Home Mathematics On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
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On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications

  • Xue-Xiao You , Muhammad Aamir Ali EMAIL logo , Humaira Kalsoom , Jarunee Soontharanon and Thanin Sitthiwirattham EMAIL logo
Published/Copyright: August 26, 2022
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From the journal Open Mathematics Volume 20 Issue 1

Abstract

In this study, we establish some new Hermite-Hadamard type inequalities for s-convex functions in the second sense using the post-quantum calculus. Moreover, we prove a new ( p , q ) -integral identity to prove some new Ostrowski type inequalities for ( p , q ) -differentiable functions. We also show that the newly discovered results are generalizations of comparable results in the literature. Finally, we give application to special means of real numbers using the newly proved inequalities.

Keywords: Hermite-Hadamard inequality; Ostrowski inequality; (p, q)-integral; post-quantum calculus; s-convex functions
MSC 2010: 26D10; 26D15; 26A51

1 Introduction

The Hermite-Hadamard (HH) inequality, which was independently found by Hermite and Hadamard (see, also [1], and [2, p. 137]), is particularly important in convex functions theory:

(1) ϝ π 1 + π 2 2 ≤ 1 π 2 − π 1 ∫ π 1 π 2 ϝ ( x ) d x ≤ ϝ ( π 1 ) + ϝ ( π 2 ) 2 ,

where ϝ is a convex function on [ π 1 , π 2 ] in this case. The aforementioned inequality is true in reverse order for concave mappings.

In [3], Hudzik and Maligranda defined s-convex functions in the second sense as follows: a mapping ϝ : R + → R , where R + = [ 0 , ∞ ) is called s-convex in the second sense if

ϝ ( t x + ( 1 − t ) y ) ≤ t s ϝ ( x ) + ( 1 − t ) s ϝ ( y )

for all x , y ∈ R + and t ∈ [ 0 , 1 ] and s ∈ ( 0 , 1 ] . Dragomir and Fitzpatrick [4] then used this newly discovered class of functions to prove the HH inequality on [ π 1 , π 2 ] as follows:

(2) 2 s − 1 ϝ π 1 + π 2 2 ≤ 1 π 2 − π 1 ∫ π 1 π 2 ϝ ( x ) d x ≤ ϝ ( π 1 ) + ϝ ( π 2 ) s + 1 .

On the other hand, several works in the field of q -analysis are being carried out, beginning with Euler, to achieve mastery in the mathematics that underpins quantum computing. The link between physics and mathematics is referred to as q -calculus. It has a wide range of applications in different areas of pure and applied mathematics [5,6]. Euler is thought to be the inventor of this significant branch of mathematics. In Newton’s work on infinite series, he used the q parameter. Later, Jackson [7,8] presented the q -calculus that knew without limits calculus in a logical approach. Al-Salam [9] presented the q -analogue of the q -fractional integral and the q -Riemann-Liouville fractional in 1966. Since then, the amount of study in this area has steadily expanded. In particular, in 2013, Tariboon and Ntouyas introduced D q π 1 -difference operator and q π 1 -integral in [10]. In 2020, Bermudo et al. introduced the notion of D q π 2 derivative and q π 2 -integral in [11]. Sadjang generalized to quantum calculus and introduced the notions of post-quantum calculus or shortly ( p , q ) -calculus in [12]. Soontharanon and Sitthiwirattham [13] introduced the notions of fractional ( p , q ) -calculus later on. In [14], Tunç and Göv gave the post-quantum variant of D q π 1 -difference operator and q π 1 -integral. Recently, in 2021, Vivas-Cortez et al. introduced the notions of D p , q π 2 derivative and ( p , q ) π 2 -integral in [15].

Many integral inequalities have been studied using quantum integrals for various types of functions. For example, in [16,17,18, 19,11,20, 21,22,23], the authors used D q π 1 , D q π 2 -derivatives and q π 1 , q π 2 -integrals to prove HH integral inequalities and their left-right estimates for convex and coordinated convex functions. In [24], Noor et al. presented a generalized version of quantum HH integral inequalities. For generalized quasi-convex functions, Nwaeze and Tameru proved certain parameterized quantum integral inequalities in [25]. Khan et al. proved quantum HH inequality using the green function in [26]. Budak et al. [27], Ali et al. [28,29] and Vivas-Cortez et al. [30] developed new quantum Simpson’s and quantum Newton’s type inequalities for convex and coordinated convex functions. For quantum Ostrowski’s inequalities for convex and co-ordinated convex functions, readers refer to [31,32,33]. Kunt et al. [34] generalized the results of [18] and proved Hermite-Hadamard type inequalities and their left estimates using D p , q π 1 -difference operator and ( p , q ) π 1 -integral. Recently, Latif et al. [35] found the right estimates of Hermite-Hadamard type inequalities proved by Kunt et al. [34].

Inspired by these ongoing studies, in the context of ( p , q ) -calculus, we prove several new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in the second sense.

The following is the structure of this article: Section 2 provides a brief overview of the fundamentals of q -calculus as well as other related studies in this field. In Section 3, we go over some basic ( p , q ) -calculus notions and inequalities. In Section 4, we show the relationship between the results presented here and related results in the literature by proving post-quantum HH inequalities for s-convex functions in the second sense. Post-quantum Ostrowski type inequalities for s-convex functions in the second are presented in Section 5. In Section 6, we present some applications to special means of real numbers for newly established inequalities. Section 7 concludes with some recommendations for future research.

2 Preliminaries of q -calculus and some inequalities

In this section, we revisit several previously regarded ideas. In addition, throughout the paper, s ∈ ( 0 , 1 ] , and we use the following notations (see, [6]):

[ n ] q = 1 − q n 1 − q = 1 + q + q 2 + ⋯ + q n − 1 , q ∈ ( 0 , 1 ) .

In [8], Jackson gave the q -Jackson integral from 0 to π 2 for 0 < q < 1 as follows:

(3) ∫ 0 π 2 ϝ ( x ) d q x = ( 1 − q ) π 2 ∑ n = 0 ∞ q n ϝ ( π 2 q n )

provided the sum converge absolutely.

Definition 1

[10] For a function ϝ : [ π 1 , π 2 ] → R , the q π 1 -derivative of ϝ at x ∈ [ π 1 , π 2 ] is characterized by the expression:

(4) D q π 1 ϝ ( x ) = ϝ ( x ) − ϝ ( q x + ( 1 − q ) π 1 ) ( 1 − q ) ( x − π 1 ) , x ≠ π 1 .

If x = π 1 , we define D q π 1 ϝ ( π 1 ) = lim x → π 1 D q π 1 ϝ ( x ) if it exists, and it is finite.

