = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) 2 [ A 1 + A 2 + A 3 + A 4 ] .

Here,

(89) A 1 = − ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ( β ε − u ) ( u τ + θ − γ ε τ + θ ) ν d u + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 ( β ς + γ ς 2 ) β ε ( β ε − u ) ( u τ + θ − γ ε τ + θ ) ν d u = − ϒ 2 ν ( β ε , γ ε , τ + θ )

A 2 = ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ( β ε − u ) ( ( β ε + γ ε − u ) τ + θ − γ ε τ + θ ) ν d u + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ( β ε − u ) ( ( β ε + γ ε − u ) τ + θ − γ ε τ + θ ) ν d u , A 3 = − ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ( β ε − u ) ( β ε τ + θ − ( β ε + γ ε − u ) τ + θ ) ν d u + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 ( β ς + γ ς 2 ) β ε ( β ε − u ) ( β ε τ + θ − ( β ε + γ ε − u ) τ + θ ) ν d u .

By the same change of variables as adopted earlier, we obtain

(90) A 2 = ϒ 2 ν ( γ ε , γ ε , τ + θ ) , A 3 = − ϒ 1 ν ( γ ε , β ε , τ + θ ) ,

(91) A 4 = ∫ γ ε ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ( β ε − u ) ( β ε τ + θ − u τ + θ ) ν d u + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ( β ε − u ) ( β ε τ + θ − u τ + θ ) ν d u = ϒ 1 ν ( β ε , β ε , τ + θ ) .

By using (89), (90), and (91) in (88), we deduce

(92) ∫ 0 1 η ∣ ∇ τ + θ ν ( η ) ∣ d η = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) 2 [ ϒ 1 ν ( β ε , β ε , τ + θ ) − ϒ 1 ν ( γ ε , β ε , τ + θ ) + ϒ 2 ν ( γ ε , γ ε , τ + θ ) − ϒ 2 ν ( β ε , γ ε , τ + θ ) ] .

Similarly,

(93) ∫ 0 1 ( 1 − η ) ∣ ∇ τ + θ ν ( η ) ∣ d η = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) 2 [ ϒ 1 ν ( β ε , β ε , τ + θ ) − ϒ 1 ν ( γ ε , β ε , τ + θ ) + ϒ 2 ν ( γ ε , γ ε , τ + θ ) − ϒ 2 ν ( β ε , γ ε , τ + θ ) ] .

By using (87), (92), and (93) in (78), we obtain (76).□

Remark 6

  1. For τ + θ = 1 , the inequality (76) reduces to inequality (36) in [48];

  2. For τ + θ = 1 and ε = 2 , the inequality (76) reduces to inequality ( 3.4 ) in [30];

  3. For τ + θ = 1 , ε = 2 , β 1 = α 1 , and γ 1 = α 2 , the inequality (76) reduces to inequality (3.5) in [31].

Theorem 10

Let f be a differentiable function defined on I and α = ( α 1 , α 2 , … , α ε ) , β = ( β 1 , β 2 , … , β ε ) , and γ = ( γ 1 , γ 2 , … , γ ε ) be three tuples, where β ε > γ ε and α ς , β ς , γ ς ∈ I = [ γ ε , β ε ] , for all ς = 1 , 2 , … , ε . Let ν > 0 and τ ∈ R , θ ∈ ( 0 , 1 ] such that τ + θ > 0 . If q > 1 , β ≺ α , γ ≺ α and ∣ f ′ ∣ q is convex on I , then

