# Ostrowski Type Inequalities fors-Convex Functions via q-Integrals

Authors: Khuram Ali Khan; Allah Ditta; Ammara Nosheen; Khalid Mahmood Awan; Rostin Matendo Mabela
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The new outcomes of the present paper areq-analogues (q stands for quantum calculus) of Hermite-Hadamard type inequality, Montgomery identity, and Ostrowski type inequalities for s-convex mappings. Some new bounds of Ostrowski type functionals are obtained by using Hölder, Minkowski, and power mean inequalities via quantum calculus. Special cases of new results include existing results from the literature.


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## Body

## 1. Introduction

Integral inequalities provide a notable role in both pure and applied mathematics in the light of their wide applications in numerous regular and human sociologies, while convexity hypothesis has stayed a significant apparatus in the foundation of the theory of integral inequalities. The classical inequalities are helpful in numerous down-to-earth issues. In recent years, many authors (see [1–12]) proved numerous inequalities associated with the functions of bounded variation, Lipschitzian, monotone, absolutely continuous, convex functions, s-convex, h-convex, and n-times differentiable mappings with error estimates. Integral inequalities have been studied extensively by several researchers either in classical analysis or in the quantum one. In many practical problems, it is important to bound one quantity by another quantity. The classical inequalities including Hermite-Hadamard and Ostrowski type inequalities are very useful for this purpose (see [13–24]). Ostrowski type inequalities are well known to study the upper bounds for approximation of the integral average by the value of the function. In [25], Dragomir and Fitzpatrick have constructed Hermite-Hadamard’s inequality which is specified to s-convex functions in the second sense as follows:Theorem 1.
SupposeΦ:ℝ+⟶ℝ is an s-convex function in second sense, s∈0,1, and suppose ℘,υ∈ℝ+,℘<υ. If Φ′∈L1℘,υ, then the integral inequality is valid:
(1)2s−1Φ℘+υ2≤1υ−℘∫℘υΦwdw≤Φ℘+Φυs+1,where ℝ+=w∈ℝ∣w≥0.The following Montgomery equality is established by Alomari (see [26]):Lemma 2.
Assume thatΦ:J⊂ℝ+⟶ℝ is differentiable function on ℘,υ in which ℘,υ∈J for ℘<υ. If Φ′∈L℘,υ, then we have the equality:
(2)Φw−1υ−℘∫℘υΦζdζ=w−℘2υ−℘∫01ζΦ′ζw+1−ζ℘dζ−υ−w2υ−℘∫01ζΦ′ζw+1−ζυdζ,for each w∈℘,υ.
By using Lemma2, Alomari et al. in [26] had proved the Ostrowski type inequality, which holds for s-convex mappings in second sense as follows:Theorem 3.
AssumeΦ:J⊂ℝ+⟶ℝ is a differentiable on ℘,υ and Φ′∈L℘,υ such that ℘,υ∈Jfor℘<υ. If Φ′ is s-convex mapping in the second sense on ℘,υ unique s∈0,1 and Φ′w≤M,w∈℘,υ, then the following result holds:
(3)Φw−1υ−℘∫℘υΦζdζ≤Mυ−℘w−℘2+υ−w2s+1,for each w∈℘,υ.Theorem 4.
Suppose thatΦ:J⊂ℝ+⟶ℝ is the differentiable on ℘,υ and Φ′∈L℘,υ, where ℘,υ∈J with ℘<υ. If absolute value of Φ′m is s-convex function in the second sense in ℘,υ for unique s∈0,1, m>1, n=m/m−1 and Φ′w≤M,w∈℘,υ, then following integral inequality holds:
(4)Φw−1υ−℘∫℘υΦζdζ≤M1+n1/n2s+11/mw−℘2+υ−w2υ−℘,for each w∈℘,υ.Theorem 5.
Suppose thatΦ:J⊂ℝ+⟶ℝ is differentiable on ℘,υ and Φ′∈L℘,υ, in which ℘,υ∈J for ℘<υ. If the absolute value of Φ′m is s-convex function in ℘,υ for static s∈0,1,m≥1 and Φ′w≤M,w∈℘,υ, then the following integral inequality holds:
(5)Φw−1υ−℘∫℘υΦζdζ≤M2s+11/mw−℘2+υ−w22υ−℘,for each w∈℘,υ.Theorem 6.
SupposeΦ:J⊂ℝ+⟶ℝ be the differentiable on ℘,υ and Φ′∈L℘,υ, in which ℘,υ∈J for ℘<υ. If absolute value of Φ′m is a s-convex mapping in second sense on ℘,υ for static s∈0,1,m>1 and n=m/m−1, we have
(6)Φw−1υ−℘∫℘υΦζdζ≤2s−1/m1+n1/nυ−℘w−℘2Φ′w+℘2+υ−w2Φ′υ+w2,
for eachw∈℘,υ.The renowned mathematician Euler started the investigation ofq-calculus in the eighteenth century by presenting Newton’s work of limitless series. This subject has gotten extraordinary consideration by numerous specialists, and consequently, it is considered an in-corporative subject among math and material science. In the mid-20th century, Jackson (1910) has begun a symmetric investigation of calculus and presented q-distinct integrals. The subject of quantum analytic has various applications in different spaces of arithmetic and physical science like number hypothesis, combinatorics, symmetrical polynomials, essential hypermathematical functions, quantum theory, and mechanics and in the hypothesis of relativity. Quantum calculus can be seen as a scaffold among arithmetic and material science. It has been shown that quantum calculus is a subfield of the more general mathematical field of time scales calculus. Time scales provide a unified framework for studying dynamic equations on both discrete and continuous domains. In [27, 28], q-Bernoulli and dynamic inequalities associated with Leibniz integral rule on time scales were studied. In studying quantum calculus, we are concerned with a specific time scale, called the q-time scale. The study of q-integral inequalities is also of great importance. Integral inequalities have been studied extensively by several researchers either in classical analysis or in the quantum one.The followingq-Hermite-Hadamard and q-Ostrowski type integral inequalities were proved by Tariboon and Ntouyas (see Theorems 3.2 and 3.5 [29]):Theorem 7.
LetΦ:J⟶ℝ be a q-differentiable function with DqΦ continuous on ℘,υ and 0<q<1. Then, we have
(7)Φ℘+υ2≤1υ−℘∫℘υΦζ℘dqζ≤Φ℘+qΦυq+1.Theorem 8.
SupposeΦ:J⟶ℝ, where ℘,υ⊆ℝ is an interval, be a q-differentiable in open interval ℘,υ belonging to interior I for ℘<υ. If DqΦw≤M for all w∈℘,υ and 0<q<1, then the integral inequality is valid:
(8)Φw−1υ−℘∫℘υΦζ℘dqζ≤M2q1+qw−3q−1℘+1+qυ/4qυ−℘2+−q2+6q−18q1+q,
for allw∈℘,υ. The least value of constant on RHS of inequality (8) is −q2+6q−1/8q1+q.
The followingq-Ostrowski type integral inequalities for convex functions were proved by Noor et al. (see [30]):Theorem 9.
LetΦ:J⊂ℝ+⟶ℝ be q-differentiable mapping for ℘,υ and DqΦ∈L℘,υ, in which ℘,υ∈J for ℘<υ. If DqΦ is convex mapping ℘,υ for some static q∈0,1 and DqΦw≤M,w∈℘,υ, then we have the following q-integral inequality:
(9)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤Mυ−℘w−℘2+υ−w2q+1,for each w∈℘,υ.Theorem 10.
Assume thatΦ:J⊂ℝ+⟶ℝ is q-differentiable mapping on ℘,υ and DqΦ∈L℘,υ, in which ℘,υ∈J for ℘<υ. If DqΦm is a convex function in second sense on ℘,υ unique q∈0,1,m>1,n=m/m−1, and DqΦw≤M,w∈℘,υ, then we have the q-integral inequality:
(10)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤Mn+11/nw−℘2+υ−w2υ−℘,for each w∈℘,υ.The aim of this work is to findq-analogues of Hermite-Hadmard and Ostrowski type integral inequalities for functions whose q-derivatives are s-convex in the second sense. An interesting feature of our results is that they provide new estimates and good approximation on such types of inequalities involving q-integrals.


