# Extensions of Ostrowski Type Inequalities viah-Integrals and s-Convexity

Authors: Khuram Ali Khan; Khalid Mahmood Awan; Allah Ditta; Ammara Nosheen; Rostin Mabela Matendo
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, Hölder, Minkowski, and power mean inequalities are used to establish Ostrowski type inequalities fors-convex functions via h-calculus. The new inequalities are generalized versions of Ostrowski type inequalities available in literature.


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## Body

## 1. Introduction

In mathematics, the quantum calculus is equivalent to usual infinitesimal calculus without depending upon the concept of limit. It has two major branches,q-calculus and the h-calculus. It is really the calculus of finite differences, but a more systematic analogy with classical calculus makes it additionally transparent. The definite h-integral is a Riemann sum so that the fundamental theorem of h-calculus allows one to evaluate finite sums and h-integration by parts which is simply the Abel transform. The theory of h-discrete calculus is the rapidly developing area of great interest both from theoretical and applied point of view. This calculus is the study of the definitions, properties, and applications of the related concepts, the fractional calculus and discrete fractional calculus.


### 1.1.h-Derivative [1]

Theh-derivative is defined as follows: let ϕ:Tν⟶ℝ,Tν=ν,ν+h,ν+2h,…,h>0.(1)Dhϕθ=dhϕθdhθ=ϕθ+h−ϕθh,where, classically, limh⟶0Dhϕθ=dϕθ/dθ. Also, h-differential is dhϕθ=ϕθ+h−ϕθ, in particular dhθ=h.


### 1.2.h-Integral [1]

Theh-integral is defined as follows: let ϕ:Tν⟶ℝ,Tν=ν,ν+h,ν+2h,…,h>0.(2)∫νμϕθdhθ=∑j=0μ−ν/h−1ϕν+jhh,where ν,μ∈ℝ,ν<μ.


### 1.3. Definition ofh-Integral [1]

Ifμ−ν∈hℤ, we define h-integral to be(3)∫νμϕθdhθ=hϕν+ϕν+h+ϕν+2h+⋯+ϕμ−h,ifν<μ,0ifν=μ,−hϕμ+ϕμ+h+ϕμ+2h+⋯+ϕν−hifν>μ.With this definition, the definiteh-integral is Riemann sum of ϕθ on the interval ν,μ, which is proportioned to subintervals of equal width.


### 1.4. Formula ofh-Integration by Parts [1]

Letϕ,g:ν,μ⟶ℝ be the continuous functions and θ∈ν,μ, then the formula of h-integration by parts is stated as(4)∫νμϕθdhgθ=ϕμgμ−ϕνgν−∫vμgθ+hdhϕθ.


### 1.5. Properties ofh-Calculus [1]

Theh-analogue of a binomial expansion θ−νn is defined as(5)θ−νhn=θ−νθ−ν−h…θ−ν−n−1h.Forn⩾1andθ−νh0=1,(6)Dhθ−νhn=nθ−νhn−1,Dhν−θhn=−nν−h−θhn−1,Dh1θ−νhn=−nν+h−θhn+1,Dh1ν−θhn=nν−θhn+1.Note thath-analogues of an integer n is still n, and θ−0hn≠θn.


### 1.6.h-Fractional Function [2]

Lett,α∈ℝ, then the h-fractional function thα is defined by(7)thα=hαΓt/h+1Γt/h+1−α,where Γ is the Euler gamma function t/h∉−1,−2,−3,−4…. Note that limh⟶0thα=tα, hence we define(8)thα−1=1αDhthα.Basic inequalities have a massive role both in pure and applied sciences in the light of their wide applications in mathematics and physical sciences, while convexity theory has stayed as a significant instrument in the foundation of the hypothesis of integral inequalities.The following Ostrowski inequality [3] is notable to read.Theorem 1.
Letϕ:I⟶ℝ, where I⊆ℝ is an interval, be a mapping differentiable in the interior I° of I, and let ν,μ∈I° with ν<μ. If ϕ′θ≤M for all θ∈ν,μ, then the following inequality holds:(9)ϕθ−1μ−ν∫νμϕtdt≤Mμ−ν14+θ−ν+μ/22μ−ν2,for all θ∈ν,μ.From the invention of (9), it is being studied extensively by many researchers (see [4–9]). Generalizations, extensions, and variants of this inequality exist in literature (see [10–17]) for different classes of convex functions. More on s-convex functions and on the conformable functions can be seen in [4, 18–24].In [25], the class of functions which are called s-convex in the second sense has been introduced by Hudzik and Maligranda as follows:


