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Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization

  • Shanhe Wu , Muhammad Adil Khan EMAIL logo , Shah Faisal , Tareq Saeed EMAIL logo and Eze R. Nwaeze
Published/Copyright: August 13, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

This article is mainly concerned to link the Hermite-Hadamard and the Jensen-Mercer inequalities by using majorization theory and fractional calculus. We derive the Hermite-Hadamard-Jensen-Mercer-type inequalities in conticrete form, which serve as both discrete and continuous inequalities at the same time, for majorized tuples in the framework of the famous Atangana-Baleanu fractional operators. Also, the main inequalities encompass the previously established inequalities as special cases. Using generalized Mercer’s inequality, we also investigate the weighted forms of our major inequalities for certain monotonic tuples. Furthermore, the derivation of new integral identities enables us to construct bounds for the discrepancy of the terms concerning the main results. These bounds are constructed by incorporating the convexity of ∣ f ′ ∣ and ∣ f ′ ∣ q ( q > 1 ) and making use of power mean and Hölder inequalities along with the established identities.

Keywords: convex function; majorization; Atangana-Baleanu fractional operators
MSC 2010: 26A51; 26D15; 68P30

1 Introduction

Mathematical inequalities are immensely useful when dealing with the quantities that fall under conditions such as greater than, greater or equal to, less than, and less than or equal to. They are also needed when we are required to approximate a solution instead of giving an exact solution to a mathematical problem [1–3]. Applications of this field can be observed in different scientific fields such as Information Theory [4], Economics [5], Mathematical Statistics [6], and Engineering [7]. The theory of convex functions further strengthens the literature of the field of inequalities [8–11]. Consequently, researchers used the concept of convexity and obtained a large number of inequalities. Among these inequalities, some well-known inequalities are the Jensen [4], the Jensen-Mercer [12], Fejér [13], the Hermite-Hadamard [14,15], and the Ostrowski [16,17] inequalities. This article commences with a description of the Hermite-Hadamard and Jensen-Mercer inequalities.

The celebrated Hermite-Hadamard inequality, which applies to convex function f with θ 1 < θ 2 , is written as

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

The classical Jensen-Mercer inequality that applies to a convex function f defined on [ θ 1 , θ 2 ] and β ζ ∈ [ θ 1 , θ 2 ] , σ ζ ≥ 0 for all ζ = 1 , 2 , … , ω , with ∑ ζ = 1 ω σ ζ = 1 , is given as

(2) f θ 1 + θ 2 − ∑ ζ = 1 ω σ ζ β ζ ≤ f ( θ 1 ) + f ( θ 2 ) − ∑ ζ = 1 ω σ ζ f ( β ζ ) .

A handful amount of research has been devoted to the Hermite-Hadamard inequality in recent years [18–20]. To obtain its various shapes, researchers have either applied convexity of the function in various forms such as strong convexity [21], s-convexity [22], coordinated convexity [23], and η -convexity [24], or they have mixed convex theory with some other well-known concepts such as majorization and fractional calculus [25–30].

The area of fractional calculus has been the subject of research for many researchers due to its extensive variety of applications in numerous scientific fields such as medicine [31], biology [32], geophysics [33], and mathematical analysis [34]. Earlier, this journey began with a query about the possibility of a solution of the differential equation if the integer order of the involved derivative is replaced by fractional order. However, it has now been evolved with a variety of fractional derivatives and integral operators. These operators increase the connection between mathematics and other disciplines regarding their application areas by putting out approaches that are extremely appropriate for real-world problems. Riemann-Liouville fractional integrals played a significant role in this regard, and their discovery gave rise to numerous extended and generalized forms of fractional operators [35]. Some most commonly used operators include Caputo [36], Caputo-Fabrizio [37], k -Caputo [27,38], Hadamard [39], and Katugampola [40]. In the following, the definitions of some well-known fractional operators are discussed.

Definition 1

[35]  Riemann-Liouville fractional integral operators: Let a function f : [ θ 1 , θ 2 ] → R , gamma function Γ ( . ) , and η > 0 . Then, the below-mentioned integrals are termed as Riemann-Liouville fractional integral operators:

J θ 1 + η f ( z ) = 1 Γ ( η ) ∫ θ 1 z ( z − u ) η − 1 f ( u ) d u , z > θ 1 , (left-sided integral)

and

J θ 2 − η f ( z ) = 1 Γ ( η ) ∫ z θ 2 ( u − z ) η − 1 f ( u ) d u , z < θ 2 , (right-sided integral) .

Note that f ( z ) = J θ 2 − 0 f ( z ) = J θ 1 + 0 f ( z ) .

Definition 2

[35,36] Caputo fractional derivative operators: Let a function f : [ θ 1 , θ 2 ] → R be such that f ∈ C n [ θ 1 , θ 2 ] , gamma function Γ ( . ) and η > 0 . Then, the below-mentioned derivatives are termed as Caputo fractional derivative operators:

D θ 1 + η c f ( z ) = 1 Γ ( n − η ) ∫ θ 1 z f ( n ) ( u ) ( z − u ) η − n + 1 d u , z > θ 1 , (left-sided derivative operator)

and

D θ 2 − η c f ( z ) = ( − 1 ) n Γ ( n − η ) ∫ z θ 2 f ( n ) ( u ) ( u − z ) η − n + 1 d u , z < θ 2 , (right-sided derivative operator) .

Note that for η = 0 and n = 1 , we have

f ( z ) = ( D θ 2 − 0 c f ) ( z ) = ( D θ 1 + 0 c f ) ( z ) .

Definition 3

[37]. Caputo-Fabrizio fractional derivative operator: Let a function f ∈ H ( 0 , θ 2 ) (“H” denotes Hilbert space), θ 2 > θ 1 , η ∈ [ 0 , 1 ] , and normalization function B ( η ) . Then, the below-mentioned derivative is termed as Caputo-Fabrizio fractional derivative operator:

(3) C F D η f ( z ) = B ( η ) 1 − η ∫ θ 1 z f ′ ( u ) exp − η 1 − η ( z − u ) d u .

The associated fractional integrals to the aforementioned operator are given in [41].

The Riemann-Liouville and Caputo operators contain power law as part of the kernel. In their applications, these operators have a number of benefits. However, it is important to note that power law behavior is not always present in nature. To address this issue, a new derivative and fractional operator known as Atangana-Baleanu (AB) fractional operator was introduced [42,43]. This operator contains the Mittag-Leffler function as part of its kernel, which can best model several real-world processes. One of the attractive advantages of this operator is that this operator reduces to the original function when the value of the parameter is selected zero. This characteristic distinguishes AB fractional operator over the Caputo-Fabrizio operator, which does not reduce to the original function for specific value of the parameter. Another disguising feature of this operator is that it has not been immensely used in connection to the inequalities and specially to the Hermite-Hadamard-type inequalities. The definition of this operator is expressed as follows.