Definition 2

[11] For a function ϝ : [ π 1 , π 2 ] → R , the q π 2 -derivative of ϝ at x ∈ [ π 1 , π 2 ] is characterized by the expression:

(5) D q π 2 ϝ ( x ) = ϝ ( q x + ( 1 − q ) π 2 ) − ϝ ( x ) ( 1 − q ) ( π 2 − x ) , x ≠ π 2 .

If x = π 2 , we define D q π 2 ϝ ( π 2 ) = lim x → π 2 D q π 2 ϝ ( x ) if it exists and it is finite.

Definition 3

[10] Let ϝ : [ π 1 , π 2 ] → R be a function. Then, the q π 1 -definite integral on [ π 1 , π 2 ] is defined as follows:

(6) ∫ π 1 π 2 ϝ ( x ) d q π 1 x = ( 1 − q ) ( π 2 − π 1 ) ∑ n = 0 ∞ q n ϝ ( q n π 2 + ( 1 − q n ) π 1 ) = ( π 2 − π 1 ) ∫ 0 1 ϝ ( ( 1 − t ) π 1 + t π 2 ) d q t .

Definition 4

[11] Let ϝ : [ π 1 , π 2 ] → R be a function. Then, the q π 2 -definite integral on [ π 1 , π 2 ] is defined as follows:

(7) ∫ π 1 π 2 ϝ ( x ) d q π 2 x = ( 1 − q ) ( π 2 − π 1 ) ∑ n = 0 ∞ q n ϝ ( q n π 1 + ( 1 − q n ) π 2 ) = ( π 2 − π 1 ) ∫ 0 1 ϝ ( t π 1 + ( 1 − t ) π 2 ) d q t .

In [11], Bermudo et al. established the following quantum HH type inequality.

Theorem 1

For the convex mapping ϝ : [ π 1 , π 2 ] → R , the following inequality holds

(8) ϝ π 1 + π 2 2 ≤ 1 2 ( π 2 − π 1 ) ∫ π 1 π 2 ϝ ( x ) d q π 1 x + ∫ π 1 π 2 ϝ ( x ) d q π 2 x ≤ ϝ ( π 1 ) + ϝ ( π 2 ) 2 .

In [33], Budak et al. proved the following Ostrowski inequality by using the concepts of quantum derivatives and integrals.

Theorem 2

Let ϝ : [ π 1 , π 2 ] ⊂ R → R be a function and D q π 2 ϝ and d q π 1 ϝ be two continuous and integrable functions on [ π 1 , π 2 ] . If ∣ D q π 2 ϝ ( t ) ∣ , ∣ D q π 1 ϝ ( t ) ∣ ≤ M for all t ∈ [ π 1 , π 2 ] , then we have the following quantum quantum Ostrowski inequality:

(9) ϝ ( x ) − 1 π 2 − π 1 ∫ π 1 x ϝ ( t ) d q π 1 t + ∫ x π 2 ϝ ( t ) d q π 2 t ≤ q M ( π 2 − π 1 ) ( x − π 1 ) 2 + ( π 2 − x ) 2 [ 2 ] q .

Recently, Asawasamrit et al. [36] gave the following generalizations of inequalities (8) and (9) using the s-convexity.

Theorem 3

Assume that the mapping ϝ : [ 0 , ∞ ) → R is s-convex in the second sense and π 1 , π 2 ∈ [ 0 , ∞ ) with π 1 < π 2 , then the following inequality holds for s ∈ ( 0 , 1 ] :

(10) 2 s − 1 ϝ π 1 + π 2 2 ≤ 1 2 ( π 2 − π 1 ) ∫ π 1 π 2 ϝ ( x ) d q π 1 x + ∫ π 1 π 2 ϝ ( x ) d q π 2 x ≤ ϝ ( π 1 ) + ϝ ( π 2 ) [ s + 1 ] q .

Theorem 4

Let ϝ : [ π 1 , π 2 ] ⊂ R → R be function and D q π 2 ϝ and D q π 1 ϝ be two continuous and integrable functions on [ π 1 , π 2 ] . If ∣ D q π 2 ϝ ( t ) ∣ , ∣ D q π 1 ϝ ( t ) ∣ ≤ M for all t ∈ [ π 1 , π 2 ] , then we have the following quantum Ostrowski inequality for s-convex functions in the second sense:

(11) ϝ ( x ) − 1 π 2 − π 1 ∫ π 1 x ϝ ( t ) d q π 1 t + ∫ x π 2 ϝ ( t ) d q π 2 t ≤ M q π 2 − π 1 1 [ s + 2 ] q + Θ 11 [ ( x − π 1 ) 2 + ( π 2 − x ) 2 ] ,

where

Θ 11 = ∫ 0 1 t ( 1 − t ) d q s t .

3 Post-quantum calculus and some inequalities

In this section, we review some fundamental notions and notations of ( p , q ) -calculus.

The [ n ] p , q is said to be ( p , q )-integers and expressed as follows:

[ n ] p , q = p n − q n p − q

with 0 < q < p ≤ 1 . The [ n ] p , q ! and n k ! are called ( p , q )-factorial and ( p , q )-binomial, respectively, and expressed as follows:

[ n ] p , q ! = ∏ k = 1 n [ k ] p , q , n ≥ 1 , [ 0 ] p , q ! = 1 , n k ! = [ n ] p , q ! [ n − k ] p , q ! [ k ] p , q ! .

Definition 5

[12] The ( p , q ) -derivative of mapping ϝ : [ π 1 , π 2 ] → R is given as follows:

D p , q ϝ ( x ) = ϝ ( p x ) − ϝ ( q x ) ( p − q ) x , x ≠ 0

with 0 < q < p ≤ 1 .

Definition 6

[14] The ( p , q ) π 1 -derivative of mapping ϝ : [ π 1 , π 2 ] → R is given as follows:

(12) D p , q π 1 ϝ ( x ) = ϝ ( p x + ( 1 − p ) π 1 ) − ϝ ( q x + ( 1 − q ) π 1 ) ( p − q ) ( x − π 1 ) , x ≠ π 1

with 0 < q < p ≤ 1 . For x = π 1 , we state D p , q π 1 ϝ ( π 1 ) = lim x → π 1 D p , q π 1 ϝ ( x ) if it exists and it is finite.

Definition 7

[15] The ( p , q ) π 2 -derivative of mapping ϝ : [ π 1 , π 2 ] → R is given as follows:

(13) D p , q π 2 ϝ ( x ) = ϝ ( q x + ( 1 − q ) π 2 ) − ϝ ( p x + ( 1 − p ) π 2 ) ( p − q ) ( π 2 − x ) , x ≠ π 2 .

with 0 < q < p ≤ 1 . For x = π 2 , we state D p , q π 2 ϝ ( π 2 ) = lim x → π 2 D p , q π 2 ϝ ( x ) if it exists and it is finite.