### 1.7.s-Convex Function

A functionϕ:ℝ+⟶ℝ is said to be s-convex in second sense if(10)ϕΩθ+1−Ωϕ≤Ωsϕθ+1−Ωsϕϕ,for each θ,ϕ∈ℝ+, Ω∈0,1 and for unique s∈0,1.The integral equality is established by Alomari et al. in [26].Lemma 1.
Suppose thatϕ:J⊂ℝ⟶ℝ is a mapping such that ϕ′∈Lν,μ, then(11)ϕθ−1μ−ν∫νμϕudu=θ−ν2μ−ν∫01uϕ′uθ+1−uνdu−μ−θ2μ−ν∫01uϕ′uθ+1−uμdu,for each θ∈ν,μ.
Using Lemma1, Alomari et al. in [26] presented the following integral inequalities.Theorem 2.
Consider the functionϕ:J⊂ℝ+⟶ℝ such that ϕ′∈Lν,μ for ν,μ∈J. If ϕ′ in term of second sense is s-convex on ν,μ, for unique s∈0,1 and ϕ′θ≤M, θ∈ν,μ, the following result holds:(12)ϕθ−1μ−ν∫νμϕudu≤Mμ−νθ−ν2+μ−θ2s+1,for each θ∈ν,μ.Theorem 3.
Consider the functionϕ:J⊂ℝ+⟶ℝ such that ϕ′∈Lν,μ for ν,μ∈J. If ϕ′m is s-convex in second sense on ν,μ, for unique s∈0,1, m>1, n=m/m−1 and ϕ′θ≤M, θ∈ν,μ, the following integral inequality holds:(13)ϕθ−1μ−ν∫νμϕudu≤M1+n1/n2s+11/mθ−ν2+μ−θ2μ−ν,for each θ∈ν,μ.Theorem 4.
Consider the functionϕ:J⊂ℝ+⟶ℝ such that ϕ′∈Lν,μ for ν,μ∈J. If ϕ′m is a s-convex in second sense on ν,μ, for some static s∈0,1, m≥1 and ϕ′θ≤M, θ∈ν,μ, then the following result holds:(14)ϕθ−1μ−ν∫νμϕudu≤M2s+11/mθ−ν2+μ−θ22μ−ν,for each θ∈ν,μ.


## 1.1.h-Derivative [1]

Theh-derivative is defined as follows: let ϕ:Tν⟶ℝ,Tν=ν,ν+h,ν+2h,…,h>0.(1)Dhϕθ=dhϕθdhθ=ϕθ+h−ϕθh,where, classically, limh⟶0Dhϕθ=dϕθ/dθ. Also, h-differential is dhϕθ=ϕθ+h−ϕθ, in particular dhθ=h.


## 1.2.h-Integral [1]

Theh-integral is defined as follows: let ϕ:Tν⟶ℝ,Tν=ν,ν+h,ν+2h,…,h>0.(2)∫νμϕθdhθ=∑j=0μ−ν/h−1ϕν+jhh,where ν,μ∈ℝ,ν<μ.


## 1.3. Definition ofh-Integral [1]

Ifμ−ν∈hℤ, we define h-integral to be(3)∫νμϕθdhθ=hϕν+ϕν+h+ϕν+2h+⋯+ϕμ−h,ifν<μ,0ifν=μ,−hϕμ+ϕμ+h+ϕμ+2h+⋯+ϕν−hifν>μ.With this definition, the definiteh-integral is Riemann sum of ϕθ on the interval ν,μ, which is proportioned to subintervals of equal width.


## 1.4. Formula ofh-Integration by Parts [1]

Letϕ,g:ν,μ⟶ℝ be the continuous functions and θ∈ν,μ, then the formula of h-integration by parts is stated as(4)∫νμϕθdhgθ=ϕμgμ−ϕνgν−∫vμgθ+hdhϕθ.


## 1.5. Properties ofh-Calculus [1]

Theh-analogue of a binomial expansion θ−νn is defined as(5)θ−νhn=θ−νθ−ν−h…θ−ν−n−1h.Forn⩾1andθ−νh0=1,(6)Dhθ−νhn=nθ−νhn−1,Dhν−θhn=−nν−h−θhn−1,Dh1θ−νhn=−nν+h−θhn+1,Dh1ν−θhn=nν−θhn+1.Note thath-analogues of an integer n is still n, and θ−0hn≠θn.