(94) f ( γ ε ) + f ( β ε ) 2 − ( τ + θ ) ν Γ ( ν + 1 ) 4 ( β ε τ + θ − γ ε τ + θ ) ν [ K γ ε + ν θ τ F ( β ε ) + K β ε − ν θ τ F ( γ ε ) ] ≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 2 ( β ε τ + θ − γ ε τ + θ ) ν ( M τ + θ ν ( β ε , γ ε ) ) ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ q − N τ + θ ν ( β ε , γ ε ) 2 M τ + θ ν ( β ε , γ ε ) ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ q + ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ q 1 q ,

where

M τ + θ ν ( β ε , γ ε ) = Λ 1 ν ( β ε , τ + θ ) − Λ 2 ν ( γ ε , τ + θ ) ∑ ς = 1 ε − 1 ( γ ς − β ς )

and

N τ + θ ν ( β ε , γ ε ) = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) 2 [ ϒ 1 ν ( β ε , β ε , τ + θ ) − ϒ 1 ν ( γ ε , β ε , τ + θ ) + ϒ 2 ν ( γ ε , γ ε , τ + θ ) − ϒ 2 ν ( β ε , γ ε , τ + θ ) ] .

Proof

By using Lemma 3, we have

(95) f ( β ε ) + f ( γ ε ) 2 − ( τ + θ ) ν Γ ( ν + 1 ) 4 ( β ε τ + θ − γ ε τ + θ ) ν [ K γ ε + ν θ τ F ( β ε ) + K β ε − ν θ τ F ( γ ε ) ] = ∑ ς = 1 ε − 1 ( γ ς − β ς ) 4 ( β ε τ + θ − γ ε τ + θ ) ν ∫ 0 1 ∇ τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 ( η γ ς + ( 1 − η ) β ς ) d η ≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 ( β ε τ + θ − γ ε τ + θ ) ν ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ f ′ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 ( η γ ς + ( 1 − η ) β ς ) d η .

By applying power mean inequality, we deduce

(96) ≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 ( β ε τ + θ − γ ε τ + θ ) ν ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ d η 1 − 1 q ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ f ′ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 ( η γ ς + ( 1 − η ) β ς ) q d η 1 q .

Since ∣ f ′ ∣ q is convex. Therefore, we apply Theorem 4 for the values σ 1 = η , σ 2 = 1 − η , and n = 2 in (96) to obtain

(97) ≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 ( β ε τ + θ − γ ε τ + θ ) ν ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ d η 1 − 1 q × ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ q − η ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ q − ( 1 − η ) ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ q d η 1 q = ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 ( β ε τ + θ − γ ε τ + θ ) ν ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ d η 1 − 1 q ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ q ∫ 0 1 ∣ ∇ τ + θ ν ( η ) ∣ d η − ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ q ∫ 0 1 η ∣ ∇ τ + θ ν ( η ) ∣ d η − ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ q ∫ 0 1 ( 1 − η ) ∣ ∇ τ + θ ν ( η ) ∣ d η 1 q .

Now again by substituting (87), (92), and (93) into (97), we obtain (94).□

To obtain more new results, we give another definition as follows:

Definition 7

Let α = ( α 1 , α 2 , … , α ε ) , β = ( β 1 , β 2 , … , β ε ) , and γ = ( γ 1 , γ 2 , … , γ ε ) be three tuples, where β ε > γ ε and α ς , β ς , γ ς ∈ I = [ γ ε , β ε ] ⊆ [ 0 , ∞ ) , for all ς = 1 , 2 , … , ε , ν > 0 , τ + θ > 0 . If ξ 1 , ξ 2 ∈ I , then we define the operators:

(98) Λ 3 ν ( ξ 1 , ξ 2 , τ + θ ) = ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ∣ ξ 1 τ + θ − w τ + θ ∣ ν d w + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ∣ ( ξ 1 + ξ 2 − w ) τ + θ − ξ 2 τ + θ ∣ ν d w .

(99) Λ 4 ν ( ξ 1 , ξ 2 , τ + θ ) = ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ∣ ξ 1 − w ∣ ∣ ξ 1 τ + θ − w τ + θ ∣ ν d w + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ∣ ξ 1 − w ∣ ∣ ( ξ 1 + ξ 2 − w ) τ + θ − ξ 2 τ + θ ∣ ν d w .