Definition 4

[42] AB fractional integral operators: Let a function f ∈ H ( θ 1 , θ 2 ) with θ 2 > θ 1 and η ∈ [ 0 , 1 ] . Then, the below-mentioned integrals are termed as AB fractional integral operators:

(4) I η θ 1 A B f ( z ) = 1 − η B ( η ) f ( z ) + η B ( η ) Γ ( η ) ∫ θ 1 z ( z − u ) η − 1 f ( u ) d u .

The right-sided AB fractional integral operator is defined in [43] as follows:

(5) I θ 2 η A B f ( z ) = 1 − η B ( η ) f ( z ) + η B ( η ) Γ ( η ) ∫ z θ 2 ( u − z ) η − 1 f ( u ) d u .

Another concept that is related to our results is the majorization concept. Majorization symbolizes a partial order relation between the two tuples. It describes how far off one tuple is from the other tuple or how one tuple’s entries are almost equal to those of the other tuple. It is defined as follows:

Definition 5

[44] For any two tuples ϑ = ( ϑ 1 , … , ϑ ω ) and θ = ( θ 1 , … , θ ω ) whose components are arranged in decreasing order as ϑ [ ω ] ≤ ϑ [ ω − 1 ] ≤ ⋯ ≤ ϑ [ 1 ] , θ [ ω ] ≤ θ [ ω − 1 ] ≤ ⋯ ≤ θ [ 1 ] , then we say that the tuple ϑ majorizes the tuple θ or θ is majorized by ϑ (in symbols θ ≺ ϑ ), if:

∑ ζ = 1 k θ [ ζ ] ≤ ∑ ζ = 1 k ϑ [ ζ ] for k = 1 , 2 , … , ω − 1 ,

and

∑ ζ = 1 ω ϑ ζ = ∑ ζ = 1 ω θ ζ .

Majorization turns complex optimization-related problems into straightforward ones that are then simple to address [45,46]. Moreover, majorization has also played a crucial role in the theory of matrices and led to the derivation of a number of matrix inequalities and norm inequalities [47]. Nowadays, researchers are using the concept of majorization to construct inequalities connecting various fields. Deng et al. [4] refined the Jensen inequality using the idea of majorization and convexity of the function. In addition to the bounds for quasi-arithmetic and power means, they also provided applications for various divergences. Saeed et al. [48] presented a refinement of the integral version of the Jensen inequality by incorporating the idea of majorization and convexity. With the aid of the major obtained results, they also established an improvement and refinement of the Hermite-Hadamard and Hölder inequalities, respectively. Furthermore, applications of the major results in the field of information theory have also been discussed.

Research has been carried out continuously to construct inequalities in either discrete or continuous sense [28–30,49]. However, the derivation of compact inequalities containing both continuous and discrete inequalities simultaneously was the need of the hour. To address this issue, Faisal et al. [50–52] used the concept of majorization together with convex theory and various fractional operators and constructed such remarkable inequalities that not only serve as combined form of discrete and continuous inequalities but also produce the previously existing inequalities in the literature, as particular cases. The authors also obtained integral identities that have been further used to construct bounds for the established inequalities.

To obtain our main findings, let us first remind the below-extended form of the Jensen-Mercer inequality established with the aid of majorization.

Theorem 1

[53] Assume a convex function f : I → R , a tuple ρ = ( ρ 1 , … , ρ ω ) , and n × ω real matrix ( x i ζ ) such that ρ ζ , x i ζ ∈ I for all i = 1 , 2 , … , n , ζ = 1 , 2 , … , ω , and σ i ≥ 0 for i = 1 , 2 , … , n with ∑ i = 1 n σ i = 1 . If each row of the matrix ( x i ω ) is majorized by the tuple ρ , then

f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ∑ i = 1 n σ i x i ζ ≤ ∑ ζ = 1 ω f ( ρ ζ ) − ∑ ζ = 1 ω − 1 ∑ i = 1 n σ i f ( x i ζ ) .

This article is categorized into four sections. Section 1 consists of the derivation of new inequalities in terms of the majorization concept and fractional calculus. These results are accompanied by the supporting remarks which show that the specific values of the parameter transform the freshly constructed inequalities to other classical inequalities and unified inequalities for majorized tuples that contain Riemann integrals. Section 2 provides weighted forms of the results obtained in Section 1 by incorporating Lemmas 1 and 2. In order to make the established inequalities more powerful and attractive, we derive integral identities in Section 3. These identities work as a tool that bound the major results when applied along with the Hölder and power mean inequalities and assumptions of convexity of the function. Section 5 contains a detailed conclusion.

2 Main results

Our first major finding in the context of AB fractional integral operators is demonstrated as follows:

Theorem 2

Assume a convex function f : [ ν ω , μ ω ] → R and three tuples ρ = ( ρ 1 , ρ 2 , … , ρ ω ) , μ = ( μ 1 , μ 2 , … , μ ω ) , ν = ( ν 1 , ν 2 , … , ν ω ) such that ρ ζ , μ ζ , ν ζ ∈ [ ν ω , μ ω ] for all ζ = 1 , 2 , … , ω and η > 0 . Also, if μ ≺ ρ and ν ≺ ρ , then

(6) f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 ≤ B ( η ) Γ ( η ) 2 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η [ I η ν ω A B f ( μ ω ) + I μ ω η A B f ( ν ω ) ] − 1 − η B ( η ) [ f ( μ ω ) + f ( ν ω ) ] ≤ 1 2 [ f ( μ ω ) + f ( ν ω ) ] ≤ ∑ ζ = 1 ω f ( ρ ζ ) − 1 2 ∑ ζ = 1 ω − 1 f ( μ ζ ) + ∑ ζ = 1 ω − 1 f ( ν ζ ) .

Proof

By the fact that μ ≺ ρ and ν ≺ ρ , we have

∑ ζ = 1 ω ρ ζ = ∑ ζ = 1 ω μ ζ and ∑ ζ = 1 ω ρ ζ = ∑ ζ = 1 ω ν ζ ,

which implies

∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ = μ ω and ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ν ζ = ν ω .

Now, we start by considering that

(7) f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 = f μ ω + ν ω 2 = f ξ μ ω + ( 1 − ξ ) ν ω + ξ ν ω + ( 1 − ξ ) μ ω 2 .