## 1.6.h-Fractional Function [2]

Lett,α∈ℝ, then the h-fractional function thα is defined by(7)thα=hαΓt/h+1Γt/h+1−α,where Γ is the Euler gamma function t/h∉−1,−2,−3,−4…. Note that limh⟶0thα=tα, hence we define(8)thα−1=1αDhthα.Basic inequalities have a massive role both in pure and applied sciences in the light of their wide applications in mathematics and physical sciences, while convexity theory has stayed as a significant instrument in the foundation of the hypothesis of integral inequalities.The following Ostrowski inequality [3] is notable to read.Theorem 1.
Letϕ:I⟶ℝ, where I⊆ℝ is an interval, be a mapping differentiable in the interior I° of I, and let ν,μ∈I° with ν<μ. If ϕ′θ≤M for all θ∈ν,μ, then the following inequality holds:(9)ϕθ−1μ−ν∫νμϕtdt≤Mμ−ν14+θ−ν+μ/22μ−ν2,for all θ∈ν,μ.From the invention of (9), it is being studied extensively by many researchers (see [4–9]). Generalizations, extensions, and variants of this inequality exist in literature (see [10–17]) for different classes of convex functions. More on s-convex functions and on the conformable functions can be seen in [4, 18–24].In [25], the class of functions which are called s-convex in the second sense has been introduced by Hudzik and Maligranda as follows:


## 1.7.s-Convex Function

A functionϕ:ℝ+⟶ℝ is said to be s-convex in second sense if(10)ϕΩθ+1−Ωϕ≤Ωsϕθ+1−Ωsϕϕ,for each θ,ϕ∈ℝ+, Ω∈0,1 and for unique s∈0,1.The integral equality is established by Alomari et al. in [26].Lemma 1.
Suppose thatϕ:J⊂ℝ⟶ℝ is a mapping such that ϕ′∈Lν,μ, then(11)ϕθ−1μ−ν∫νμϕudu=θ−ν2μ−ν∫01uϕ′uθ+1−uνdu−μ−θ2μ−ν∫01uϕ′uθ+1−uμdu,for each θ∈ν,μ.
Using Lemma1, Alomari et al. in [26] presented the following integral inequalities.Theorem 2.
Consider the functionϕ:J⊂ℝ+⟶ℝ such that ϕ′∈Lν,μ for ν,μ∈J. If ϕ′ in term of second sense is s-convex on ν,μ, for unique s∈0,1 and ϕ′θ≤M, θ∈ν,μ, the following result holds:(12)ϕθ−1μ−ν∫νμϕudu≤Mμ−νθ−ν2+μ−θ2s+1,for each θ∈ν,μ.Theorem 3.
Consider the functionϕ:J⊂ℝ+⟶ℝ such that ϕ′∈Lν,μ for ν,μ∈J. If ϕ′m is s-convex in second sense on ν,μ, for unique s∈0,1, m>1, n=m/m−1 and ϕ′θ≤M, θ∈ν,μ, the following integral inequality holds:(13)ϕθ−1μ−ν∫νμϕudu≤M1+n1/n2s+11/mθ−ν2+μ−θ2μ−ν,for each θ∈ν,μ.Theorem 4.
Consider the functionϕ:J⊂ℝ+⟶ℝ such that ϕ′∈Lν,μ for ν,μ∈J. If ϕ′m is a s-convex in second sense on ν,μ, for some static s∈0,1, m≥1 and ϕ′θ≤M, θ∈ν,μ, then the following result holds:(14)ϕθ−1μ−ν∫νμϕudu≤M2s+11/mθ−ν2+μ−θ22μ−ν,for each θ∈ν,μ.