## 2. Basic Essentials

### 2.1. Convex Function

LetΦ be the function; it is said to be convex function on intervalJ if
(11)ΦΩw+1−Ωρ≤ΩΦw+1−ΩΦρholds for all w,ρ∈J and Ω∈0,1.In [31], s-convex functions in the second sense have been introduced by Hudzik and Maligranda as follows:


### 2.2.s-Convex Function

A mappingΦ:ℝ+⟶ℝ is said to be s-convex if
(12)ΦΩw+1−Ωρ≤ΩsΦw+1−ΩsΦρ,for each w,ρ∈ℝ+, Ω∈0,1 and for unique s∈0,1.


### 2.3.q-Derivative [32]

For a continuous mappingΦ:℘,υ⟶ℝq-derivative at w∈℘,υ is
(13)D℘qΦw=Φw−Φqw+1−q℘1−qw−℘w≠℘.Also, forn≥1, one may find the following evaluations:
(14)w−℘qn=w−℘w−q℘w−q2℘⋯w−qn−1℘,℘−wqn=℘−qw℘−q2w⋯℘−qn−1w,Dqw−℘qn=nw−℘qn−1,Dq℘−wqn=−n℘−qwqn−1,℘−qwqn=−1n+1Dq℘−wqn+1,Dq℘−wqn=−n℘−qwqn−1,∫℘−wqndqw=−q℘−q−1wqn+1n+1℘≠−1.Here,(15)n=qn−1q−1,and also, we have
(16)1−℘qn=∏j=0n1−qj℘.