(100) Λ 5 ν ( ξ 1 , ξ 2 , τ + θ ) = ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ∣ w − ξ 2 ∣ ∣ ξ 1 τ + θ − w τ + θ ∣ ν d w + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ∣ w − ξ 2 ∣ ∣ ( ξ 1 + ξ 2 − w ) τ + θ − ξ 2 τ + θ ∣ ν d w .

On the basis of Lemma 4, we derive the following results.

Theorem 11

Let all the conditions in the hypotheses of Theorem 9  hold true, then

(101) Γ ( ν + 1 ) ( τ + θ ) ν 2 Q τ + θ ν ( β ε , γ ε ) K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 + ν θ τ F ( β ε ) + K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 − ν θ τ F ( γ ε ) − f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ 1 Q τ + θ ν ( β ε , γ ε ) ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ R τ + θ ν ( β ε , γ ε ) ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ − 1 2 S τ + θ ν ( β ε , γ ε ) ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ + ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ ,

where

R τ + θ ν ( β ε , γ ε ) = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) Λ 3 ν ( β ε , γ ε , τ + θ )

and

S τ + θ ν ( β ε , γ ε ) = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) 2 [ Λ 4 ν ( β ε , γ ε , τ + θ ) + Λ 5 ν ( β ε , γ ε , τ + θ ) ] .

Proof

By using Lemma 4, we have

(102) Γ ( ν + 1 ) ( τ + θ ) ν 2 Q τ + θ ν ( β ε , γ ε ) K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 + ν θ τ F ( β ε ) + K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 − ν θ τ F ( γ ε ) − f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 = ∑ ς = 1 ε − 1 ( γ ς − β ς ) 4 Q τ + θ ν ( β ε , γ ε ) ∫ 0 1 P τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς − f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς d η , ≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 Q τ + θ ν ( β ε , γ ε ) ∫ 0 1 ∣ P τ + θ ν ( η ) ∣ f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς d η + ∫ 0 1 ∣ P τ + θ ν ( η ) ∣ f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς d η = ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 Q τ + θ ν ( β ε , γ ε ) ∫ 0 1 P τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς d η + ∫ 0 1 P τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς d η .

By using Theorem 5 for the values n = 2 , σ 1 = η 2 , and σ 2 = 2 − η 2 in (102), we have

(103) ≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 Q τ + θ ν ( β ε , γ ε ) ∫ 0 1 P τ + θ ν ( η ) ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ − η 2 ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ − 2 − η 2 ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ d η + ∫ 0 1 P τ + θ ν ( η ) ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ − η 2 ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ − 2 − η 2 ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ d η . = ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 Q τ + θ ν ( β ε , γ ε ) 2 ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ ∫ 0 1 P τ + θ ν ( η ) d η − 1 2 ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ + ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ ∫ 0 1 η P τ + θ ν ( η ) d η − 1 2 ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ + ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ ∫ 0 1 ( 2 − η ) P τ + θ ν ( η ) d η = ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 Q τ + θ ν ( β ε , γ ε ) 2 ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ ∫ 0 1 P τ + θ ν ( η ) d η − 1 2 ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ + ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ × ∫ 0 1 η P τ + θ ν ( η ) d η + ∫ 0 1 ( 2 − η ) P τ + θ ν ( η ) d η .

By using the substitution u = ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς , we have

(104) ∫ 0 1 P τ + θ ν ( η ) d η = 2 ∑ ς = 1 ε − 1 ( γ ς − β ς ) ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ( β ε τ + θ − u τ + θ ) ν d u + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ( ( β ε + γ ε − u ) τ + θ − γ ε τ + θ ) ν d u = 2 ∑ ς = 1 ε − 1 ( γ ς − β ς ) Λ 3 ν ( β ε , γ ε , τ + θ ) ,