Using convexity of f in (7), we have

(8) f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 ≤ 1 2 [ f ( ξ μ ω + ( 1 − ξ ) ν ω ) + f ( ξ ν ω + ( 1 − ξ ) μ ω ) ] .

Multiplying (8) by η B ( η ) Γ ( η ) ξ η − 1 and then integrating, we obtain

(9) 1 B ( η ) Γ ( η ) f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 ≤ 1 2 η B ( η ) Γ ( η ) ∫ 0 1 ξ η − 1 f ( ξ μ ω + ( 1 − ξ ) ν ω ) d ξ + η B ( η ) Γ ( η ) ∫ 0 1 ξ η − 1 f ( ξ ν ω + ( 1 − ξ ) μ ω ) d ξ .

Using AB fractional integrals in (9), we obtain

(10) f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 ≤ B ( η ) Γ ( η ) 2 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η [ I η ν ω A B f ( μ ω ) + I μ ω η A B f ( ν ω ) ] − 1 − η B ( η ) [ f ( μ ω ) + f ( ν ω ) ] .

The proof for the first part of inequality (6) is now finished. In order to establish the second inequality, we make use of the convexity of f as shown in the following:

(11) f ( ξ μ ω + ( 1 − ξ ) ν ω ) ≤ ξ f ( μ ω ) + ( 1 − ξ ) f ( ν ω ) ,

(12) f ( ξ ν ω + ( 1 − ξ ) μ ω ) ≤ ξ f ( ν ω ) + ( 1 − ξ ) f ( μ ω ) .

Adding (11) and (12), and using Theorem 1 with n = 1 and σ 1 = 1 , we obtain

(13) f ( ξ μ ω + ( 1 − ξ ) ν ω ) + f ( ξ ν ω + ( 1 − ξ ) μ ω ) ≤ f ( μ ω ) + f ( ν ω ) ≤ 2 ∑ ζ = 1 ω f ( ρ ζ ) − ∑ ζ = 1 ω − 1 f ( μ ζ ) + ∑ ζ = 1 ω − 1 f ( ν ζ ) .

By integrating the inequality that results from multiplying (13) with η B ( η ) Γ ( η ) ξ η − 1 , we derive the remaining inequalities in (6).□

Remark 1

  1.   By considering the hypotheses of Theorem 2 and taking η = 1 , we obtain the following inequality given in [50]:

    (14) f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 ≤ 1 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) ∫ ∑ ζ = 1 ω − 1 μ ζ ∑ ζ = 1 ω − 1 ν ζ f ∑ ζ = 1 ω ρ ζ − u d u ≤ ∑ ζ = 1 ω f ( ρ ζ ) − 1 2 ∑ ζ = 1 ω − 1 f ( μ ζ ) + ∑ ζ = 1 ω − 1 f ( ν ζ ) .

  2.   By choosing ω = 2 , inequality (6) produces the following inequality given in [54]:

    (15) f ρ 1 + ρ 2 − μ 1 + ν 1 2 ≤ B ( η ) Γ ( η ) 2 ( ν 1 − μ 1 ) η [ [ I η ( ρ 1 + ρ 2 − ν 1 ) A B f ( ρ 1 + ρ 2 − μ 1 ) + I ( ρ 1 + ρ 2 − μ 1 ) η A B f ( ρ 1 + ρ 2 − ν 1 ) ] − 1 − η B ( η ) [ f ( ρ 1 + ρ 2 − μ 1 ) + f ( ρ 1 + ρ 2 − ν 1 ) ] ≤ f ( ρ 1 ) + f ( ρ 2 ) − 1 2 [ f ( μ 1 ) + f ( ν 1 ) ] .

  3. If we choose ω = 2 and η = 1 in (6), then we obtain the succeeding result obtained by Kian and Moslehian [55]:

    (16) f ρ 1 + ρ 2 − μ 1 + ν 1 2 ≤ 1 ν 1 − μ 1 ∫ μ 1 ν 1 f ( ρ 1 + ρ 2 − u ) d u ≤ f ( ρ 1 ) + f ( ρ 2 ) − 1 2 [ f ( μ 1 ) + f ( ν 1 ) ] .

As previously mentioned, we construct our next finding as follows:

Theorem 3

Assuming that all the conditions specified in Theorem 2 are satisfied, then

(17) f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 ≤ 2 η − 1 B ( η ) Γ ( η ) ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η I ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 η A B f ( ν ω ) + I η ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 A B f ( μ ω ) − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] ≤ ∑ ζ = 1 ω f ( ρ ζ ) − 1 2 ∑ ζ = 1 ω − 1 f ( μ ζ ) + ∑ ζ = 1 ω − 1 f ( ν ζ ) .

Proof

For ξ ∈ [ 0 , 1 ] , it can be expressed that

(18) f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 = f μ ω + ν ω 2 = f 1 2 ∑ ζ = 1 ω ρ ζ − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ∑ ζ = 1 ω ρ ζ − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ .

Using the convexity of f in (18), we establish that

(19) f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 ≤ 1 2 f ∑ ζ = 1 ω ρ ζ − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + f ∑ ζ = 1 ω ρ ζ − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ .

Multiplying (19) by η B ( η ) Γ ( η ) ξ η − 1 and then integrating yield

(20) 1 B ( η ) Γ ( η ) f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 ≤ 1 2 η B ( η ) Γ ( η ) ∫ 0 1 ξ η − 1 f ∑ ζ = 1 ω ρ ζ − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ d ξ + η B ( η ) Γ ( η ) ∫ 0 1 ξ η − 1 f ∑ ζ = 1 ω ρ ζ − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ d ξ .

Using AB fractional integrals in (20), we arrive at

(21) f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 ≤ 2 η − 1 B ( η ) Γ ( η ) ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η I ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 η A B f ( ν ω ) + I η ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 A B f ( μ ω ) − ( 1 − η ) Γ ( η ) 2 [ f ( μ ω ) + f ( ν ω ) ] .

Thus, the first part of Inequality (17) is completed. In order to demonstrate the second inequality of (17), we employ Theorem 1 with σ 1 = ξ 2 , σ 2 = 2 − ξ 2 and n = 2 in the following manner:

(22) f ∑ ζ = 1 ω ρ ζ − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ ≤ ∑ ζ = 1 ω f ( ρ ζ ) − ξ 2 ∑ ζ = 1 ω − 1 f ( μ ζ ) + 2 − ξ 2 ∑ ζ = 1 ω − 1 f ( ν ζ )

and

(23) f ∑ ζ = 1 ω ρ ζ − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ ≤ ∑ ζ = 1 ω f ( ρ ζ ) − ξ 2 ∑ ζ = 1 ω − 1 f ( ν ζ ) + 2 − ξ 2 ∑ ζ = 1 ω − 1 f ( μ ζ ) .