Remark 1

It is clear that if we use p = 1 in (12) and (13), then the equalities (12) and (13) reduce to (4) and (5), respectively.

Definition 8

[14] The definite ( p , q ) π 1 -integral of mapping ϝ : [ π 1 , π 2 ] → R on [ π 1 , π 2 ] is stated as follows:

(14) ∫ π 1 x ϝ ( τ ) d p , q π 1 τ = ( p − q ) ( x − π 1 ) ∑ n = 0 ∞ q n p n + 1 ϝ q n p n + 1 x + 1 − q n p n + 1 π 1

with 0 < q < p ≤ 1 .

Definition 9

[15] The definite ( p , q ) π 2 -integral of mapping ϝ : [ π 1 , π 2 ] → R on [ π 1 , π 2 ] is stated as follows:

(15) ∫ x π 2 ϝ ( τ ) d p , q π 2 τ = ( p − q ) ( π 2 − x ) ∑ n = 0 ∞ q n p n + 1 ϝ q n p n + 1 x + 1 − q n p n + 1 π 2

with 0 < q < p ≤ 1 .

Remark 2

It is evident that if we pick p = 1 in (14) and (15), then the equalities (14) and (15) change into (6) and (7), respectively.

Remark 3

If we take π 1 = 0 and x = π 2 = 1 in (14), then we have

∫ 0 1 ϝ ( τ ) d p , q 0 τ = ( p − q ) ∑ n = 0 ∞ q n p n + 1 ϝ q n p n + 1 .

Similarly, by taking x = π 1 = 0 and π 2 = 1 in (15), then we obtain that

∫ 0 1 ϝ ( τ ) d p , q 1 τ = ( p − q ) ∑ n = 0 ∞ q n p n + 1 ϝ 1 − q n p n + 1 .

Lemma 1

[15] We have the following equalities:

∫ π 1 π 2 ( π 2 − x ) α d p , q π 2 x = ( π 2 − π 1 ) α + 1 [ α + 1 ] p , q

∫ π 1 π 2 ( x − π 1 ) α d p , q π 1 x = ( π 2 − π 1 ) α + 1 [ α + 1 ] p , q ,

where α ∈ R − { − 1 } .

Recently, Vivas-Cortez et al. [15] proved the following HH type inequalities for convex functions using the ( p , q ) π 2 -integral:

Theorem 5

[15] For a convex mapping ϝ : [ π 1 , π 2 ] → R , which is differentiable on [ π 1 , π 2 ] , the following inequalities hold for ( p , q ) π 2 -integral:

(16) ϝ p π 1 + q π 2 [ 2 ] p , q ≤ 1 p ( π 2 − π 1 ) ∫ p π 1 + ( 1 − p ) π 2 π 2 ϝ ( x ) d p , q π 2 x ≤ p ϝ ( π 1 ) + q ϝ ( π 2 ) [ 2 ] p , q ,

where 0 < q < p ≤ 1 .

Theorem 6

[15] For a convex function ϝ : [ π 1 , π 2 ] → R , the following inequality holds:

(17) ϝ π 1 + π 2 2 ≤ 1 2 p ( π 2 − π 1 ) ∫ π 1 p π 2 + ( 1 − p ) π 1 ϝ ( x ) d p , q π 1 x + ∫ p π 1 + ( 1 − p ) π 2 π 2 ϝ ( x ) d p , q π 2 x ≤ ϝ ( π 1 ) + ϝ ( π 2 ) 2 ,

where 0 < q < p ≤ 1 .

4 Hermite-Hadamard inequalities

In this section, we prove HH inequalities for s-convex functions in the second kind using the post-quantum integrals.

Theorem 7

Assume that the mapping ϝ : [ 0 , ∞ ) → R is s-convex in the second sense and π 1 , π 2 ∈ [ 0 , ∞ ) with π 1 < π 2 , then the following inequality holds for s ∈ ( 0 , 1 ] :

(18) 2 s − 1 ϝ π 1 + π 2 2 ≤ 1 2 p ( π 2 − π 1 ) ∫ π 1 π 2 ϝ ( x ) d p , q π 1 x + ∫ π 1 π 2 ϝ ( x ) d p , q π 2 x ≤ ϝ ( π 1 ) + ϝ ( π 2 ) [ s + 1 ] p , q .

Proof

We have s-convexity, as we know from s-convexity

(19) 2 s ϝ x + y 2 ≤ ϝ ( x ) + ϝ ( y ) .

We obtain the following by putting x = t π 2 + ( 1 − t ) π 1 and y = t π 1 + ( 1 − t ) π 2 in (19)

2 s ϝ π 1 + π 2 2 ≤ ϝ ( t π 2 + ( 1 − t ) π 1 ) + ϝ ( t π 1 + ( 1 − t ) π 2 ) .

From Definitions 8 and 9, we have

2 s − 1 ϝ π 1 + π 2 2 ≤ 1 2 p ( π 2 − π 1 ) ∫ π 1 p π 2 + ( 1 − p ) π 1 ϝ ( x ) d p , q π 1 x + ∫ p π 1 + ( 1 − p ) π 2 π 2 ϝ ( x ) d p , q π 2 x ,

and the first inequality in (18) is proved.

To prove the second inequality, we use the s-convexity, and we have

(20) ϝ ( t π 2 + ( 1 − t ) π 1 ) ≤ t s ϝ ( π 2 ) + ( 1 − t ) s ϝ ( π 1 )

and

(21) ϝ ( t π 1 + ( 1 − t ) π 2 ) ≤ t s ϝ ( π 1 ) + ( 1 − t ) s ϝ ( π 2 ) .

By adding (20) and (21), from Definitions 8 and 9, we have

1 2 p ( π 2 − π 1 ) ∫ π 1 p π 2 + ( 1 − p ) π 1 ϝ ( x ) d p , q π 1 x + ∫ p π 1 + ( 1 − p ) π 2 π 2 ϝ ( x ) d p , q π 2 x ≤ ϝ ( π 1 ) + ϝ ( π 2 ) [ s + 1 ] p , q ,

and the proof is completed.□

Example 1

For s-convex function f ( x ) = x s , from inequality (18) with a = s = 1 , b = 2 , p = 1 2 , and q = 1 4 , we have

2 s − 1 ϝ π 1 + π 2 2 = 3 2 ,

1 2 p ( π 2 − π 1 ) ∫ π 1 π 2 ϝ ( x ) d p , q π 1 x + ∫ π 1 π 2 ϝ ( x ) d p , q π 2 x = 1 2 − 1 4 ∑ n = 0 ∞ 1 4 n 1 2 n + 1 1 4 n 1 2 n + 1 2 + 1 − 1 4 n 1 2 n + 1 + 1 2 − 1 4 ∑ n = 0 ∞ 1 4 n 1 2 n + 1 1 4 n 1 2 n + 1 + 1 − 1 4 n 1 2 n + 1 2 = 3

and

ϝ ( π 1 ) + ϝ ( π 2 ) [ 1 + s ] p , q = 1 + 2 1 4 + 1 2 = 4 .