## 2. Main Results

Initially, we establish the following identity.Lemma 2.
Supposeϕ:J⟶ℝ be h-differentiable mapping on interior of interval J in which ν,μ∈J and ν<μ. If Dhϕ∈Lν,μ, then the following h-integral equality is valid:(15)ϕθ+hθ−νϕν+μ−θϕμμ−ν−1μ−ν∫νμϕuνdhu=θ−ν2μ−ν∫01uDhϕuθ+1−uν0dhu−μ−θ2μ−ν∫01uDhϕuθ+1−uμ0dhu,for each θ∈ν,μ.Proof.
By formula (4) of h-integration by parts, the first term of right hand side of (15) becomes(16)θ−ν2μ−ν∫01uDhϕuθ+1−uνdhuθ−ν2μ−νuϕuθ+1−uνθ−ν|01−∫01ϕθu+h+1−u+hνθ−νdhu=θ−νμ−νϕθ−θ−νμ−νh∑j=01/h−1ϕθjh+h+1−jh+hν=θ−νμ−νϕθ−θ−νμ−νh∑j=01/h−1ϕθjh+h+1−jh+hν=θ−νμ−νϕθ−θ−νμ−νh∑j=11/h−1ϕjhθ+1−jhν=θ−νμ−νϕθ−1μ−νhθ−ν∑j=11/h−1ϕν+θ−νjh−hθ−νϕνμ−ν+hθ−νϕνμ−ν=θ−νμ−νϕθ−1μ−νhθ−ν∑j=11/h−1ϕν+θ−νjh+θ−νϕνμ−ν+θ−νϕνμ−ν=θ−νμ−νϕθ−1μ−νhθ−ν∑j=01/h−1ϕν+θ−νjh+hθ−νϕνμ−ν=θ−νμ−νϕθ−1μ−ν∫vθϕudhu+hθ−νϕνμ−ν,and the second term of right hand side of (15) becomes(17)μ−θ2μ−ν∫01uDhϕuθ+1−uμdhu=μ−θ2μ−νuϕuθ+1−uμθ−μ|01+∫01ϕθu+h+1−u+hμμ−θdhu=−μ−θμ−νϕθ+μ−θμ−νh∑j=01/h−1ϕθjh+h+1−jh+hμ=−μ−θμ−νϕθ+μ−θμ−νh∑j=01/h−1ϕθjh+h+1−jh+hμ=θ−νμ−νϕθ−θ−νμ−νh∑j=11/h−1ϕjhθ+1−jhμ=θ−νμ−νϕθ−1μ−νhθ−μ∑j=11/h−1ϕμ+θ−μjh−hθ−μϕμμ−ν+hθ−μϕμμ−ν=−μ−θμ−νϕθ+1μ−νhθ−μ∑j=11/h−1ϕμ+θ−μjh+θ−μϕνμ−ν−hθ−μϕμμ−ν=−μ−θμ−νϕθ+1μ−νhθ−μ∑j=01/h−1ϕν+θ−μjh−hθ−μϕμμ−ν=−μ−θμ−νϕθ−1μ−ν∫μθϕudhu+hθ−μϕμμ−ν.
From (16) and (17),(18)θ−νμ−νϕθ−1μ−ν∫νθϕudhu+hθ−νϕνμ−ν+μ−θμ−νϕθ−1μ−ν∫θμϕudhu+hμ−θϕμμ−νθ−ν+μ−θμ−νϕθ+hθ−νϕν+hμ−θϕμμ−ν−1μ−ν∫νμϕudhu=ϕθ+hθ−νϕν+hμ−θϕμμ−ν−1μ−ν∫νμϕudhu,which is the required result.Using Lemma2, we prove the following results.Theorem 5.
Supposeϕ:J⟶ℝ be a h-differentiable mapping on interior of a positive interval J in such a way that Dhϕ∈Lν,μ, for ν,μ∈J. If Dhϕ is the s-convex in second sense on ν,μ for some fixed s∈0,1 and Dhϕθ≤M, θ∈ν,μ, we have the following h-integral inequality:(19)ϕθ+hθ−νϕν+μ−θϕμμ−ν−1μ−ν∫νμϕuνdhu⩽Mθ−ν2+μ−θ2μ−ν−hs+1s+1+1s+1−hs+2+1+hs+2s+2+1s+2,for each θ∈ν,μ.Proof.
Using Lemma2 for s−convex mapping Dhϕ defined on ν,μ, we see(20)ϕθ+hθ−νϕν+μ−θϕμμ−ν−1μ−ν∫νμϕuνdhu⩽θ−ν2μ−ν∫01uDhϕuθ+1−uνdhu+μ−θ2μ−ν∫01uDhϕuθ+1−uμdhu⩽θ−ν2μ−ν∫01uhs+1Dhϕθdhu+∫01u1−usDhϕνdhu+μ−θ2μ−ν∫01uhs+1Dhϕθdqu+∫01u1−usDhϕμdhu⩽θ−ν2μ−ν∫01uhs+1Dhϕθdhu∫01u1−usDhϕνdhu+μ−θ2μ−ν∫01uhs+1Dhϕθdhu+∫01u1−usDhϕμdhu=Mθ−ν2μ−ν∫01uhs+1dhu+∫01u1−usdhu+μ−θ2μ−ν∫01uhs+1dhu+∫01u1−usdhu=Mθ−ν2+μ−θ2μ−ν∫01uhs+1dhu+∫01u1−usdhu∫01uhs+1dhu=1s+2∫01u1−usdhu=u.−1−u+hs+1s+101+1s+1∫011−u+h+hs+1dhu=−hs+1s+1+1s+1∫011−us+1dhu=−hs+1s+1+1s+1−1s+21+h−us+101=−hs+1s+1+1s+1−hs+2s+2+1+hs+2s+2≤Mθ−ν2+μ−θ2μ−ν−hs+1s+1+1s+1−hs+2+1+hs+2s+2+1s+2.Theorem 6.