(105) ∫ 0 1 η P τ + θ ν ( η ) d η = 4 ∑ ς = 1 ε − 1 ( γ ς − β ς ) 2 ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ( β ε − u ) ( β ε τ + θ − u τ + θ ) ν d u + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ( β ε − u ) [ ( β ε + γ ε − u ) τ + θ − ( γ ε ) τ + θ ] ν d u = 4 ∑ ς = 1 ε − 1 ( γ ς − β ς ) 2 Λ 4 ν ( β ε , γ ε , τ + θ ) ,

and

(106) ∫ 0 1 ( 2 − η ) P τ + θ ν ( η ) d η = 4 ∑ ς = 1 ε − 1 ( γ ς − β ς ) 2 ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ( u − γ ε ) ( β ε τ + θ − u τ + θ ) ν d u + ∫ ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 β ε ( u − β ε ) ( ( β ε + γ ε − u ) τ + θ − γ ε τ + θ ) ν d u = 4 ∑ ς = 1 ε − 1 ( γ ς − β ς ) 2 Λ 5 ν ( β ε , γ ε , τ + θ ) .

By substituting (104), (105), and (106) in (103), we obtain (101).□

Theorem 12

Let all the conditions in the hypotheses of Theorem 10  hold true, then

(107) Γ ( ν + 1 ) ( τ + θ ) ν 2 Q τ + θ ν ( β ε , γ ε ) K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 + ν θ τ F ( β ε ) + K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 − ν θ τ F ( γ ε ) − f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 ≤ 1 2 Q τ + θ ν ( β ε , γ ε ) ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ R τ + θ ν ( β ε , γ ε ) ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ q − ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ q B τ + θ ν ( β ε , γ ε ) − ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ q C τ + θ ν ( β ε , γ ε ) 1 q + ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ q − ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ q B τ + θ ν ( β ε , γ ε ) − ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ q C τ + θ ν ( β ε , γ ε ) 1 q ,

where

R τ + θ ν ( β ε , γ ε ) = 1 ∑ ς = 1 ε − 1 ( γ ς − β ς ) Λ 3 ν ( β ε , γ ε , τ + θ ) B τ + θ ν ( β ε , γ ε ) = Λ 4 ν ( β ε , γ ε , τ + θ ) Λ 3 ν ( β ε , γ ε , τ + θ ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) C τ + θ ν ( β ε , γ ε ) = Λ 5 ν ( β ε , γ ε , τ + θ ) Λ 3 ν ( β ε , γ ε , τ + θ ) ∑ ς = 1 ε − 1 ( γ ς − β ς ) .

Proof

By using Lemma 4, we have

(108) Γ ( ν + 1 ) ( τ + θ ) ν 2 Q τ + θ ν ( β ε , γ ε ) K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 + ν θ τ F ( β ε ) + K ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 − ν θ τ F ( γ ε ) − f ∑ ς = 1 ε α ς − ∑ ς = 1 ε − 1 β ς + γ ς 2 = ∑ ς = 1 ε − 1 ( γ ς − β ς ) 4 Q τ + θ ν ( β ε , γ ε ) ∫ 0 1 P τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς − f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς d η , ≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 Q τ + θ ν ( β ε , γ ε ) ∫ 0 1 ∣ P τ + θ ν ( η ) ∣ f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς d η + ∫ 0 1 ∣ P τ + θ ν ( η ) ∣ f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς d η = ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 Q τ + θ ν ( β ε , γ ε ) ∫ 0 1 P τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς d η + ∫ 0 1 P τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς d η .

By using power mean inequality, we have

≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 Q τ + θ ν ( β ε , γ ε ) ∫ 0 1 P τ + θ ν ( η ) 1 − 1 q ∫ 0 1 P τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς q d η 1 q + ∫ 0 1 P τ + θ ν ( η ) 1 − 1 q ∫ 0 1 P τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς q d η 1 q .

(109) ≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 Q τ + θ ν ( β ε , γ ε ) ∫ 0 1 P τ + θ ν ( η ) 1 − 1 q ∫ 0 1 P τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 γ ς + 2 − η 2 ∑ ς = 1 ε − 1 β ς q d η 1 q + ∫ 0 1 P τ + θ ν ( η ) f ′ ∑ ς = 1 ε α ς − η 2 ∑ ς = 1 ε − 1 β ς + 2 − η 2 ∑ ς = 1 ε − 1 γ ς q d η 1 q .