### 2.4.q-Antiderivative [32]

Suppose thatΦ:℘,υ⟶ℝ be the continuous mapping. Then, q-definite integral on ℘,υ is stated as
(17)∫℘wΦζ℘dqζ=1−qw−℘∑n=0∞qnΦqnw+1−qn℘,for w∈℘,υ.


### 2.5. The Formula ofq-Integration by Parts [29]

LetΦ,g:℘,υ⟶ℝ be the continuous functions ℘∈ℝ and w,c∈℘,υ, Then, the formula of q-integration by parts is stated as
(18)∫cwΦζ℘Dqgζdqζ=Φwgw−Φcgc−∫cwgqζ+1−q℘℘DqΦζdqζ.Theorem 11.q-Hölder Inequality ([4], Theorem 2).
LetΦ and g be q-integrable on ℘,υ and 0<q<1 and 1/n+1/m=1 with m>1; then, one may obtain the following:
(19)∫℘υΦζgζ℘dqζ≤∫℘υΦζn℘dqζ1/n∫℘υgζm℘dqζ1/m.
Using (19), the following is valid.


### 2.6.q-Minkowski’s Inequality

Let℘,υ∈ℝ and n>1 be a real number then for continuous functions Φ,g:℘,υ⟶ℝ,
(20)∫℘υΦζ+gζn℘dqζ1/n≤∫℘υΦζn℘dqζ1/n+∫℘υgζn℘dqζ1/n.Proof.
(21)∫υ℘Φ+gζndqζ=∫υ℘Φ+gζn−1Φ+gζdqζ≤∫υ℘Φ+gζn−1Φζdqζ+∫υ℘Φ+gζn−1gζdqζ≤∫υ℘Φζndqζ1/n∫υ℘Φ+gζmn−1dqζ1/n+∫υ℘gζndqζ1/n∫υ℘Φ+gζmn−1dqζ1/m=∫υ℘Φζndqζ1/n+∫υ℘gζndqζ1/n∫υ℘Φ+gζmn−1dqζ1/m,which gives the required result for positive real numbers m,n such that 1/m+1/n=1.
The classical power mean inequality for integrals has the following form forq-integral.


### 2.7.q-Power Mean Inequality

Let1/n+1/m=1 for real numbers n,m>1. Let ℘,υ∈ℝ and Φ,g:℘,υ⟶ℝ be continuous functions; then,
(22)∫℘υΦζgζ℘dqζ≤∫℘υΦζ℘dqζ1−1/m∫℘υΦζgζm℘dqζ1/m.Proposition 12. [33].
For eachk,r∈ℕorℤq∈ℝ×, we have
(23)k+rq=kq+qkrq.


## 2.1. Convex Function

LetΦ be the function; it is said to be convex function on intervalJ if
(11)ΦΩw+1−Ωρ≤ΩΦw+1−ΩΦρholds for all w,ρ∈J and Ω∈0,1.In [31], s-convex functions in the second sense have been introduced by Hudzik and Maligranda as follows:


## 2.2.s-Convex Function

A mappingΦ:ℝ+⟶ℝ is said to be s-convex if
(12)ΦΩw+1−Ωρ≤ΩsΦw+1−ΩsΦρ,for each w,ρ∈ℝ+, Ω∈0,1 and for unique s∈0,1.


## 2.3.q-Derivative [32]

For a continuous mappingΦ:℘,υ⟶ℝq-derivative at w∈℘,υ is
(13)D℘qΦw=Φw−Φqw+1−q℘1−qw−℘w≠℘.Also, forn≥1, one may find the following evaluations:
(14)w−℘qn=w−℘w−q℘w−q2℘⋯w−qn−1℘,℘−wqn=℘−qw℘−q2w⋯℘−qn−1w,Dqw−℘qn=nw−℘qn−1,Dq℘−wqn=−n℘−qwqn−1,℘−qwqn=−1n+1Dq℘−wqn+1,Dq℘−wqn=−n℘−qwqn−1,∫℘−wqndqw=−q℘−q−1wqn+1n+1℘≠−1.Here,(15)n=qn−1q−1,and also, we have
(16)1−℘qn=∏j=0n1−qj℘.


## 2.4.q-Antiderivative [32]

Suppose thatΦ:℘,υ⟶ℝ be the continuous mapping. Then, q-definite integral on ℘,υ is stated as
(17)∫℘wΦζ℘dqζ=1−qw−℘∑n=0∞qnΦqnw+1−qn℘,for w∈℘,υ.