Supposeϕ:J⊂ℝ+⟶ℝ be a h-differentiable mapping on J° in such a way that Dhϕ∈Lν,μ, in which ν,μ∈J for ν<μ. If Dhϕm is s-convex ν,μ for some static s∈0,1, m>1, n=m/m−1, and Dhϕθ≤M, θ∈ν,μ, then we have the h-integral inequality in discrete calculus:(21)ϕθ+hθ−νϕν+μ−θϕμμ−ν−1μ−ν∫νμϕuνdhu⩽Mθ−ν2+μ−θ2μ−ν1n+11/n1−hs+1+1+hs+1s+11/m,for each θ∈ν,μ.Proof.
From Lemma2 and keeping the familiar Hölder inequality, we have(22)ϕθ+hθ−νϕν+μ−θϕμμ−ν−1μ−ν∫νμϕuνdhu⩽θ−ν2μ−ν∫01uDhϕuθ+1−uνdhu+μ−θ2μ−ν∫01uDhϕuθ+1−uμdhu⩽θ−ν2μ−ν∫01uhndhu1/n∫01Dhϕuθ+1−uνmdhu1/m+μ−θ2μ−ν∫01∫01uhndhu1n∫01Dhϕuθ+1−uμmdhu1/m∫01Dhϕuθ+1−uνmdhu⩽∫01uhsDhϕθmdhu+∫011−usDhϕνmdhu⩽Mm∫01uhsdhu+∫011−usdhu⩽Mmuhs+1s+101+−1+h−us+1s+101=Mm1s+1−hs+1s+1+1+hs+1s+1=Mm1−hs+1+1+hs+1s+1, (23)∫01Dhϕuθ+1−uμmdhu⩽∫01uhsDhϕθmdhu+∫011−usDhϕμmdhu⩽Mmuhs+1s+101+−1+h−us+1s+101=Mm1s+1−hs+1s+1+1+hs+1s+1=Mm1−hs+1+1+hs+1s+1⩽θ−ν2+μ−θ2μ−ν∫01uhndhu1/nMm∫01uhsdqu+∫011−usdhu1/m⩽θ−ν2μ−ν+μ−θ2μ−ν1n+11/nMm1−hs+1+1+hs+1s+11/m⩽Mθ−ν2μ−ν+μ−θ2μ−ν1n+11/n1−hs+1+1+hs+1s+11/m,which completes the proof.Theorem 7.
Supposeϕ:J⟶ℝ be a h-differentiable mapping on interior of a positive interval J in such a way that Dhϕ∈Lν,μ, for ν,μ∈J. If Dhϕm is a s-convex in second sense on ν,μ for some static s∈0,1, m≥1 and Dhϕθ≤M, θ∈ν,μ, then we have the following h-integral inequality:(24)ϕθ+hθ−νϕν+μ−θϕμμ−ν−1μ−ν∫νμϕuνdhu⩽Mθ−ν2+μ−θ2μ−ν121−1/m−hs+1s+1+1s+1−hs+2+1+hs+2s+2+1s+21/m,for each θ∈ν,μ.Proof.
From Lemma2 and keeping in view the familiar power mean inequality, we get(25)ϕθ+hθ−νϕν+μ−θϕμμ−ν−1μ−ν∫νμϕuνdhu⩽θ−ν2μ−ν∫01uDhϕuθ+1−uνdhu+μ−θ2μ−ν∫01uDhϕuθ+1−uμdhu⩽θ−ν2μ−ν∫01udhu1−1/m∫01Dhϕuθ+1−uνmdhu1/m+μ−θ2μ−ν∫01udhu1−1/m∫01Dhϕuθ+1−uμmdhu1/m⩽Mθ−ν2μ−ν∫01udhu1−1/m∫01uhs+1dhu+∫01u1−usdhu1/m+μ−θ2μ−ν∫01udhu1−1/m∫01uhs+1dhu+∫01u1−usdhu1/m∫01Dhϕuθ+1−uνmdhu⩽∫01uhs+1Dhϕθmdhu+∫01u1−usDhϕνmdhu⩽Mm∫01uhs+1dhu+∫01u1−usdhu∫01u1−usdhu=u⋅−1−u+hs+1s+101+1s+1∫011−u+h+hs+1dhu=−hs+1s+1+1s+1∫011−us+1dhu=−hs+1s+1+1s+1−1s+21+h−us+201=−hs+1s+1+1s+1−hs+2s+2+1+hs+2s+2,⩽Mm−hs+1s+1+1s+1−hs+2s+2+1+hs+2s+2+1s+2, (26)∫01Dhϕuθ+1−uμmdhu⩽∫01uhs+1Dhϕθmdhu+∫01u1−usDhϕμmdhu⩽Mm∫01uhs+1dhu+∫01u1−usdhu⩽Mm−hs+1s+1+1s+1−hs+2s+2+1+hs+2s+2+1s+2⩽Mθ−ν2+μ−θ2μ−ν121−1/m−hs+1s+1+1s+1−hs+2+1+hs+2s+2+1s+21/m.Remark 1.