Adding (22) and (23), we obtain

(24) f ∑ ζ = 1 ω ρ ζ − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + f ∑ ζ = 1 ω ρ ζ − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ ≤ 2 ∑ ζ = 1 ω f ( ρ ζ ) − ∑ ζ = 1 ω − 1 f ( μ ζ ) + ∑ ζ = 1 ω − 1 f ( ν ζ ) .

By multiplying (24) by η B ( η ) Γ ( η ) ξ η − 1 and then integrating the resulting inequality, we obtain the second inequality of (17).□

Remark 2

Inequality (17) produces the below result obtained by Kian and Moslehian given in [55] by choosing ω = 2 and η = 1 :

(25) f ρ 1 + ρ 2 − μ 1 + ν 1 2 ≤ 1 ν 1 − μ 1 ∫ μ 1 ν 1 f ( ρ 1 + ρ 2 − u ) d u ≤ f ( ρ 1 ) + f ( ρ 2 ) − 1 2 [ f ( μ 1 ) + f ( ν 1 ) ] .

3 Weighted forms of the main results

This section of the manuscript is devoted to the derivation of weighted forms of our major results. We obtain these weighted forms by taking into account certain monotonic tuples.

To obtain our new results in weighted forms, we employ the following lemmas [50].

Lemma 1

Assume a convex function f : I → R , two tuples ρ = ( ρ 1 , … , ρ ω ) , r = ( r 1 , … , r ω ) , and n × ω real matrix ( x i ζ ) such that ρ ζ , x i ζ ∈ I , r ζ ≥ 0 with r ω ≠ 0 , ε = 1 r ω for all ζ = 1 , … , ω , i = 1 , 2 , … , n and σ i ≥ 0 for i = 1 , 2 , … , n with ∑ i = 1 n σ i = 1 . If for every i = 1 , 2 , … , n , the tuple ( x i 1 , … , x i ω ) is decreasing and

∑ ζ = 1 k r ζ x i ζ ≤ ∑ ζ = 1 k r ζ ρ ζ for k = 1 , 2 , … , ω − 1 , ∑ ζ = 1 ω r ζ ρ ζ = ∑ ζ = 1 ω r ζ x i ζ ,

then

f ∑ ζ = 1 ω ε r ζ ρ ζ − ∑ ζ = 1 ω − 1 ∑ i = 1 n ε σ i r ζ x i ζ ≤ ∑ ζ = 1 ω ε r ζ f ( ρ ζ ) − ∑ ζ = 1 ω − 1 ∑ i = 1 n ε σ i r ζ f ( x i ζ ) .

Lemma 2

Assume a convex function f : I → R , two tuples ρ = ( ρ 1 , … , ρ ω ) , r = ( r 1 , … , r ω ) , and n × ω real matrix ( x i ζ ) such that ρ ζ , x i ζ ∈ I , r ζ ≥ 0 with r ω ≠ 0 , ε = 1 r ω for all ζ = 1 , … , ω , i = 1 , 2 , … , n and σ i ≥ 0 for i = 1 , 2 , … , n with ∑ i = 1 n σ i = 1 . If x i ζ and ( ρ ζ − x i ζ ) are showing the same monotonic behavior for each i = 1 , … , n , and

∑ ζ = 1 ω r ζ ρ ζ = ∑ ζ = 1 ω r ζ x i ζ ,

then

f ∑ ζ = 1 ω ε r ζ ρ ζ − ∑ ζ = 1 ω − 1 ∑ i = 1 n ε σ i r ζ x i ζ ≤ ∑ ζ = 1 ω ε r ζ f ( ρ ζ ) − ∑ ζ = 1 ω − 1 ∑ i = 1 n ε σ i r ζ f ( x i ζ ) .

The following result is based on Lemma 1.

Theorem 4

Assume a convex function f : [ ν ω , μ ω ] → R and four tuples ρ = ( ρ 1 , ρ 2 , … , ρ ω ) , μ = ( μ 1 , μ 2 , … , μ ω ) , ν = ( ν 1 , ν 2 , … , ν ω ) , and r = ( r 1 , r 2 , … , r ω ) such that ρ ζ , μ ζ , ν ζ ∈ [ ν ω , μ ω ] , r ζ ≥ 0 with r ω ≠ 0 for all ζ = 1 , 2 , … , ω and ε = 1 r ω , η > 0 . Also, if the tuples μ and ν are decreasing and

∑ ζ = 1 k r ζ μ ζ ≤ ∑ ζ = 1 k r ζ ρ ζ , ∑ ζ = 1 k r ζ ν ζ ≤ ∑ ζ = 1 k r ζ ρ ζ , f o r k = 1 , … , ω − 1 ,

∑ ζ = 1 ω r ζ ρ ζ = ∑ ζ = 1 ω r ζ μ ζ , ∑ ζ = 1 ω r ζ ρ ζ = ∑ ζ = 1 ω r ζ ν ζ ,

then

(26) f ∑ ζ = 1 ω ε r ζ ρ ζ − ε ∑ ζ = 1 ω − 1 r ζ μ ζ + r ζ ν ζ 2 ≤ B ( η ) Γ ( η ) 2 ( μ ω − ν ω ) η [ I η ν ω A B f ( μ ω ) + I μ ω η A B f ( ν ω ) ] − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] ≤ 1 2 [ f ( μ ω ) + f ( ν ω ) ] ≤ ∑ ζ = 1 ω ε r ζ f ( ρ ζ ) − 1 2 ∑ ζ = 1 ω − 1 ε r ζ f ( μ ζ ) + ∑ ζ = 1 ω − 1 ε r ζ f ( ν ζ ) .

Proof

By taking

∑ ζ = 1 ω r ζ ρ ζ = ∑ ζ = 1 ω r ζ μ ζ and ∑ ζ = 1 ω r ζ ρ ζ = ∑ ζ = 1 ω r ζ ν ζ ,

we arrive at

∑ ζ = 1 ω ε r ζ ρ ζ − ∑ ζ = 1 ω − 1 ε r ζ μ ζ = μ ω and ∑ ζ = 1 ω ε r ζ ρ ζ − ∑ ζ = 1 ω − 1 ε r ζ ν ζ = ν ω .