## 2.5. The Formula ofq-Integration by Parts [29]

LetΦ,g:℘,υ⟶ℝ be the continuous functions ℘∈ℝ and w,c∈℘,υ, Then, the formula of q-integration by parts is stated as
(18)∫cwΦζ℘Dqgζdqζ=Φwgw−Φcgc−∫cwgqζ+1−q℘℘DqΦζdqζ.Theorem 11.q-Hölder Inequality ([4], Theorem 2).
LetΦ and g be q-integrable on ℘,υ and 0<q<1 and 1/n+1/m=1 with m>1; then, one may obtain the following:
(19)∫℘υΦζgζ℘dqζ≤∫℘υΦζn℘dqζ1/n∫℘υgζm℘dqζ1/m.
Using (19), the following is valid.


## 2.6.q-Minkowski’s Inequality

Let℘,υ∈ℝ and n>1 be a real number then for continuous functions Φ,g:℘,υ⟶ℝ,
(20)∫℘υΦζ+gζn℘dqζ1/n≤∫℘υΦζn℘dqζ1/n+∫℘υgζn℘dqζ1/n.Proof.
(21)∫υ℘Φ+gζndqζ=∫υ℘Φ+gζn−1Φ+gζdqζ≤∫υ℘Φ+gζn−1Φζdqζ+∫υ℘Φ+gζn−1gζdqζ≤∫υ℘Φζndqζ1/n∫υ℘Φ+gζmn−1dqζ1/n+∫υ℘gζndqζ1/n∫υ℘Φ+gζmn−1dqζ1/m=∫υ℘Φζndqζ1/n+∫υ℘gζndqζ1/n∫υ℘Φ+gζmn−1dqζ1/m,which gives the required result for positive real numbers m,n such that 1/m+1/n=1.
The classical power mean inequality for integrals has the following form forq-integral.


## 2.7.q-Power Mean Inequality

Let1/n+1/m=1 for real numbers n,m>1. Let ℘,υ∈ℝ and Φ,g:℘,υ⟶ℝ be continuous functions; then,
(22)∫℘υΦζgζ℘dqζ≤∫℘υΦζ℘dqζ1−1/m∫℘υΦζgζm℘dqζ1/m.Proposition 12. [33].
For eachk,r∈ℕorℤq∈ℝ×, we have
(23)k+rq=kq+qkrq.


## 3. Main Results

### 3.1.q-Hermite-Hadamard Inequality

Theorem 13.
SupposeΦ:ℝ+⟶ℝ is a s-convex mapping in the second sense, in which s,q∈0,1, and let ℘,υ∈ℝ+,℘<υ. If DqΦ∈La,b, then the integral inequality is valid:
(24)2s−1Φ℘+υ2≤1υ−℘∫℘υΦζ℘dqζ≤Φυq1−1−q−1s+1+Φ℘s+1.Proof.
By definition ofs-convex functions,
(25)Φζ℘+1−ζυ≤ζsΦ℘+1−ζsΦυ∫01Φζ℘+1−ζυ0dqζ≤Φ℘∫01ζs0dqζ+Φυ∫011−ζs0dqζ,∫01Φζ℘+1−ζυ0dqζ=1−qυ−℘υ−℘∑n=0∞qnΦqn℘+1−qnυ=1υ−℘∫℘υΦζ0dqζ≤Φ℘∫01ζs0dqζ+Φυ∫011−ζsd0qζ=Φ℘+q1−1−q−1s+1Φυs+1.
Hence,(26)1υ−℘∫℘υΦζ0dqζ≤Φ℘+q1−1−q−1s+1Φυs+1.
Letw=ζ℘+1−ζυ and ζ=ζυ+1−ζ℘ in Φw+ζ/2≤Φw+Φζ/2s to get
(27)Φζ℘+1−ζυ+ζυ+1−ζ℘2≤Φζ℘+1−ζυ+Φζυ+1−ζ℘2s,Φ℘+υ2≤12s∫01Φζ℘+1−ζυ0dqζ+∫01Φζυ+1−ζ℘0dqζ=12s1℘−υ∫υ℘Φζ0dqζ+1υ−℘∫℘υΦζ0dqζ,(28)2s−1Φ℘+υ2≤1υ−℘∫℘υΦζ0dqζ,
From (26) and (28), the desired result is
(29)2s−1Φ℘+υ2≤1υ−℘∫℘υΦζ℘dqζ≤Φ℘+q1−1−q−1s+1Φυs+1.