Thus,

3 2 < 3 < 4 ,

which shows that the inequality proved in Theorem 7 is true.

Remark 4

If we set s = 1 in Theorem 7, then we recapture the inequality (17).

Remark 5

In Theorem 7, if we take the limit as p = 1 , then inequality (18) becomes the inequality (10).

Remark 6

In Theorem 7, if we take p = 1 and later take the limit as q → 1 − , then inequality (18) becomes the inequality (2).

5 Ostrowski’s inequalities

In this section, we prove post-quantum Ostrowski type inequalities for s-convex functions in the second sense.

We begin with the following identity.

Lemma 2

Let ϝ : [ π 1 , π 2 ] ⊂ R → R be a function. If D p , q π 2 ϝ and π 1 D p , q ϝ are two continuous and integrable functions on [ π 1 , π 2 ] , then for all x ∈ [ π 1 , π 2 ] , we have

(22) ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t = q ( x − π 1 ) 2 π 2 − π 1 ∫ 0 1 t D p , q π 1 ϝ ( t x + ( 1 − t ) π 1 ) d p , q 0 t − q ( π 2 − x ) 2 π 2 − π 1 ∫ 0 1 t D p , q π 2 ϝ ( t x + ( 1 − t ) π 2 ) d p , q 1 t .

Proof

From Definitions 6 and 7, we have

D p , q π 1 ϝ ( t x + ( 1 − t ) π 1 ) = ϝ ( p t x + ( 1 − p t ) π 1 ) − ϝ ( q t x + ( 1 − q t ) π 1 ) t ( x − π 1 ) ( p − q )

and

D p , q π 2 ϝ ( t x + ( 1 − t ) π 2 ) = ϝ ( q t x + ( 1 − q t ) π 2 ) − ϝ ( p t x + ( 1 − p t ) π 2 ) t ( π 2 − x ) ( p − q ) .

By using Definition 9, we have

(23) I 1 = ∫ 0 1 t D p , q π 2 ϝ ( t x + ( 1 − t ) π 2 ) d p , q 1 t = 1 ( π 2 − x ) ( p − q ) ∫ 0 1 [ ϝ ( q t x + ( 1 − q t ) π 2 ) − ϝ ( p t x + ( 1 − p t ) π 2 ) ] d p , q 1 t = 1 π 2 − x ∑ n = 0 ∞ q n p n + 1 ϝ q n + 1 p n + 1 x + 1 − q n + 1 p n + 1 π 2 − ∑ n = 0 ∞ q n p n + 1 ϝ q n p n x + 1 − q n p n π 2 = 1 π 2 − x 1 q ∑ n = 0 ∞ q n + 1 p n + 1 ϝ q n + 1 p n + 1 x + 1 − q n + 1 p n + 1 π 2 − 1 p ∑ n = 0 ∞ q n p n ϝ q n p n x + 1 − q n p n π 2 = 1 π 2 − x 1 q − 1 p ∑ n = 0 ∞ q n p n ϝ q n p n x + 1 − q n p n π 2 − 1 q ϝ ( x ) = 1 π 2 − x p − q p q ∑ n = 0 ∞ q n p n ϝ q n p n x + 1 − q n p n π 2 − 1 q ϝ ( x ) = 1 π 2 − x 1 p q ( π 2 − x ) ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( x ) d p , q π 2 x − 1 q ϝ ( x ) .

Similarly, from Definition 8, we have

(24) I 2 = ∫ 0 1 t D p , q π 1 ϝ ( t x + ( 1 − t ) π 1 ) d p , q 0 t = 1 x − π 1 1 q ϝ ( x ) − 1 p q ( x − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( x ) d p , q π 1 x .

Thus, we obtain the resultant equality (22) by subtracting (23) from (24).□

Remark 7

In Lemma 2, if we set p = 1 , then we obtain the equality:

ϝ ( x ) − 1 ( π 2 − π 1 ) ∫ π 1 x ϝ ( t ) d q π 1 t + ∫ x π 2 ϝ ( t ) d q π 2 t = q ( x − π 1 ) 2 π 2 − π 1 ∫ 0 1 t D q π 1 ϝ ( t x + ( 1 − t ) π 1 ) d q 0 t − q ( π 2 − x ) 2 π 2 − π 1 ∫ 0 1 t D q π 2 ϝ ( t x + ( 1 − t ) π 2 ) d q 1 t ,

which is proved by Budak et al. in [33].

Remark 8

In Lemma 2, if we set p = 1 and later taking the limit as q → 1 − , then we obtain [37, Lemma 1].

Theorem 8

Assume that the mapping ϝ : I ⊂ [ 0 , ∞ ) → R is differentiable and π 1 , π 2 ∈ I with π 1 < π 2 . If ∣ D p , q π 1 ϝ ∣ and ∣ D p , q π 2 ϝ ∣ are s-convex mappings in the second sense, then the following inequality holds:

(25) ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t

≤ q ( x − π 1 ) 2 π 2 − π 1 1 [ s + 2 ] p , q ∣ D p , q π 1 ϝ ( x ) ∣ + Θ 1 ∣ D p , q π 1 ϝ ( π 1 ) ∣ + q ( π 2 − x ) 2 π 2 − π 1 1 [ s + 2 ] p , q ∣ D p , q π 2 ϝ ( x ) ∣ + Θ 2 ∣ D p , q π 2 ϝ ( π 2 ) ∣ ,

where

Θ 1 = ∫ 0 1 t ( 1 − t ) s d p , q 0 t

and

Θ 2 = ∫ 0 1 t ( 1 − t ) s d p , q 1 t .

Proof

From Lemma 2 and properties of the modulus, we have

(26) ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t ≤ q ( x − π 1 ) 2 π 2 − π 1 ∫ 0 1 t ∣ D p , q π 1 ϝ ( t x + ( 1 − t ) π 1 ) ∣ d p , q 0 t + q ( π 2 − x ) 2 π 2 − π 1 ∫ 0 1 t ∣ D p , q π 2 ϝ ( t x + ( 1 − t ) π 2 ) ∣ d p , q 1 t .