By applying Theorem 5 for the values n = 2 , σ 1 = η 2 , and σ 2 = 2 − η 2 in (109), we have

(110) ≤ ∑ ς = 1 ε − 1 ∣ γ ς − β ς ∣ 4 Q τ + θ ν ( β ε , γ ε ) ∫ 0 1 P τ + θ ν ( η ) 1 − 1 q × ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ q ∫ 0 1 P τ + θ ν ( η ) d η − 1 2 ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ q ∫ 0 1 η P τ + θ ν ( η ) d η − 1 2 ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ q ∫ 0 1 ( 2 − η ) P τ + θ ν ( η ) d η 1 q + ∑ ς = 1 ε ∣ f ′ ( α ς ) ∣ q ∫ 0 1 P τ + θ ν ( η ) − 1 2 ∑ ς = 1 ε − 1 ∣ f ′ ( β ς ) ∣ q ∫ 0 1 η P τ + θ ν ( η ) d η − 1 2 ∑ ς = 1 ε − 1 ∣ f ′ ( γ ς ) ∣ q ∫ 0 1 ( 2 − η ) P τ + θ ν ( η ) d η 1 q .

By substituting (104), (105), and (106) in (110), we obtain (107).□

5 Conclusion

New conticrete versions of the Hermite-Hadamard-Jensen-Mercer inequalities were obtained for generalized conformable fractional integral operators. These new inequalities were established by using majorized tuples and convexity of the function along with fractional operators. As particular cases, new and old versions of the Hermite-Hadamard-Jensen-Mercer inequalities were also presented for various fractional operators such as Katugampola, Hadamard, Riemann-Liouville, conformable, and Riemann integrals. The obtained results adopted a more beautiful look when they were expressed in a weighted form. These weighted versions were acquired for two types of tuples, i.e., for decreasing tuples and the tuples which show same monotonic behavior. Furthermore, two new identities were investigated by using a differentiable function and three tuples. Then by applying these newly obtained identities and assuming the convexity of ∣ f ′ ∣ and ∣ f ′ ∣ q ( q > 1 ) , we constructed bounds for the absolute difference of terms involved in the main results. The current work can also be seen as application of majorization theory and produces new set of inequalities when combined with convex theory and fractional operators.

Acknowledgments

This research work was funded by Institutional Fund Projects under grant no. (IFPIP:321-130-1443). The authors gratefully acknowledge technical and financial support provided by the Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.

  1. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  2. Conflict of interest: The authors state that there is no conflict of interest.

References

[1] T.-H. Zhao, B.-C. Zhou, M.-K. Wang, and Y.-M. Chu, On approximating the quasi-arithmetic mean, J. Inequal. Appl. 2019 (2019), Article ID 42, 1–12. 10.1186/s13660-019-1991-0Search in Google Scholar

[2] T.-H. Zhao, Z.-H. Yang, and Y.-M. Chu, Monotonicity properties of a function involving the psi function with applications, J. Inequal. Appl. 2015 (2015), Article ID 193, 1–10. 10.1186/s13660-015-0724-2Search in Google Scholar

[3] T.-H. Zhao, Z.-Y. He, and Y.-M. Chu, On some refinements for inequalities involving zero-balanced hypergeometric function, AIMS Math. 5 (2020), no. 6, 6479–6495. 10.3934/math.2020418Search in Google Scholar

[4] M. J. Cloud, B. C. Drachman, and L. P. Lebedev, Inequalities with Applications to Engineering, Springer, Cham Heidelberg, New York Dordrecht London, 2014. 10.1007/978-3-319-05311-0Search in Google Scholar

[5] Q. Lin, Jensen inequality for superlinear expectations, Stat. Probabil. Lett. 151 (2019), 79–83. 10.1016/j.spl.2019.03.006Search in Google Scholar