### 3.2.q-Ostrowski Type Inequalities

To prove someq-Ostrowski type inequalities, it needs to establish the following Montgomery identity for q-integrals:Lemma 14.
LetΦ:J⊂ℝ⟶ℝ be a q-differentiable on J∘ in which ℘,υ∈J for ℘<υ. If DqΦ∈L℘,υ, we have the following q-integral equality which is valid:
(30)1qΦw−1υ−℘∫℘υΦζ℘dqζ=w−℘2υ−℘∫01ζDqΦζw+1−ζ℘0dqζ−υ−w2υ−℘∫01ζDqΦζw+1−ζυ0dqζ,
for eachw∈℘,υ.
By using Lemma14, we have constructed the following Ostrowski type inequalities, which hold for s-convex functions in the second sense:Theorem 15.
LetΦ:J⊂ℝ+⟶ℝ is a q-differentiable mapping on J∘ and DqΦ∈L℘,υ, in which ℘,υ∈J for ℘<υ. If the absolute value of DqΦw is s-convex in second sense on ℘,υ for unique s∈0,1 and DqΦw is bounded by M, w∈℘,υ, we have been seeing that the following q-integral inequality is valid:
(31)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤Mw−℘2+υ−w2υ−℘−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+2+1s+2,
for eachw∈℘,υ.Proof.
SinceDqΦ is s-convex function in the second sense on ℘,υ, therefore, Lemma 14 gives the following:
(32)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤w−℘2υ−℘∫01ζDqΦζw+1−ζ℘0dqζ+υ−w2υ−℘∫01ζDqΦζw+1−ζυ0dqζ≤w−℘2υ−℘∫01ζqs+1DqΦw0dqζ+∫01ζ1−ζqsDqΦ℘0dqζ+υ−w2υ−℘∫01ζqs+1DqΦw0dqζ+∫01ζ1−ζqsDqΦυ0dqζ=Mw−℘2υ−℘∫01ζqs+10dqζ+∫01ζ1−ζqs0dqζ+υ−w2υ−℘∫01ζqs+10dqζ+∫01ζ1−ζqs0dqζ=Mw−℘2+υ−w2υ−℘∫01ζs+10dqζ+∫01ζ1−ζqs0dqζ,∫01ζqs+10dqζ=1s+2,(33)=−qs+1∫01ζDq1−q−1ζqs+10dqζ=−1qs+1ζ1−q−1ζqs+101−∫011−ζqs+1.10dqζ=−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+2=−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+2+1s+2=Mw−℘2+υ−w2υ−℘−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+2+1s+2.Theorem 16.
Suppose thatΦ:J⊂ℝ+⟶ℝ is a q-differentiable on J∘ and DqΦ∈L℘,υ, in which ℘,υ∈J for ℘<υ. If DqΦm is a s-convex function in second sense on ℘,υ for some static s∈0,1,m>1,n=m/m−1 and DqΦw is bounded by M, w∈℘,υ, then the q-integral inequality is valid:
(34)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤Mn+11/n1+q1−1−q−1s+1s+11/m×w−℘2+υ−w2υ−℘,for each w∈℘,υ.Proof.
From Lemma14 and keeping in view the well-known q-analogue of Hölder inequality, we have
(35)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤w−℘2υ−℘∫01ζDqΦζw+1−ζ℘0dqζ+υ−w2υ−℘∫01ζDqΦζw+1−ζυ0dqζ≤w−℘2υ−℘∫01ζqn0dqζ1/n∫01DqΦζw+1−ζ℘m0dqζ1/m+υ−w2υ−℘∫01∫01ζqn0dqζ1/n∫01DqΦζw+1−ζυm0dqζ1/m,∫01DqΦζw+1−ζ℘m0dqζ≤∫01ζqsDqΦwm0dqζ+∫011−ζqsDqΦ℘m0dqζ≤Mmζqs+1s+101−q1−q−1ζqs+1s+101=Mm1s+1−q1−q−1qs+1s+1+qs+1=Mm1+q1−1−q−1qs+1s+1,∫01DqΦζw+1−ζυm0dqζ≤∫01ζqsDqΦwm0dqζ+∫011−ζqsDqΦυm0dqζ≤Mmζqs+1s+101−q1−q−1ζqs+1s+101=Mm1s+1−q1−q−1qs+1s+1+qs+1=Mm1+q1−1−q−1qs+1s+1≤M11+n1/n1+q1−1−q−1qs+1s+11/mw−℘2+υ−w2υ−℘.
It completes the proof.Theorem 17.
LetΦ:J⊂ℝ+⟶ℝ is a q-differentiable mapping on J∘ such as DqΦ∈L℘,υ, in which ℘,υ∈J for ℘<υ. If the absolute value of DqΦwm is a s-convex mapping in the second sense on ℘,υ for unique s∈0,1, m≥1, and DqΦw≤M, w∈℘,υ, we have seen that the q-integral inequality is valid:
(36)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤Mw−℘2+υ−w2υ−℘121−1/m−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+21/m,for each w∈℘,υ.Proof.
Lemma14 and keeping in view the well-known q-analogue of power-mean inequality, we have
(37)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤w−℘2υ−℘∫01ζDqΦζw+1−ζ℘0dqζ+υ−w2υ−℘∫01ζDqΦζw+1−ζυ0dqζ≤w−℘2υ−℘∫01ζ0dqζ1−1/m∫01ζDqΦζw+1−ζ℘m0dqζ1/m+υ−w2υ−℘∫01ζ0dqζ1−1/m∫01ζDqΦζw+1−ζυm0dqζ1/m,∫01ζDqΦζw+1−ζ℘m0dqζ≤∫01ζqs+1DqΦwm0dqζ+∫01ζ1−ζqsDqΦ℘m0dqζ≤Mm∫01ζ0dqζ1−1/m∫01ζqs+10dqζ+∫01ζ1−ζqs0dqζ=Mw−℘2+υ−w2υ−℘∫01ζ0dqζ1−1/m∫01ζs+10dqζ+∫01ζ1−ζqs0dqζ,∫01ζqs+10dqζ=1s+2,−qs+1∫01ζDq1−q−1ζqs+10dqζ=−qs+1ζ1−q−1ζqs+101−∫011−ζqs+1.10dqζ=−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+2=−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+2=Mw−℘2+υ−w2υ−℘121−1/m−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+21/m.
It completes the proof.Theorem 18.
Suppose thatΦ:J⊂ℝ+⟶ℝ is a q-differentiable mapping on J∘ such that DqΦ∈L℘,υ, in which ℘,υ∈J for ℘<υ. If DqΦm is s-convex function in second sense on ℘,υ for some s∈0,1,q>1 and m>1 and n=m/m−1, then the q-integral inequality is valid:
(38)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤2s−1/m1+n1/nυ−℘w−℘2DqΦw+℘2+υ−w2DqΦυ+w2,for each w∈℘,υ.Proof.
Lemma 3.1 and keeping in view the familiarq-analogue of Hölder inequality, we have
(39)1qΦw−1℘+υ∫℘υΦζ℘dqζ≤w−℘2υ−℘∫01ζDqΦζw+1−ζ℘0dqζ+υ−w2υ−℘∫01ζDqΦζw+1−ζυ0dqζ≤w−℘2υ−℘∫01ζqn0dqζ1/n∫01DqΦζw+1−ζ℘m0dqζ1/m+υ−w2υ−℘∫01∫01ζqn0dqζ1/n∫01DqΦζw+1−ζυm0dqζ1/m,∫01DqΦζw+1−ζ℘mdζ≤2s−1DqΦw+℘2m∫01DqΦζw+1−ζυmdζ≤2s−1DqΦυ+w2m≤2s−1/m1+n1/nυ−℘w−℘2DqΦw+℘2+υ−w2DqΦυ+w2.Remark 19.
In Theorem13, if we choose q=1, then (24) diminishes the inequality (1) of Theorem 1.Remark 20.
In Theorem13, if we choose s=1, then (24) diminishes the inequality (7) of Theorem 7.Remark 21.
In Theorem15, if we fixed q=1, then (31) reduces the inequality (3) of Theorem 3.Remark 22.
In Theorem15, if we take s=1, then (31) diminishes the inequality (9) of Theorem 9.Remark 23.
In Theorem16, if we take q=1, then (34) reduces the inequality (4) of Theorem 4.Remark 24.
In Theorem16, if we choose s=1, then (34) diminishes the inequality (10) of Theorem 10.Remark 25.
In Theorem17, if we take q=1, then (36) diminishes the inequality (5) of Theorem 5.Remark 26.
In Theorem18, if we take q=1, then (38) diminishes the inequality (6) of Theorem 6.