Since the mapping ∣ D p , q π 1 ϝ ∣ and ∣ D p , q π 2 ϝ ∣ are s-convexities in the second sense, therefore

(27) ∫ 0 1 t ∣ D p , q π 1 ϝ ( t x + ( 1 − t ) π 1 ) ∣ d p , q 0 t ≤ ∫ 0 1 t s + 1 ∣ D p , q π 1 ϝ ( x ) ∣ d p , q 0 t + ∫ 0 1 t ( 1 − t ) s ∣ D p , q π 1 ϝ ( π 1 ) ∣ d p , q 0 t = 1 [ s + 2 ] p , q ∣ D p , q π 1 ϝ ( x ) ∣ + Θ 1 ∣ D p , q π 1 ϝ ( π 1 ) ∣

and

(28) ∫ 0 1 t ∣ D p , q π 2 ϝ ( t x + ( 1 − t ) π 2 ) ∣ d p , q 1 t ≤ ∫ 0 1 t s + 1 ∣ D p , q π 2 ϝ ( x ) ∣ d p , q 1 t + ∫ 0 1 t ( 1 − t ) s ∣ D p , q π 2 ϝ ( π 2 ) ∣ d p , q 1 t = 1 [ s + 2 ] p , q ∣ D p , q π 2 ϝ ( x ) ∣ + Θ 2 ∣ D p , q π 2 ϝ ( π 2 ) ∣ .

We obtain the resultant inequality (25) by putting (27) and (28) in (26).□

Corollary 1

If we set s = 1 in Theorem 8, then we obtain the following new Ostrowski type inequality for convex functions:

ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t ≤ q ( x − π 1 ) 2 π 2 − π 1 1 [ 3 ] p , q ∣ D p , q π 1 ϝ ( x ) ∣ + [ 3 ] p , q − [ 2 ] p , q [ 3 ] p , q [ 2 ] p , q ∣ D p , q π 1 ϝ ( π 1 ) ∣ + q ( π 2 − x ) 2 π 2 − π 1 1 [ 3 ] p , q ∣ D p , q π 2 ϝ ( x ) ∣ + [ 3 ] p , q − [ 2 ] p , q [ 3 ] p , q [ 2 ] p , q ∣ D p , q π 2 ϝ ( π 2 ) ∣ .

Remark 9

In Theorem 8, if we set p = 1 , then Theorem 8 reduces to [36, Theorem 4.1].

Remark 10

In Corollary 1, if we set p = 1 , then we obtain the following inequality:

ϝ ( x ) − 1 π 2 − π 1 ∫ π 1 x ϝ ( t ) d q π 1 t + ∫ x π 2 ϝ ( t ) d q π 2 t ≤ q ( π 2 − π 1 ) [ 2 ] q [ 3 ] q [ ( x − π 1 ) 2 ( [ 2 ] q ∣ D q π 1 ϝ ( x ) ∣ + q 2 ∣ D q π 1 ϝ ( π 1 ) ∣ ) + ( π 2 − x ) 2 ( [ 2 ] q ∣ D q π 2 ϝ ( x ) ∣ + q 2 ∣ D q π 2 ϝ ( π 2 ) ∣ ) ] ,

which is given by Budak et al. in [33].

Corollary 2

If we assume ∣ D p , q π 1 ϝ ( x ) ∣ , ∣ D p , q π 1 ϝ ( x ) ∣ ≤ M in Theorem 8, then we have following post-quantum Ostrowski type inequality for s-convex functions in the second sense:

(29) ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t ≤ M q ( x − π 1 ) 2 π 2 − π 1 1 [ s + 2 ] p , q + Θ 1 + M q ( π 2 − x ) 2 π 2 − π 1 1 [ s + 2 ] p , q + Θ 2 .

Remark 11

In Corollary 2, if we set p = 1 , then Corollary 2 reduces to [36, Corollary 4.1].

Remark 12

If we set s = p = 1 in Corollary 2, then we recapture inequality (9).

Remark 13

In Corollary 2, if we set p = 1 and later take the limit as q → 1 − , then Corollary 2 reduces to [38, Theorem 2].

Theorem 9

Assume that the mapping ϝ : I ⊂ [ 0 , ∞ ) → R is differentiable and π 1 , π 2 ∈ I with π 1 < π 2 . If ∣ D p , q π 1 ϝ ∣ p 1 and ∣ D p , q π 2 ϝ ∣ p 1 , p 1 ≥ 1 are s-convex mappings in the second sense, then the following inequality holds:

(30) ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t ≤ q π 2 − π 1 1 [ 2 ] p , q 1 − 1 p 1 ( x − π 1 ) 2 1 [ s + 2 ] p , q ∣ D p , q π 1 ϝ ( x ) ∣ p 1 + Θ 1 ∣ D p , q π 1 ϝ ( π 1 ) ∣ p 1 1 p 1 + ( π 2 − x ) 2 1 [ s + 2 ] p , q ∣ D p , q π 2 ϝ ( x ) ∣ p 1 + Θ 2 ∣ D p , q π 2 ϝ ( π 2 ) ∣ p 1 1 p 1 .

Proof

From Lemma 2, by using properties of the modulus and power mean inequality, we have

(31) ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t ≤ q ( x − π 1 ) 2 π 2 − π 1 ∫ 0 1 t ∣ D p , q π 1 ϝ ( t x + ( 1 − t ) π 1 ) ∣ d p , q 0 t + q ( π 2 − x ) 2 π 2 − π 1 ∫ 0 1 t ∣ D p , q π 2 ϝ ( t x + ( 1 − t ) π 2 ) ∣ d p , q 1 t

≤ q ( x − π 1 ) 2 π 2 − π 1 ∫ 0 1 t d p , q 0 t 1 − 1 p 1 ∫ 0 1 t ∣ D p , q π 1 ϝ ( t x + ( 1 − t ) π 1 ) ∣ p 1 d p , q 0 t 1 p 1 + q ( π 2 − x ) 2 π 2 − π 1 ∫ 0 1 t d p , q 1 t 1 − 1 p 1 ∫ 0 1 t ∣ D p , q π 2 ϝ ( t x + ( 1 − t ) π 2 ) ∣ p 1 d p , q 1 t 1 p 1 .