[6] J. G. Liao and A. Berg, Sharpening Jensen’s inequality, Amer. Statistician. 73 (2019), no. 3, 278–281. 10.1080/00031305.2017.1419145Search in Google Scholar

[7] H.-H. Chu, T.-H. Zhao, and Y.-M. Chu, Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means, Math. Slovaca 70 (2020), no. 5, 1097–1112. 10.1515/ms-2017-0417Search in Google Scholar

[8] S. S. Dragomir, M. Adil Khan, and A. Abathun, Refinement of the Jensen integral inequality, Open Math. 14 (2016), 221–228. 10.1515/math-2016-0020Search in Google Scholar

[9] M. Adil Khan, Z. Al-sahwi, and Y.-M. Chu, New estimations for Shannon and Zipf-Mandelbrot entropies, Entropy 20 (2018), no. 8, 608. 10.3390/e20080608Search in Google Scholar PubMed PubMed Central

[10] M. Adil Khan, M. Hanif, Z. A. Khan, K. Ahmad, and Y.-M. Chu, Association of Jensen’s inequality for s-convex function with Csiszár divergence, J. Inequal. Appl. 2019 (2019), no. 1, 1–14. 10.1186/s13660-019-2112-9Search in Google Scholar

[11] T.-H. Zhao, M.-K. Wang, and Y.-M. Chu, Concavity and bounds involving generalized elliptic integral of the first kind, J. Math. Inequal. 15 (2019), no. 1, 701–724. 10.7153/jmi-2021-15-50Search in Google Scholar

[12] T.-H. Zhao, M.-K. Wang, and Y.-M. Chu, Monotonicity and convexity involving generalized elliptic integral of the first kind, Rev. R. Acad. Cienc. Exactas Fiiiis. Nat. Ser. A Mat. RACSAM 115 (2021), no. 2, 1–13. 10.1007/s13398-020-00992-3Search in Google Scholar

[13] T.-H. Zhao, Z.-Y. He, and Y.-M. Chu, Sharp bounds for the weighted Hölder mean of the zero-balanced generalized complete elliptic integrals, Comput. Methods Funct. Theory 21 (2021), no. 3, 413–426. 10.1007/s40315-020-00352-7Search in Google Scholar

[14] S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11 (1998), no. 5, 91–95. 10.1016/S0893-9659(98)00086-XSearch in Google Scholar

[15] M. Adil Khan, S. Khan, and Y.-M. Chu, New estimates for the Jensen gap using s-convexity with applications, Front. Phys. 8 (2020), 313. 10.3389/fphy.2020.00313Search in Google Scholar

[16] A. McD. Mercer, A variant of Jensen inequality, J. Ineq. Pure Appl. Math. 4 (2003), no. 4, Article 73. Search in Google Scholar

[17] M. E. Kiris, M. Eyüp, and M. Z. Sarikaya, On Ostrowski type inequalities and Čebyšev type inequalities with applications, Filomat 29 (2015), no. 8, 1695–1713. 10.2298/FIL1508695KSearch in Google Scholar

[18] M. Z. Sarikaya, On new Hermite-Hadamard Fejér type integral inequalities, Stud. Univ. Babes-Bolyai Math. 57 (2012), no. 3, 377–386. Search in Google Scholar

[19] S. S. Dragomir and S. Fitzpatrick, The Hadamardas inequality for s-convexn functions in the second sense, Demonstratio Math. 32 (1999), no. 4, 687–696. 10.1515/dema-1999-0403Search in Google Scholar

[20] S. S. Dragomir, On the Hadamard’s type inequality for convex functions on the co-ordinates in a rectangle from the plane, Taiwan. J. Math. 5 (2001), 775–788. 10.11650/twjm/1500574995Search in Google Scholar

[21] N. Merentes and K. Nikodem, Remarks on strongly convex functions, Aequat. Math. 80 (2010), 193–199. 10.1007/s00010-010-0043-0Search in Google Scholar