## 3.1.q-Hermite-Hadamard Inequality

Theorem 13.
SupposeΦ:ℝ+⟶ℝ is a s-convex mapping in the second sense, in which s,q∈0,1, and let ℘,υ∈ℝ+,℘<υ. If DqΦ∈La,b, then the integral inequality is valid:
(24)2s−1Φ℘+υ2≤1υ−℘∫℘υΦζ℘dqζ≤Φυq1−1−q−1s+1+Φ℘s+1.Proof.
By definition ofs-convex functions,
(25)Φζ℘+1−ζυ≤ζsΦ℘+1−ζsΦυ∫01Φζ℘+1−ζυ0dqζ≤Φ℘∫01ζs0dqζ+Φυ∫011−ζs0dqζ,∫01Φζ℘+1−ζυ0dqζ=1−qυ−℘υ−℘∑n=0∞qnΦqn℘+1−qnυ=1υ−℘∫℘υΦζ0dqζ≤Φ℘∫01ζs0dqζ+Φυ∫011−ζsd0qζ=Φ℘+q1−1−q−1s+1Φυs+1.
Hence,(26)1υ−℘∫℘υΦζ0dqζ≤Φ℘+q1−1−q−1s+1Φυs+1.
Letw=ζ℘+1−ζυ and ζ=ζυ+1−ζ℘ in Φw+ζ/2≤Φw+Φζ/2s to get
(27)Φζ℘+1−ζυ+ζυ+1−ζ℘2≤Φζ℘+1−ζυ+Φζυ+1−ζ℘2s,Φ℘+υ2≤12s∫01Φζ℘+1−ζυ0dqζ+∫01Φζυ+1−ζ℘0dqζ=12s1℘−υ∫υ℘Φζ0dqζ+1υ−℘∫℘υΦζ0dqζ,(28)2s−1Φ℘+υ2≤1υ−℘∫℘υΦζ0dqζ,
From (26) and (28), the desired result is
(29)2s−1Φ℘+υ2≤1υ−℘∫℘υΦζ℘dqζ≤Φ℘+q1−1−q−1s+1Φυs+1.