Since the mapping ∣ D p , q π 1 ϝ ∣ p 1 and ∣ D p , q π 2 ϝ ∣ p 1 are s-convexities in the second sense, therefore

(32) ∫ 0 1 t d p , q 0 t 1 − 1 p 1 ∫ 0 1 t ∣ D p , q π 1 ϝ ( t x + ( 1 − t ) π 1 ) ∣ p 1 d p , q 0 t 1 p 1 ≤ 1 [ 2 ] p , q 1 − 1 p 1 1 [ s + 2 ] p , q ∣ D p , q π 1 ϝ ( x ) ∣ p 1 + Θ 1 ∣ D p , q π 1 ϝ ( π 1 ) ∣ p 1 1 p 1

and

(33) ∫ 0 1 t d p , q 1 t 1 − 1 p 1 ∫ 0 1 t ∣ D p , q π 2 ϝ ( t x + ( 1 − t ) π 2 ) ∣ p 1 d p , q 1 t 1 p 1 ≤ 1 [ 2 ] p , q 1 − 1 p 1 1 [ s + 2 ] p , q ∣ D p , q π 2 ϝ ( x ) ∣ p 1 + Θ 2 ∣ D p , q π 2 ϝ ( π 2 ) ∣ p 1 1 p 1 .

We obtain the resultant inequality (30) by putting (32) and (33) in (31).□

Corollary 3

If we set s = 1 in Theorem 9, then we obtain the following new Ostrowski type inequality for convex functions:

ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t ≤ q π 2 − π 1 1 [ 2 ] p , q 1 − 1 p 1 ( x − π 1 ) 2 1 [ 3 ] p , q ∣ D p , q π 1 ϝ ( x ) ∣ p 1 + [ 3 ] p , q − [ 2 ] p , q [ 3 ] p , q [ 2 ] p , q ∣ D p , q π 1 ϝ ( π 1 ) ∣ p 1 1 p 1 + ( π 2 − x ) 2 1 [ 3 ] p , q ∣ D p , q π 2 ϝ ( x ) ∣ p 1 + [ 3 ] p , q − [ 2 ] p , q [ 3 ] p , q [ 2 ] p , q ∣ D p , q π 2 ϝ ( π 2 ) ∣ p 1 1 p 1 .

Remark 14

In Theorem 9, if we set p = 1 , then Theorem 9 reduces to [36, Theorem 4.2].

Remark 15

In Corollary 3, if we set p = 1 , then we obtain the following inequality:

ϝ ( x ) − 1 π 2 − π 1 ∫ π 1 x ϝ ( t ) d q π 1 t + ∫ x π 2 ϝ ( t ) d q π 2 t ≤ q ( π 2 − π 1 ) [ 2 ] q ( x − π 1 ) 2 [ 2 ] q D q π 1 ϝ ( x ) p + q 2 D q π 1 ϝ ( π 1 ) p [ 3 ] q 1 p + ( π 2 − x ) 2 [ 2 ] q D q π 2 ϝ ( x ) p + q 2 D q π 2 ϝ ( π 2 ) p [ 3 ] q 1 p ,

which is proved by Budak et al. in [33].

Corollary 4

If we assume D p , q π 1 ϝ ( x ) , D p , q π 1 ϝ ( x ) ≤ M in Theorem 9, then we have following post-quantum Ostrowski type inequality for s-convex functions in the second sense:

ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t ≤ M q π 2 − π 1 1 [ 2 ] p , q 1 − 1 p 1 ( x − π 1 ) 2 1 [ s + 2 ] p , q + Θ 1 1 p 1 + ( π 2 − x ) 2 1 [ s + 2 ] p , q + Θ 2 1 p 1 .

Remark 16

In Corollary 4, if we set p = 1 , then Corollary 4 reduces to [36, Corollary 4.2].

Remark 17

In Corollary 4, if we set p = 1 and later take the limit as q → 1 − , then Corollary 4 reduces to [38, Theorem 4].

Theorem 10

Assume that the mapping ϝ : I ⊂ [ 0 , ∞ ) → R is differentiable and π 1 , π 2 ∈ I with π 1 < π 2 . If ∣ D p , q π 1 ϝ ∣ p 1 and ∣ D p , q π 2 ϝ ∣ p 1 , p 1 > 1 are s-convex mappings in the second sense, then the following inequality holds:

(34) ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t ≤ q π 2 − π 1 1 [ r 1 + 1 ] p , q 1 r 1 ( x − π 1 ) 2 1 [ s + 1 ] p , q ( ∣ D p , q π 1 ϝ ( x ) ∣ p 1 + ∣ D p , q π 1 ϝ ( π 1 ) ∣ p 1 ) 1 p 1 + ( π 2 − x ) 2 1 [ s + 1 ] p , q ( ∣ D p , q π 2 ϝ ( x ) ∣ p 1 + ∣ D p , q π 2 ϝ ( π 2 ) ∣ p 1 ) 1 p 1 ,

where r 1 − 1 + p 1 − 1 = 1 .

Proof

From Lemma 2, by using properties of the modulus and Hölder’s inequality, we have

(35) ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t ≤ q ( x − π 1 ) 2 π 2 − π 1 ∫ 0 1 t ∣ D p , q π 1 ϝ ( t x + ( 1 − t ) π 1 ) ∣ d p , q 0 t + q ( π 2 − x ) 2 π 2 − π 1 ∫ 0 1 t ∣ D p , q π 2 ϝ ( t x + ( 1 − t ) π 2 ) ∣ d p , q 1 t ≤ q ( x − π 1 ) 2 π 2 − π 1 ∫ 0 1 t r 1 d p , q 0 t 1 r 1 ∫ 0 1 ∣ D p , q π 1 ϝ ( t x + ( 1 − t ) π 1 ) ∣ p 1 d p , q 0 t 1 p 1 + q ( π 2 − x ) 2 π 2 − π 1 ∫ 0 1 t r 1 d p , q 1 t 1 r 1 ∫ 0 1 ∣ D p , q π 2 ϝ ( t x + ( 1 − t ) π 2 ) ∣ p 1 d p , q 1 t 1 p 1 .

Since the mapping ∣ D p , q π 1 ϝ ∣ p 1 and ∣ D p , q π 2 ϝ ∣ p 1 are s-convexities in the second sense, therefore

(36) ∫ 0 1 t r 1 d p , q 0 t 1 r 1 ∫ 0 1 ∣ D p , q π 1 ϝ ( t x + ( 1 − t ) π 1 ) ∣ p 1 d p , q 0 t 1 p 1 ≤ 1 [ r 1 + 1 ] p , q 1 r 1 1 [ s + 1 ] p , q ( ∣ D p , q π 1 ϝ ( x ) ∣ p 1 + ∣ D p , q π 1 ϝ ( π 1 ) ∣ p 1 )

and

(37) ∫ 0 1 t r 1 d p , q 1 t 1 r 1 ∫ 0 1 ∣ D p , q π 2 ϝ ( t x + ( 1 − t ) π 2 ) ∣ p 1 d p , q 1 t 1 p 1 ≤ 1 [ r 1 + 1 ] p , q 1 r 1 1 [ s + 1 ] p , q ( ∣ D p , q π 2 ϝ ( x ) ∣ p 1 + ∣ D p , q π 2 ϝ ( π 2 ) ∣ p 1 ) .