(a)
In Theorem5, if we take h=0, then (19) diminishes the inequality (12) of Theorem 2(b)
In Theorem6, if we take h=0, then (21) diminishes the inequality (13) of Theorem 3(c)
In Theorem7, if we take h=0, then (24) diminishes the inequality (14) of Theorem 4
In [27], if ⊤=ℤ,(27)∫νμφuΔu=∑u=νμ−1φu,ifν<μ,0,ifν=μ,−∑u=μν−1φu,ifν>μ.Example 1.
In case of time scale⊤=ℤ in Lemma 2, we have(28)φθ+θ−νφν+μ−θφμμ−ν−1μ−ν∫νμφuνd1u=θ−ν2μ−ν∫01uD1φuθ+1−uν0d1u−μ−θ2μ−ν∫01uD1φuθ+1−uμ0d1u.
By takingh=1, in Theorem 5, we have(29)φθ+θ−νφν+μ−θφμμ−ν−1μ−ν∫νμφuνd1u≤Mθ−ν2+μ−θ2μ−ν−1s+1+−1+2s+2s+1s+2+1s+2.
By takingh=1, in Theorem 6, we have(30)φθ+θ−νφν+μ−θφμμ−ν−1μ−ν∫νμφuνd1u≤Mθ−ν2+μ−θ2μ−ν1n+11/n2s+1s+11/m.
By takingh=1, in Theorem 7, we have(31)φθ+θ−νφν+μ−θφμμ−ν−1μ−ν∫νμφuνd1u≤Mθ−ν2+μ−θ2μ−ν121−1/m−1s+1+−1+2s+2s+1s+2+1s+21/m.
By takingh=1 and s=1 in Theorem 5, we have(32)φθ+θ−νφν+μ−θφμμ−ν−1μ−ν∫νμφuνd1u≤Mθ−ν2+μ−θ2μ−ν.
By takingh=1 and s=1 in Theorem 6, we have(33)φθ+θ−νφν+μ−θφμμ−ν−1μ−ν∫νμφuνd1u≤M21/mθ−ν2+μ−θ2μ−ν1n+11/n.
By takingh=1 and s=1 in Theorem 7, we have(34)φθ+θ−νφν+μ−θφμμ−ν−1μ−ν∫νμφuνd1u≤Mθ−ν2+μ−θ2μ−ν121−1/m.


## 3. Conclusion

Our results extend and generalize the results of Alomari et al. In this work, some important Ostrowski type inequalities are established in the context ofh-calculus. The derived results constitute contributions to the theory of h-integral and can be specialized to yield numerous interesting integral inequalities including some known results. An interesting feature of our results is that they provide new estimates and best approximation on Ostrowski type of inequalities for h-integral. If we take limit h⟶0, Ostrowski type of h-integral inequalities reduces to simple inequalities present in [26]. The presented results motivate scientists to stimulate more work in such directions.



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