Now, for ξ ∈ [ 0 , 1 ] , we may write that

(27) f ∑ ζ = 1 ω ε r ζ ρ ζ − ε ∑ ζ = 1 ω − 1 r ζ μ ζ + ν ζ 2 = f μ ω + ν ω 2 = f ξ μ ω + ( 1 − ξ ) ν ω + ξ ν ω + ( 1 − ξ ) μ ω 2 .

Using the convexity of f in (27), we obtain

(28) f ∑ ζ = 1 ω ε r ζ ρ ζ − ε ∑ ζ = 1 ω − 1 r ζ μ ζ + ν ζ 2 ≤ 1 2 [ f ( ξ μ ω + ( 1 − ξ ) ν ω ) + f ( ξ ν ω + ( 1 − ξ ) μ ω ) ] .

Multiplying (28) by η B ( η ) Γ ( η ) ξ η − 1 and then integrating yield

(29) 1 B ( η ) Γ ( η ) f ∑ ζ = 1 ω ε r ζ ρ ζ − ε ∑ ζ = 1 ω − 1 r ζ μ ζ + ν ζ 2 ≤ 1 2 η B ( η ) Γ ( η ) ∫ 0 1 ξ η − 1 f ( ξ μ ω + ( 1 − ξ ) ν ω ) d ξ + η B ( η ) Γ ( η ) ∫ 0 1 ξ η − 1 f ( ξ ν ω + ( 1 − ξ ) μ ω ) d ξ .

Using AB fractional integrals in (29), we obtain

(30) f ∑ ζ = 1 ω ε r ζ ρ ζ − ε ∑ ζ = 1 ω − 1 r ζ μ ζ + ν ζ 2 ≤ B ( η ) Γ ( η ) 2 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η [ I η ν ω A B f ( μ ω ) + I μ ω η A B f ( ν ω ) ] − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] .

Hence, we have successfully accomplished the first part of (26). To obtain the second part, we use the convexity of f to express that

(31) f ( ξ μ ω + ( 1 − ξ ) ν ω ) ≤ ξ f ( μ ω ) + ( 1 − ξ ) f ( ν ω ) ,

(32) f ( ξ ν ω + ( 1 − ξ ) μ ω ) ≤ ξ f ( ν ω ) + ( 1 − ξ ) f ( μ ω ) .

Adding (31) and (32) and then using Lemma 1 for n = 2 , σ 1 = ξ , and σ 2 = 1 − ξ , we obtain

(33) f ( ξ μ ω + ( 1 − ξ ) ν ω ) + f ( ξ ν ω + ( 1 − ξ ) μ ω ) ≤ f ( μ ω ) + f ( ν ω ) ≤ 2 ∑ ζ = 1 ω ε r ζ f ( ρ ζ ) − ∑ ζ = 1 ω − 1 ε r ζ f ( μ ζ ) + ∑ ζ = 1 ω − 1 ε r ζ f ( ν ζ ) .

By integrating the inequality that results from multiplying (33) with η B ( η ) Γ ( η ) ξ η − 1 , we obtain the remaining inequalities in (26).□

Remark 3

By setting η = 1 , Inequality (26) transforms into the underlying inequality given in [50]:

(34) f ∑ ζ = 1 ω ε r ζ ρ ζ − ε ∑ ζ = 1 ω − 1 r ζ μ ζ + r ζ ν ζ 2 ≤ 1 ∑ ζ = 1 ω − 1 ( ε r ζ ν ζ − ε r ζ μ ζ ) ∫ ∑ ζ = 1 ω − 1 ε r ζ μ ζ ∑ ζ = 1 ω − 1 ε r ζ ν ζ f ∑ ζ = 1 ω ε r ζ ρ ζ − u d u ≤ ∑ ζ = 1 ω ε r ζ f ( ρ ζ ) − 1 2 ∑ ζ = 1 ω − 1 ε r ζ f ( μ ζ ) + ∑ ζ = 1 ω − 1 ε r ζ f ( ν ζ ) .

Lemma 2 is used to establish the following result.

Theorem 5

Assume a convex function f : [ ν ω , μ ω ] → R and four tuples ρ = ( ρ 1 , ρ 2 , … , ρ ω ) , μ = ( μ 1 , μ 2 , … , μ ω ) , ν = ( ν 1 , ν 2 , … , ν ω ) , and r = ( r 1 , r 2 , … , r ω ) such that ρ ζ , μ ζ , ν ζ ∈ [ ν ω , μ ω ] , r ζ ≥ 0 with r ω ≠ 0 for all ζ = 1 , 2 , … , ω , and ε = 1 r ω , η > 0 . If ρ − μ , μ , ρ − ν , and ν are monotonically showing the similar behavior and

∑ ζ = 1 ω r ζ ρ ζ = ∑ ζ = 1 ω r ζ μ ζ , ∑ ζ = 1 ω r ζ ρ ζ = ∑ ζ = 1 ω r ζ ν ζ ,

then

(35) f ∑ ζ = 1 ω ε r ζ ρ ζ − ε ∑ ζ = 1 ω − 1 r ζ μ ζ + r ζ ν ζ 2 ≤ B ( η ) Γ ( η ) 2 ( μ ω − ν ω ) η [ I η ν ω A B f ( μ ω ) + I μ ω η A B f ( ν ω ) ] − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] ≤ 1 2 [ f ( μ ω ) + f ( ν ω ) ] ≤ ∑ ζ = 1 ω ε r ζ f ( ρ ζ ) − 1 2 ∑ ζ = 1 ω − 1 ε r ζ f ( μ ζ ) + ∑ ζ = 1 ω − 1 ε r ζ f ( ν ζ ) .

Proof

We prove (35) by using Lemma 2 and similar steps taken in the proof of Theorem 4. Thus, the proof is complete.□

Remark 4

Theorems 4 and 5 serve as weighted versions of Theorem 2, and corresponding weighted forms for Theorem 3 can be obtained in the similar manner.

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

Integral identities are the tools that lead us to more powerful inequalities when combined with some assumptions. In this section, first, we derive the identities for a differentiable function and then using these identities, we construct bounds for the major inequalities.

Lemma 3

Assume a differentiable function f on [ ν ω , μ ω ] and three tuples ρ = ( ρ 1 , ρ 2 , … , ρ ω ) and μ = ( μ 1 , μ 2 , … , μ ω ) , ν = ( ν 1 , ν 2 , … , ν ω ) such that ρ ζ , μ ζ , ν ζ ∈ [ ν ω , μ ω ] for all ζ = 1 , 2 , … , ω , η > 0 and ξ ∈ [ 0 , 1 ] . If f ′ ∈ L ( I ) , then

(36) 1 2 [ f ( μ ω ) + f ( ν ω ) ] − B ( η ) Γ ( η ) 2 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η [ I η ν ω A B f ( μ ω ) + I μ ω η A B f ( ν ω ) ] − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] = ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) 2 ∫ 0 1 ( ξ η − ( 1 − ξ ) η ) f ′ ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ .

Proof

In order to demonstrate our desired result, we take into consideration that

(37) I = ∫ 0 1 ( ξ η − ( 1 − ξ ) η ) f ′ ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ = ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ − ∫ 0 1 ( 1 − ξ ) η f ′ ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ = I 1 − I 2 .

Here,

(38) I 1 = ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ = ξ η f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) 0 1 − η ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) × ∫ 0 1 ξ η − 1 f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ = f ( μ ω ) ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) − η ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) ∫ 0 1 ξ η − 1 f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ .

Likewise,

(39) I 2 = ∫ 0 1 ( 1 − ξ ) η f ′ ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ = ( 1 − ξ ) η f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) 0 1 + η ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) × ∫ 0 1 ( 1 − ξ ) η − 1 f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ = − f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ν ζ ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) + η ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) ∫ 0 1 ( 1 − ξ ) η − 1 f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ .

By inserting (38) and (39) into (37) and then using the definition of AB fractional integrals and multiplying both sides by ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) 2 , we obtain (36).□

Remark 5

  1. For ω = 2 , Equality (36) transforms into the following equality, which has been proved in [54]:

    (40) 1 2 [ f ( μ 1 ) + f ( ν 1 ) ] − B ( η ) Γ ( η ) 2 ( ν 1 − μ 1 ) η [ [ I η ( ρ 1 + ρ 2 − ν 1 ) A B f ( ρ 1 + ρ 2 − μ 1 ) + I ( ρ 1 + ρ 2 − μ 1 ) η A B f ( ρ 1 + ρ 2 − ν 1 ) ] − 1 − η B ( η ) [ f ( ρ 1 + ρ 2 − μ 1 ) + f ( ρ 1 + ρ 2 − ν 1 ) ] = ( ν 1 − μ 1 ) 2 ∫ 0 1 ( ξ η − ( 1 − ξ ) η ) f ′ ( ρ 1 + ρ 2 − ( ξ μ 1 + ( 1 − ξ ) ν 1 ) ) d ξ .

  2. By taking η = 1 , ω = 2 , μ 1 = ρ 1 , and ν 1 = ρ 2 in (36), we obtain the following inequality given in [14]:

    (41) f ( ρ 1 ) + f ( ρ 2 ) 2 − 1 ρ 2 − ρ 1 ∫ ρ 1 ρ 2 f ( u ) d u = ρ 2 − ρ 1 2 ∫ 0 1 ( 2 ξ − 1 ) f ′ ( ξ ρ 2 + ( 1 − ξ ) ρ 1 ) d ξ .

Lemma 4

Assuming that all the conditions specified in Lemma 3 are satisfied, then

(42) 2 η − 1 B ( η ) Γ ( η ) ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η I ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 η A B f ( ν ω ) + I η ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 A B f ( μ ω ) − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] − f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 = ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) 4 ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ − f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ d ξ .

Proof

It is very simple to derive Equality (42) by applying similar steps as presented in the proof of Lemma 3.□

Now, we present the following result based on Lemma 3.

Theorem 6

Assume a differentiable function f on [ ν ω , μ ω ] and three tuples ρ = ( ρ 1 , ρ 2 , … , ρ ω ) , μ = ( μ 1 , μ 2 , … , μ ω ) , and ν = ( ν 1 , ν 2 , … , ν ω ) such that ρ ζ , μ ζ , ν ζ ∈ [ ν ω , μ ω ] for all ζ = 1 , 2 , … , ω and η > 0 . If ∣ f ′ ∣ is convex, μ ≺ ρ and ν ≺ ρ , then

(43) 1 2 [ f ( μ ω ) + f ( ν ω ) ] − B ( η ) Γ ( η ) 2 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η [ I η ν ω A B f ( μ ω ) + I μ ω η A B f ( ν ω ) ] − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] ≤ ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ η + 1 1 − 1 2 η ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ − 1 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ + ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ .

Proof

Using Lemma 3, we can write

(44) 1 2 [ f ( μ ω ) + f ( ν ω ) ] − B ( η ) Γ ( η ) 2 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η [ I η ν ω A B f ( μ ω ) + I μ ω η A B f ( ν ω ) ] + ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] = 1 2 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) ∫ 0 1 ( ξ η − ( 1 − ξ ) η ) f ′ ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ ≤ 1 2 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 ∣ ( ξ η − ( 1 − ξ ) η ) ∣ f ′ ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ .

The convexity of ∣ f ′ ∣ allows us to obtain the following by applying Theorem 1 with σ 1 = ξ , n = 2 , and σ 2 = 1 − ξ in (44):

(45) ≤ 1 2 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 ∣ ( ξ η − ( 1 − ξ ) η ) ∣ ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ − ξ ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ + ( 1 − ξ ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ d ξ , = 1 2 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 2 ( ( 1 − ξ ) η − ξ η ) ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ − ξ ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ + ( 1 − ξ ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ d ξ + ∫ 1 2 1 ( ξ η − ( 1 − ξ ) η ) ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ − ξ ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ + ( 1 − ξ ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ d ξ , = 1 2 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ( B 1 + B 2 ) .

Note that we have

(46) B 1 = ∫ 0 1 2 ( ( 1 − ξ ) η − ξ η ) ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ − ξ ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ + ( 1 − ξ ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ d ξ = ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ 1 η + 1 − 1 2 η η + 1 − ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ 1 ( η + 1 ) ( η + 2 ) − 1 2 η + 1 η + 1 + ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ 1 η + 2 − 1 2 η + 1 η + 1

and

(47) B 2 = ∫ 1 2 1 ( ξ η − ( 1 − ξ ) η ) ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ − ξ ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ + ( 1 − ξ ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ d ξ = ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ 1 η + 1 − 1 2 η η + 1 − ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ 1 η + 2 − 1 2 η + 1 η + 1 + ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ 1 ( η + 1 ) ( η + 2 ) − 1 2 η + 1 η + 1 .