We obtain the resultant inequality (34) by putting (36) and (37) in (35).□

Corollary 5

If we set s = 1 in Theorem 10, then we obtain the following new Ostrowski type inequality for convex functions:

ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t ≤ q π 2 − π 1 1 [ r 1 + 1 ] p , q 1 r 1 ( x − π 1 ) 2 1 [ 3 ] p , q ( ∣ D p , q π 1 ϝ ( x ) ∣ p 1 + ∣ D p , q π 1 ϝ ( π 1 ) ∣ p 1 ) 1 p 1 + ( π 2 − x ) 2 1 [ 3 ] p , q ( ∣ D p , q π 2 ϝ ( x ) ∣ p 1 + ∣ D p , q π 2 ϝ ( π 2 ) ∣ p 1 ) 1 p 1 .

Remark 18

In Theorem 9, if we set p = 1 , then Theorem 9 reduces to [36, Theorem 4.3].

Remark 19

In Corollary 5, if we set p = 1 , then we obtain the following inequality:

ϝ ( x ) − 1 π 2 − π 1 ∫ π 1 x ϝ ( t ) d q π 1 t + ∫ x π 2 ϝ ( t ) d q π 2 t ≤ q π 2 − π 1 1 [ r 1 + 1 ] q 1 r 1 ( x − π 1 ) 2 ∣ D q π 1 ϝ ( x ) ∣ p + q ∣ D q π 1 ϝ ( π 1 ) ∣ p [ 2 ] q 1 p + ( π 2 − x ) 2 ∣ D q π 2 ϝ ( x ) ∣ p + q ∣ D q π 2 ϝ ( π 2 ) ∣ p [ 2 ] q 1 p ,

which is proved by Budak et al. in [33].

Corollary 6

If we assume ∣ D p , q π 1 ϝ ( x ) ∣ , ∣ D p , q π 1 ϝ ( x ) ∣ ≤ M in Theorem 10, then we have following post-quantum Ostrowski type inequality for s-convex functions in the second sense:

(38) ϝ ( x ) − 1 p ( π 2 − π 1 ) ∫ π 1 p x + ( 1 − p ) π 1 ϝ ( t ) d p , q π 1 t + ∫ p x + ( 1 − p ) π 2 π 2 ϝ ( t ) d p , q π 2 t ≤ M q π 2 − π 1 1 [ r 1 + 1 ] p , q 1 r 1 2 [ s + 1 ] p , q 1 p 1 [ ( x − π 1 ) 2 + ( π 2 − x ) 2 ] .

Remark 20

In Corollary 6, if we set p = 1 , then Corollary 6 reduces to [36, Corollary 4.3].

Remark 21

In Corollary 6, if we set p = 1 and later take the limit as q → 1 − , then Corollary 6 reduces to [38, Theorem 3].

6 Applications to special means

For arbitrary positive numbers π 1 , π 2 ( π 1 ≠ π 2 ) , we consider the means as follows:

  1. The arithmetic mean

    A = A ( π 1 , π 2 ) = π 1 + π 2 2 .

  2. The logarithmic mean

    ℒ σ σ = ℒ σ σ ( π 1 , π 2 ) = π 2 σ + 1 − π 1 σ + 1 ( σ + 1 ) ( π 2 − π 1 ) .

Proposition 1

For 0 < π 1 < π 2 and 0 < q < p ≤ 1 , the following inequality is true:

1 s + 1 [ A s + 1 ( π 1 , π 2 ) − A ( k 1 , k 2 ) ] ≤ q ( π 2 − π 1 ) 2 1 [ s + 2 ] p , q ℒ s s q π 2 − π 1 2 + π 1 , p π 2 − π 1 2 + π 1 + ℒ s s π 2 − q π 2 − π 1 2 , π 2 − p π 2 − π 1 2 + Θ 1 A ( π 1 s , π 2 s ) + Θ 2 A ( π 1 s , π 2 s ) ,

where

k 1 = ( p − q ) ∑ n = 0 ∞ q n p n + 1 q n p n π 2 − π 1 2 + π 1 s + 1 , k 2 = ( p − q ) ∑ n = 0 ∞ q n p n + 1 π 2 − q n p n π 2 − π 1 2 s + 1 .

Proof

The inequality (25) in Theorem 8 with x = π 1 + π 2 2 for ϝ ( x ) = x s + 1 s + 1 , where x > 0 leads to this conclusion.□

Proposition 2

For 0 < π 1 < π 2 and 0 < q < p ≤ 1 , the following inequality is true:

1 s + 1 [ A s + 1 ( π 1 , π 2 ) − A ( k 1 , k 2 ) ] ≤ M q ( π 2 − π 1 ) 4 2 [ s + 2 ] p , q + Θ 1 + Θ 2 .

Proof

The inequality (29) in Corollary 2 with x = π 1 + π 2 2 for ϝ ( x ) = x s + 1 s + 1 , where x > 0 leads to this conclusion.□

Proposition 3

For 0 < π 1 < π 2 and 0 < q < p ≤ 1 , the following inequality is true:

1 s + 1 [ A s + 1 ( π 1 , π 2 ) − A ( k 1 , k 2 ) ] ≤ q ( π 2 − π 1 ) 2 1 [ 2 ] p , q 1 − 1 p 1 1 [ s + 2 ] p , q ℒ s s q π 2 − π 1 2 + π 1 , p π 2 − π 1 2 + π 1 p 1 + Θ 1 ∣ π 1 s ∣ p 1 1 p 1 + 1 [ s + 2 ] p , q ℒ s s π 2 − q π 2 − π 1 2 , π 2 − p π 2 − π 1 2 p 1 + Θ 2 ∣ π 2 s ∣ p 1 1 p 1 .

Proof

The inequality (30) in Theorem 9 with x = π 1 + π 2 2 for ϝ ( x ) = x s + 1 s + 1 , where x > 0 leads to this conclusion.□

Proposition 4

For 0 < π 1 < π 2 and 0 < q < p ≤ 1 , the following inequality is true:

1 s + 1 [ A s + 1 ( π 1 , π 2 ) − A ( k 1 , k 2 ) ] ≤ q ( π 2 − π 1 ) 2 1 [ r 1 + 1 ] p , q 1 r 1 1 [ s + 1 ] p , q ℒ s s q π 2 − π 1 2 + π 1 , p π 2 − π 1 2 + π 1 p 1 + ∣ π 1 s ∣ p 1 1 p 1 + 1 [ s + 1 ] p , q ℒ s s π 2 − q π 2 − π 1 2 , π 2 − p π 2 − π 1 2 p 1 + ∣ π 2 s ∣ p 1 1 p 1 .