[22] M. R. Delavar and M. De La Sen, Some generalizations of Hermite-Hadamard type inequalities, SpringerPlus 5 (2016), no.1, 1–9. 10.1186/s40064-016-3301-3Search in Google Scholar PubMed PubMed Central

[23] F. Cesarone, M. Caputo, and C. Cametti, Memory formalism in the passive diffusion across a biological membrane, J. Membr. Sci. 250 (2004), 79–84. 10.1016/j.memsci.2004.10.018Search in Google Scholar

[24] M. Caputo, Modeling social and economic cycles, Elgar, Cheltenham, UK, 2014. Search in Google Scholar

[25] M. El-Shahed, Fractional calculus model of the semilunar heart valve vibrations, in: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 37033, 2003, 711–714. 10.1115/DETC2003/VIB-48384Search in Google Scholar

[26] G. Iaffaldano, M. Caputo, and S. Martino, Experimental and theoretical memory diffusion of water in the sand, Hydrol. Earth Syst. Sc. 10 (2006), 93–100. 10.5194/hess-10-93-2006Search in Google Scholar

[27] M. Naeem, H. Razazadeh, A. A. Khammash, R. Shah, and S. Zaland, Analysis of the fuzzy fractional-order solitary wave solutions for the KdV equation in the sense of Caputo-Fabrizio derivative, J. Mathematics 2022 (2022), Article ID 3688916, 1–12. 10.1155/2022/3688916Search in Google Scholar

[28] U. Younas, J. Ren, L. Akinyemi, and H. Razazadeh, On the multiple explicit exact solutions to the double-chain DNA dynamical system, Math. Methods Appl. Sci. 46 (2023), no. 6, 6309–6323. 10.1002/mma.8904Search in Google Scholar

[29] D. Kumar, A. Yildirim, M. K. A. Kabaar, H. Razazadeh, and M. E. Samei, Exploration of some novel solutions to a coupled Schrödinger-KdV equations in the interactions of capillary-gravity waves, Math. Sci. (2022), 1–13, DOI: https://doi.org/10.1007/s40096-022-00501-0. 10.1007/s40096-022-00501-0Search in Google Scholar

[30] H. Öagülmüş and M. Z. Sarikaya, Hermite-Hadamard-Mercer type inequalities for fractional integrals, Filomat 35 (2021), 2425–2436. 10.2298/FIL2107425OSearch in Google Scholar

[31] M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Baa̧k, Hermite-Hadamard inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model. 57 (2013), no. 9–10, 2403–2407. 10.1016/j.mcm.2011.12.048Search in Google Scholar

[32] J. Zhao, S. I. Butt, J. Nasir, Z. Wang, and I. Tlili, Hermite-Jensen-Mercer type inequalities for Caputo fractional derivatives, J. Func. Spaces 2020 (2020), no. 1, 1–11. 10.1155/2020/7061549Search in Google Scholar

[33] S. Zhao, S. I. Butt, W. Nazeer, J. Nasir, M. Umar, and Y. Liu, Some Hermite-Jensen-Mercer type inequalities for k-Caputo-fractional derivatives and related results, Adv. Differential Equations 2020 (2020), no. 1, 1–17. 10.1186/s13662-020-02693-ySearch in Google Scholar

[34] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Aapplications of Fractional Differential Equations, Elsevier, Amsterdam, vol. 204, 2006. Search in Google Scholar

[35] J. Hadamard, Essai sur laetude des fonctions donnees par leur developpment de Taylor, J. Pure Appl. Math. 4 (1892), no. 8, 101–186. Search in Google Scholar

[36] P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002), no. 2, 387–400. 10.1016/S0022-247X(02)00049-5Search in Google Scholar

[37] U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218 (2011), no. 3, 860–865. 10.1016/j.amc.2011.03.062Search in Google Scholar

[38] M. Jleli, D. O’Regan, and B. Sameet, On Hermite-Hadamard type inequalities via generalized fractional integrals, Turk. J. Math. 40 (2016), no. 6, 1221–1230. 10.3906/mat-1507-79Search in Google Scholar