Substituting (46) and (47) into (45), we obtain (43).□

Another result is obtained as follows:

Theorem 7

Assume a differentiable function f on [ ν ω , μ ω ] and three tuples ρ = ( ρ 1 , ρ 2 , … , ρ ω ) , μ = ( μ 1 , μ 2 , … , μ ω ) , and ν = ( ν 1 , ν 2 , … , ν ω ) such that ρ ζ , μ ζ , ν ζ ∈ [ ν ω , μ ω ] for all ζ = 1 , 2 , … , ω and η > 0 . If ∣ f ′ ∣ q ( q > 1 ) is convex, μ ≺ ρ and ν ≺ ρ , then

(48) 1 2 [ f ( μ ω ) + f ( ν ω ) ] − B ( η ) Γ ( η ) 2 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η [ I η ν ω A B f ( μ ω ) + I μ ω η A B f ( ν ω ) ] − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] ≤ ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ η + 1 1 − 1 2 η ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − 1 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q + ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q 1 q .

Proof

Using Lemma 3, we have

1 2 [ f ( μ ω ) + f ( ν ω ) ] − B ( η ) Γ ( η ) 2 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η [ I η ν ω A B f ( μ ω ) + I μ ω η A B f ( ν ω ) ] − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] = 1 2 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) ∫ 0 1 ( ξ η − ( 1 − ξ ) η ) f ′ ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ ≤ 1 2 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 ∣ ξ η − ( 1 − ξ ) η ∣ f ′ ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) d ξ .

The utilization of the power mean inequality to the aforementioned integral yields

(49) ≤ 1 2 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 ∣ ξ η − ( 1 − ξ ) η ∣ d ξ 1 − 1 q ∫ 0 1 ∣ ξ η − ( 1 − ξ ) η ∣ f ′ ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) q d ξ 1 q = 1 2 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 2 ( ( 1 − ξ ) η − ξ η ) d ξ + ∫ 1 2 1 ( ξ η − ( 1 − ξ ) η ) d ξ 1 − 1 q ∫ 0 1 ∣ ξ η − ( 1 − ξ ) η ∣ f ′ ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 ( ξ μ ζ + ( 1 − ξ ) ν ζ ) q d ξ 1 q .

The convexity of ∣ f ′ ∣ q allows us to obtain the following by applying Theorem 1 with σ 1 = ξ , n = 2 , and σ 2 = 1 − ξ in (49):

(50) = 1 2 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 2 ( ( 1 − ξ ) η − ξ η ) d ξ + ∫ 1 2 1 ( ξ η − ( 1 − ξ ) η ) d ξ 1 − 1 q × ∫ 0 1 ∣ ξ η − ( 1 − ξ ) η ∣ ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − ξ ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q + ( 1 − ξ ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q d ξ 1 q = 1 2 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 2 ( ( 1 − ξ ) η − ξ η ) d ξ + ∫ 1 2 1 ( ξ η − ( 1 − ξ ) η ) d ξ 1 − 1 q × ∫ 0 1 2 ( ( 1 − ξ ) η − ξ η ) ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − ξ ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q + ( 1 − ξ ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q d ξ + ∫ 1 2 1 ( ξ η − ( 1 − ξ ) η ) ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − ξ ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q + ( 1 − ξ ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q d ξ 1 q = 1 2 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ 2 ( 1 − 1 2 η ) η + 1 1 − 1 q ( C 1 + C 2 ) 1 q .

Note that we have

(51) C 1 = ∫ 0 1 2 ( ( 1 − ξ ) η − ξ η ) ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − ξ ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q + ( 1 − ξ ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q d ξ = ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q 1 η + 1 − 1 2 η η + 1 − ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q 1 ( η + 1 ) ( η + 2 ) − 1 2 η + 1 η + 1 + ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q 1 η + 2 − 1 2 η + 1 η + 1

and

(52) C 2 = ∫ 1 2 1 ( ξ η − ( 1 − ξ ) η ) ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − ξ ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q + ( 1 − ξ ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q d ξ = ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q 1 η + 1 − 1 2 η η + 1 − ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q 1 η + 2 − 1 2 η + 1 η + 1 + ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q 1 ( η + 1 ) ( η + 2 ) − 1 2 η + 1 η + 1 .

By inserting (51) and (52) into (50), we obtain (48).□

By considering Lemma 4, we establish the following results.

Theorem 8

Assuming that all the conditions specified in Theorem 6  are satisfied, then

(53) 2 η − 1 B ( η ) Γ ( η ) ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η I ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 η A B f ( ν ω ) + I η ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 A B f ( μ ω ) − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] − f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 ≤ ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ 2 ( η + 1 ) ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ − 1 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ + ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ .

Proof

Using Lemma 4, we have

(54) 2 η − 1 B ( η ) Γ ( η ) ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η I ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 η A B f ( ν ω ) + I η ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 A B f ( μ ω ) − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] − f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 = 1 4 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ d ξ − ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ d ξ ≤ 1 4 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 ∣ ξ η ∣ f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ d ξ + ∫ 0 1 ∣ ξ η ∣ f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ d ξ = 1 4 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ d ξ + ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ d ξ .

Using Theorem 1 for n = 2 , σ 1 = 2 − ξ 2 , and σ 2 = ξ 2 in (54), we obtain

≤ 1 4 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 ξ η ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ + ξ 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ d ξ + ∫ 0 1 ξ η ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ + ξ 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ d ξ = 1 4 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ η + 1 − ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ η + 1 + ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ 2 ( η + 2 ) − ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ 2 ( η + 2 ) + ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ η + 1 − ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ η + 1 + ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ 2 ( η + 2 ) − ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ 2 ( η + 2 ) = ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ 2 ( η + 1 ) ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ − 1 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ + ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ .

Henceforth, the proof is complete.□

Theorem 9

Assume a differentiable function f on [ ν ω , μ ω ] and three tuples ρ = ( ρ 1 , ρ 2 , … , ρ ω ) , μ = ( μ 1 , μ 2 , … , μ ω ) , and ν = ( ν 1 , ν 2 , … , ν ω ) such that ρ ζ , μ ζ , ν ζ ∈ [ ν ω , μ ω ] for all ζ = 1 , 2 , … , ω and η > 0 . If ∣ f ′ ∣ q is convex, q > 1 , 1 p + 1 q = 1 , μ ≺ ρ , and ν ≺ ρ , then

(55) 2 η − 1 B ( η ) Γ ( η ) ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η I ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 η A B f ( ν ω ) + I η ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 A B f ( μ ω ) − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] − f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 ≤ ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ 4 ( η p + 1 ) 1 p ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − 1 4 3 ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q + ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q 1 q + ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − 1 4 3 ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q + ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q 1 q .

Proof

Using Lemma 4, we can write

2 η − 1 B ( η ) Γ ( η ) ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η I ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 η A B f ( ν ω ) + I η ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 A B f ( μ ω ) − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] − f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 = ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) 4 ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ d ξ − ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ d ξ ≤ ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ 4 ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ d ξ + ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ d ξ ≤ ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ 4 ∫ 0 1 ∣ ξ η ∣ f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ d ξ + ∫ 0 1 ∣ ξ η ∣ f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ d ξ .