Proof

The inequality (34) in Theorem 10 with x = π 1 + π 2 2 for ϝ ( x ) = x s + 1 s + 1 , where x > 0 leads to this conclusion.□

Proposition 5

For 0 < π 1 < π 2 and 0 < q < p ≤ 1 , the following inequality is true:

1 s + 1 [ A s + 1 ( π 1 , π 2 ) − A ( k 1 , k 2 ) ] ≤ M q ( π 2 − π 1 ) 2 1 [ r 1 + 1 ] p , q 1 r 1 2 [ s + 1 ] p , q 1 p 1 .

Proof

The inequality (38) in Corollary 6 with x = π 1 + π 2 2 for ϝ ( x ) = x s + 1 s + 1 , where x > 0 leads to this conclusion.□

7 Conclusion

In this work, we proved some new variants of post-quantum Hermite-Hadamard and Ostrowski type inequalities using the ( p , q ) -differentiable s-convex functions in the second sense. We also proved that the newly established results are strong generalizations of the related existing results. Finally, we presented various applications based on the newly established inequalities to demonstrate the utility of our findings. It is a new and interesting problem that upcoming researchers can obtain similar inequalities for different kinds of convexity in their future work.

Acknowledgements

The authors are thankful to the Nanjing Normal University for wonderful research environment provided to the researchers.

  1. Funding information: The work was supported by Philosophy and Social Sciences of Educational Commission of Hubei Province of China (20Y109), and Foundation of Hubei Normal University (2021YJSKCSZY06, 2021056). This work was also supported by King Mongkut’s University of Technology North Bangkok (Contract no. KMUTNB-63-KNOW-021).

  2. Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors declare that they do not have any conflict of interests.

  4. Data availability statement: Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

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Received: 2021-09-23
Revised: 2022-03-24
Accepted: 2022-04-08
Published Online: 2022-08-26

© 2022 Xue-Xiao You et al., published by De Gruyter

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  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
https://doi.org/10.1515/math-2022-0037
Keywords for this article
Hermite-Hadamard inequality; Ostrowski inequality; (p, q)-integral; post-quantum calculus; s-convex functions
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Articles in the same Issue

  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
  5. The family of random attractors for nonautonomous stochastic higher-order Kirchhoff equations with variable coefficients
  6. The intersection graph of graded submodules of a graded module
  7. Isoperimetric and Brunn-Minkowski inequalities for the (p, q)-mixed geominimal surface areas
  8. On second-order fuzzy discrete population model
  9. On certain functional equation in prime rings
  10. General complex Lp projection bodies and complex Lp mixed projection bodies
  11. Some results on the total proper k-connection number
  12. The stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects
  13. Well posedness of magnetohydrodynamic equations in 3D mixed-norm Lebesgue space
  14. Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems
  15. Generic uniqueness of saddle point for two-person zero-sum differential games
  16. Relational representations of algebraic lattices and their applications
  17. Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms
  18. The equivalent condition of G-asymptotic tracking property and G-Lipschitz tracking property
  19. Arithmetic convolution sums derived from eta quotients related to divisors of 6
  20. Dynamical behaviors of a k-order fuzzy difference equation
  21. The transfer ideal under the action of orthogonal group in modular case
  22. The multinomial convolution sum of a generalized divisor function
  23. Extensions of Gronwall-Bellman type integral inequalities with two independent variables
  24. Unicity of meromorphic functions concerning differences and small functions
  25. Solutions to problems about potentially Ks,t-bigraphic pair
  26. Monotonicity of solutions for fractional p-equations with a gradient term
  27. Data smoothing with applications to edge detection
  28. An ℋ-tensor-based criteria for testing the positive definiteness of multivariate homogeneous forms
  29. Characterizations of *-antiderivable mappings on operator algebras
  30. Initial-boundary value problem of fifth-order Korteweg-de Vries equation posed on half line with nonlinear boundary values
  31. On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions
  32. On split twisted inner derivation triple systems with no restrictions on their 0-root spaces
  33. Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry
  34. Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting
  35. A posteriori error estimates of characteristic mixed finite elements for convection-diffusion control problems
  36. Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect
  37. An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem
  38. Connectivity with respect to α-discrete closure operators
  39. Khasminskii-type theorem for a class of stochastic functional differential equations
  40. On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications
  41. New properties for the Ramanujan R-function
  42. Shooting method in the application of boundary value problems for differential equations with sign-changing weight function
  43. Ground state solution for some new Kirchhoff-type equations with Hartree-type nonlinearities and critical or supercritical growth
  44. Existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays
  45. Ambrosetti-Prodi-type results for a class of difference equations with nonlinearities indefinite in sign
  46. Research of cooperation strategy of government-enterprise digital transformation based on differential game
  47. Malmquist-type theorems on some complex differential-difference equations
  48. Disjoint diskcyclicity of weighted shifts
  49. Construction of special soliton solutions to the stochastic Riccati equation
  50. Remarks on the generalized interpolative contractions and some fixed-point theorems with application
  51. Analysis of a deteriorating system with delayed repair and unreliable repair equipment
  52. On the critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields
  53. The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods
  54. Regularity of models associated with Markov jump processes
  55. Multiplicity solutions for a class of p-Laplacian fractional differential equations via variational methods
  56. Minimal period problem for second-order Hamiltonian systems with asymptotically linear nonlinearities
  57. Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound
  58. Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩
  59. On the general position number of two classes of graphs
  60. A posteriori regularization method for the two-dimensional inverse heat conduction problem
  61. Orbital stability and Zhukovskiǐ quasi-stability in impulsive dynamical systems
  62. Approximations related to the complete p-elliptic integrals
  63. A note on commutators of strongly singular Calderón-Zygmund operators
  64. Generalized Munn rings
  65. Double domination in maximal outerplanar graphs
  66. Existence and uniqueness of solutions to the norm minimum problem on digraphs
  67. On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation
  68. Robust estimation for varying coefficient partially functional linear regression models based on exponential squared loss function
  69. Hessian equations of Krylov type on compact Hermitian manifolds
  70. Class fields generated by coordinates of elliptic curves
  71. The lattice of (2, 1)-congruences on a left restriction semigroup
  72. A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition
  73. On stochastic accelerated gradient with convergence rate
  74. Displacement structure of the DMP inverse
  75. Dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter-dependent boundary conditions
  76. Existence of positive solutions of discrete third-order three-point BVP with sign-changing Green's function
  77. Some new fixed point theorems for nonexpansive-type mappings in geodesic spaces
  78. Generalized 4-connectivity of hierarchical star networks
  79. Spectra and reticulation of semihoops
  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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