[39] T. U. Khan and M. A. Khan, Generalized conformable fractional operators, J. Comput. Appl. Math. 346 (2019), 378–389. 10.1016/j.cam.2018.07.018Search in Google Scholar

[40] T. U. Khan and M. Adil Khan, Hermite-Hdadamard inequality for new generalized conformable fractional operators, AIMS. Math. 6 (2020), no. 1, 23–38. 10.3934/math.2021002Search in Google Scholar

[41] B.-Y. Wang, Foundations of Majorization Inequalities, Beijing Normal University Press, Beijing, 1990. Search in Google Scholar

[42] M. Niezgoda, A generalization of Merceras result on convex functions, Nonlinear Anal. 71 (2009), 2771–2779. 10.1016/j.na.2009.01.120Search in Google Scholar

[43] T.-H. Zhao, M.-K. Wang, and Y.-M. Chu, A sharp double inequality involving generalized complete elliptic integral of the first kind, AIMS Math. 5 (2020), no. 5, 4512–4528. 10.3934/math.2020290Search in Google Scholar

[44] S. Rashid, S. Sultana, Y. Karaca, A. Khalid, and Y.-M. Chu, Some further extensions considering discrete proportional fractional operators, Fractals 30 (2022), no. 1, 1–12. 10.1142/S0218348X22400266Search in Google Scholar

[45] T.-H. Zhao, L. Shi, and Y.-M. Chu, Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means, RACSAM 114 (2020), no. 2, 1–14. 10.1007/s13398-020-00825-3Search in Google Scholar

[46] T.-H. Zhao, M.-K. Wang, W. Zhang, and Y.-M. Chu, Quadratic transformation inequalities for Gaussian hypergeometric function, J. Inequal. Appl. 2018 (2018), no. 1, 1–15. 10.1186/s13660-018-1848-ySearch in Google Scholar PubMed PubMed Central

[47] Y.-M. Chu and T.-H. Zhao, Concavity of the error function with respect to Hölder means, Math. Inequal. Appl. 19 (2016), no. 2, 589–595. 10.7153/mia-19-43Search in Google Scholar

[48] S. Faisal, M. Adil Khan, T. U. Khan, T. Saeed, A. M. Alshehri, and E. R. Nwaeze, New “Conticrete” Hermite-Hadamard-Jensen-Mercer fractional inequalities, Symmetry 14 (2022), no. 2, 294. 10.3390/sym14020294Search in Google Scholar

[49] S. Faisal, M. Adil Khan, and S. Iqbal, Generalized Hermite-Hadamard-Mercer type inequalities via majorization, Filomat 36 (2022), no. 2, 469–483. 10.2298/FIL2202469FSearch in Google Scholar

[50] S. Faisal, M. Adil Khan, T. U. Khan, T. Saeed, and Z. M. M. M. Syed, Unifications of continuous and discrete fractional inequalities of the Hermite-Hadamard-Jensen-Mercer type via majorization, J. Funct. Spaces 2022 (2022), 24. 10.1155/2022/6964087Search in Google Scholar

[51] M. Kian and M. S. Moslehian, Refinements of the operator Jensen-Mercer inequality, Electron. J. Linear Algebra 26 (2013), 742–753. 10.13001/1081-3810.1684Search in Google Scholar
Received: 2022-10-04
Revised: 2023-03-03
Accepted: 2023-04-03
Published Online: 2023-05-11

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
  26. Feature fusion-based text information mining method for natural scenes
  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
  88. Petri net analysis of a queueing inventory system with orbital search by the server
  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
https://doi.org/10.1515/dema-2022-0225
Keywords for this article
Jensen inequality; Mercer inequality; Hermite-Hadamard inequality; Hölder inequality; majorization theory
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
  26. Feature fusion-based text information mining method for natural scenes
  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
  88. Petri net analysis of a queueing inventory system with orbital search by the server
  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 25.1.2026 from https://www.degruyterbrill.com/document/doi/10.1515/dema-2022-0225/html
Scroll to top button