Using Hölder’s inequality to the aforementioned integral, yields

(56) ≤ 1 4 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 ∣ ξ η ∣ p d ξ 1 p ∫ 0 1 f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ q d ξ 1 q + ∫ 0 1 ∣ ξ η ∣ p d ξ 1 p ∫ 0 1 f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ q d ξ 1 q = 1 4 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 ξ η p d ξ 1 p ∫ 0 1 f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ q d ξ 1 q + ∫ 0 1 ξ η p d ξ 1 p ∫ 0 1 f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ q d ξ 1 q = 1 4 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 ξ η p d ξ 1 p ∫ 0 1 f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ q d ξ 1 q + ∫ 0 1 f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ q d ξ 1 q .

The convexity of ∣ f ′ ∣ q allows us to obtain the following by applying Theorem 1 with σ 1 = 2 − ξ 2 , n = 2 , and σ 2 = ξ 2 in (56):

= 1 4 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ 1 η p + 1 1 p ∫ 0 1 ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − 2 − ξ 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q + ξ 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q d ξ 1 q + ∫ 0 1 ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − 2 − ξ 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q + ξ 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q d ξ 1 q = ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ 4 ( η p + 1 ) 1 p ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − 1 4 3 ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q + ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q 1 q + ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − 1 4 3 ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q + ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q 1 q .

Hence, the proof is complete.□

Theorem 10

Assuming that all the conditions specified in Theorem 7 are satisfied, then

(57) 2 η − 1 B ( η ) Γ ( η ) ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η I ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 η A B f ( ν ω ) + I η ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 A B f ( μ ω ) − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] − f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 ≤ ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ 4 ( η + 1 ) 1 − 1 q 1 η + 1 ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − η + 3 2 ( η + 1 ) ( η + 2 ) ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q + 1 2 ( η + 2 ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q 1 q + 1 η + 1 ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − η + 3 2 ( η + 1 ) ( η + 2 ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q + 1 2 ( η + 2 ) ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q 1 q .

Proof

Using Lemma 4, we have

2 η − 1 B ( η ) Γ ( η ) ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) η I ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 η A B f ( ν ω ) + I η ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 A B f ( μ ω ) − ( 1 − η ) B ( η ) [ f ( μ ω ) + f ( ν ω ) ] − f ∑ ζ = 1 ω ρ ζ − ∑ ζ = 1 ω − 1 μ ζ + ν ζ 2 = 1 4 ∑ ζ = 1 ω − 1 ( ν ζ − μ ζ ) ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ d ξ − ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ d ξ ≤ 1 4 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 ∣ ξ η ∣ f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ d ξ + ∫ 0 1 ∣ ξ η ∣ f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ d ξ .

The utilization of power mean inequality to the aforementioned integral, yields

≤ 1 4 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ ∫ 0 1 ξ η d ξ 1 − 1 q ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ q d ξ 1 q + ∫ 0 1 ξ η d ξ 1 − 1 q ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ q d ξ 1 q

(58) = 1 4 ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ 1 η + 1 1 − 1 q ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 μ ζ + ξ 2 ∑ ζ = 1 ω − 1 ν ζ q d ξ 1 q + ∫ 0 1 ξ η f ′ ∑ ζ = 1 ω ρ ζ − 2 − ξ 2 ∑ ζ = 1 ω − 1 ν ζ + ξ 2 ∑ ζ = 1 ω − 1 μ ζ q d ξ 1 q .

The convexity of ∣ f ′ ∣ q allows us to achieve the following by using Theorem 1 with σ 1 = 2 − ξ 2 , n = 2 , and σ 2 = ξ 2 in (58):

= ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ 4 ( η + 1 ) 1 − 1 q ∫ 0 1 ξ η ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − 2 − ξ 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q + ξ 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q d ξ 1 q + ∫ 0 1 ξ η ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − 2 − ξ 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q + ξ 2 ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q d ξ 1 q

(59)□ = ∑ ζ = 1 ω − 1 ∣ ν ζ − μ ζ ∣ 4 ( η + 1 ) 1 − 1 q 1 η + 1 ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − η + 3 2 ( η + 1 ) ( η + 2 ) ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q + 1 2 ( η + 2 ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q 1 q + 1 η + 1 ∑ ζ = 1 ω ∣ f ′ ( ρ ζ ) ∣ q − η + 3 2 ( η + 1 ) ( η + 2 ) ∑ ζ = 1 ω − 1 ∣ f ′ ( ν ζ ) ∣ q + 1 2 ( η + 2 ) ∑ ζ = 1 ω − 1 ∣ f ′ ( μ ζ ) ∣ q 1 q .

5 Conclusion

Numerous mathematicians are presently conducting extensive study on the popular Hermite-Hadamard inequality. This inequality is used as a tool for error estimation in numerical integration. It guarantees that the convex function is integrable and offers approximations for the integral mean of the convex function. Additionally, it has been established for a variety of convex functions, with particular emphasis on s-convex, η -convex, coordinate convex, and strongly convex functions. The construction of several integral identities for this inequality has also led to the creation of numerous other inequalities. In this particular study, we have used the idea of majorization to obtain combined inequalities of the Hermite-Hadamard and the Jensen-Mercer within the framework of fractional calculus. The idea of majorization was used as a connecting mechanism between these two inequalities, and the result is a single inequality that contains both discrete and continuous inequalities. These inequalities have been constructed with the help of AB fractional operators. The given remarks demonstrated that these inequalities cover the previously known inequalities. Additionally, weighted versions of the main inequalities have been found using different kinds of majorized tuples. We have also developed bounds for the discrepancy of the terms pertaining to the major inequalities with the aid of integral identities. The convexity of ∣ f ′ ∣ and ∣ f ′ ∣ q , ( q > 1 ) as well as power mean and Hölder inequalities are used to build these bounds along with the identities that have already been established. The findings of this research would be a valuable addition to the theory of mathematical inequalities.

  1. Funding information: This work was supported by the Teaching Reform Project of Fujian Provincial Education Department of China (No. FBJG20210184), and the Natural Science Foundation of Fujian Province of China (No. 2020J01365).

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

  3. Conflict of interest: The authors declare that there are no conflicts of interest regarding the publication of this article.

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Received: 2023-12-30
Revised: 2024-04-26
Accepted: 2024-05-05
Published Online: 2024-08-13

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Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2024-0024
Keywords for this article
convex function; majorization; Atangana-Baleanu fractional operators
Creative Commons
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Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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