## 3.2.q-Ostrowski Type Inequalities

To prove someq-Ostrowski type inequalities, it needs to establish the following Montgomery identity for q-integrals:Lemma 14.
LetΦ:J⊂ℝ⟶ℝ be a q-differentiable on J∘ in which ℘,υ∈J for ℘<υ. If DqΦ∈L℘,υ, we have the following q-integral equality which is valid:
(30)1qΦw−1υ−℘∫℘υΦζ℘dqζ=w−℘2υ−℘∫01ζDqΦζw+1−ζ℘0dqζ−υ−w2υ−℘∫01ζDqΦζw+1−ζυ0dqζ,
for eachw∈℘,υ.
By using Lemma14, we have constructed the following Ostrowski type inequalities, which hold for s-convex functions in the second sense:Theorem 15.
LetΦ:J⊂ℝ+⟶ℝ is a q-differentiable mapping on J∘ and DqΦ∈L℘,υ, in which ℘,υ∈J for ℘<υ. If the absolute value of DqΦw is s-convex in second sense on ℘,υ for unique s∈0,1 and DqΦw is bounded by M, w∈℘,υ, we have been seeing that the following q-integral inequality is valid:
(31)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤Mw−℘2+υ−w2υ−℘−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+2+1s+2,
for eachw∈℘,υ.Proof.
SinceDqΦ is s-convex function in the second sense on ℘,υ, therefore, Lemma 14 gives the following:
(32)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤w−℘2υ−℘∫01ζDqΦζw+1−ζ℘0dqζ+υ−w2υ−℘∫01ζDqΦζw+1−ζυ0dqζ≤w−℘2υ−℘∫01ζqs+1DqΦw0dqζ+∫01ζ1−ζqsDqΦ℘0dqζ+υ−w2υ−℘∫01ζqs+1DqΦw0dqζ+∫01ζ1−ζqsDqΦυ0dqζ=Mw−℘2υ−℘∫01ζqs+10dqζ+∫01ζ1−ζqs0dqζ+υ−w2υ−℘∫01ζqs+10dqζ+∫01ζ1−ζqs0dqζ=Mw−℘2+υ−w2υ−℘∫01ζs+10dqζ+∫01ζ1−ζqs0dqζ,∫01ζqs+10dqζ=1s+2,(33)=−qs+1∫01ζDq1−q−1ζqs+10dqζ=−1qs+1ζ1−q−1ζqs+101−∫011−ζqs+1.10dqζ=−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+2=−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+2+1s+2=Mw−℘2+υ−w2υ−℘−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+2+1s+2.Theorem 16.
Suppose thatΦ:J⊂ℝ+⟶ℝ is a q-differentiable on J∘ and DqΦ∈L℘,υ, in which ℘,υ∈J for ℘<υ. If DqΦm is a s-convex function in second sense on ℘,υ for some static s∈0,1,m>1,n=m/m−1 and DqΦw is bounded by M, w∈℘,υ, then the q-integral inequality is valid:
(34)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤Mn+11/n1+q1−1−q−1s+1s+11/m×w−℘2+υ−w2υ−℘,for each w∈℘,υ.Proof.
From Lemma14 and keeping in view the well-known q-analogue of Hölder inequality, we have
(35)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤w−℘2υ−℘∫01ζDqΦζw+1−ζ℘0dqζ+υ−w2υ−℘∫01ζDqΦζw+1−ζυ0dqζ≤w−℘2υ−℘∫01ζqn0dqζ1/n∫01DqΦζw+1−ζ℘m0dqζ1/m+υ−w2υ−℘∫01∫01ζqn0dqζ1/n∫01DqΦζw+1−ζυm0dqζ1/m,∫01DqΦζw+1−ζ℘m0dqζ≤∫01ζqsDqΦwm0dqζ+∫011−ζqsDqΦ℘m0dqζ≤Mmζqs+1s+101−q1−q−1ζqs+1s+101=Mm1s+1−q1−q−1qs+1s+1+qs+1=Mm1+q1−1−q−1qs+1s+1,∫01DqΦζw+1−ζυm0dqζ≤∫01ζqsDqΦwm0dqζ+∫011−ζqsDqΦυm0dqζ≤Mmζqs+1s+101−q1−q−1ζqs+1s+101=Mm1s+1−q1−q−1qs+1s+1+qs+1=Mm1+q1−1−q−1qs+1s+1≤M11+n1/n1+q1−1−q−1qs+1s+11/mw−℘2+υ−w2υ−℘.
It completes the proof.Theorem 17.
LetΦ:J⊂ℝ+⟶ℝ is a q-differentiable mapping on J∘ such as DqΦ∈L℘,υ, in which ℘,υ∈J for ℘<υ. If the absolute value of DqΦwm is a s-convex mapping in the second sense on ℘,υ for unique s∈0,1, m≥1, and DqΦw≤M, w∈℘,υ, we have seen that the q-integral inequality is valid:
(36)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤Mw−℘2+υ−w2υ−℘121−1/m−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+21/m,for each w∈℘,υ.Proof.
Lemma14 and keeping in view the well-known q-analogue of power-mean inequality, we have
(37)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤w−℘2υ−℘∫01ζDqΦζw+1−ζ℘0dqζ+υ−w2υ−℘∫01ζDqΦζw+1−ζυ0dqζ≤w−℘2υ−℘∫01ζ0dqζ1−1/m∫01ζDqΦζw+1−ζ℘m0dqζ1/m+υ−w2υ−℘∫01ζ0dqζ1−1/m∫01ζDqΦζw+1−ζυm0dqζ1/m,∫01ζDqΦζw+1−ζ℘m0dqζ≤∫01ζqs+1DqΦwm0dqζ+∫01ζ1−ζqsDqΦ℘m0dqζ≤Mm∫01ζ0dqζ1−1/m∫01ζqs+10dqζ+∫01ζ1−ζqs0dqζ=Mw−℘2+υ−w2υ−℘∫01ζ0dqζ1−1/m∫01ζs+10dqζ+∫01ζ1−ζqs0dqζ,∫01ζqs+10dqζ=1s+2,−qs+1∫01ζDq1−q−1ζqs+10dqζ=−qs+1ζ1−q−1ζqs+101−∫011−ζqs+1.10dqζ=−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+2=−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+2=Mw−℘2+υ−w2υ−℘121−1/m−qs+11−q−1qs+1+q1−q−1qs+2s+2−qs+21/m.
It completes the proof.Theorem 18.
Suppose thatΦ:J⊂ℝ+⟶ℝ is a q-differentiable mapping on J∘ such that DqΦ∈L℘,υ, in which ℘,υ∈J for ℘<υ. If DqΦm is s-convex function in second sense on ℘,υ for some s∈0,1,q>1 and m>1 and n=m/m−1, then the q-integral inequality is valid:
(38)1qΦw−1υ−℘∫℘υΦζ℘dqζ≤2s−1/m1+n1/nυ−℘w−℘2DqΦw+℘2+υ−w2DqΦυ+w2,for each w∈℘,υ.Proof.
Lemma 3.1 and keeping in view the familiarq-analogue of Hölder inequality, we have
(39)1qΦw−1℘+υ∫℘υΦζ℘dqζ≤w−℘2υ−℘∫01ζDqΦζw+1−ζ℘0dqζ+υ−w2υ−℘∫01ζDqΦζw+1−ζυ0dqζ≤w−℘2υ−℘∫01ζqn0dqζ1/n∫01DqΦζw+1−ζ℘m0dqζ1/m+υ−w2υ−℘∫01∫01ζqn0dqζ1/n∫01DqΦζw+1−ζυm0dqζ1/m,∫01DqΦζw+1−ζ℘mdζ≤2s−1DqΦw+℘2m∫01DqΦζw+1−ζυmdζ≤2s−1DqΦυ+w2m≤2s−1/m1+n1/nυ−℘w−℘2DqΦw+℘2+υ−w2DqΦυ+w2.Remark 19.
In Theorem13, if we choose q=1, then (24) diminishes the inequality (1) of Theorem 1.Remark 20.
In Theorem13, if we choose s=1, then (24) diminishes the inequality (7) of Theorem 7.Remark 21.
In Theorem15, if we fixed q=1, then (31) reduces the inequality (3) of Theorem 3.Remark 22.
In Theorem15, if we take s=1, then (31) diminishes the inequality (9) of Theorem 9.Remark 23.
In Theorem16, if we take q=1, then (34) reduces the inequality (4) of Theorem 4.Remark 24.
In Theorem16, if we choose s=1, then (34) diminishes the inequality (10) of Theorem 10.Remark 25.
In Theorem17, if we take q=1, then (36) diminishes the inequality (5) of Theorem 5.Remark 26.
In Theorem18, if we take q=1, then (38) diminishes the inequality (6) of Theorem 6.


## 4. Conclusion

By the virtue ofq-calculus, some integral inequalities are proved, which provides a method to study more properties of q-integrals via other classes of integral inequalities. q-Hermite-Hadmard and q-Ostrowski type integral inequalities have provided new estimates and good approximations in comparison with existing Hermite-Hadamard and Ostrowski inequalities. In similar fashion, the same methods can be applied to other inequalities, including Simpson’s and trapezoidal inequalities for different classes of s-convex functions.



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