# New Developments on Ostrowski Type Inequalities viaq-Fractional Integrals Involving s-Convex Functions

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

---

## Abstract

In the present paper,q-fractional integral operators are used to construct quantum analogue of Ostrowski type inequalities for the class of s-convex functions. The limiting cases include the nonfractional existing cases from literature. Specially, Ostrowski type inequalities for q-integrals and Ostrowski type inequalities for convex functions are deduced.


---

## Body

## 1. Introduction

In mid of twentieth century, Jackson (1910) has begun a symmetric investigation onq-integrals. The subject of quantum analysis, depending upon q-integrals, has different applications in different branches of mathematics and material sciences like number hypothesis, combinatorics, symmetrical polynomials, essential hypermathematical capacities, quantum hypothesis, mechanics, and in the hypothesis of relativity. The perusers are suggested to Set [1], Gauchman [2], and Kac and Cheung [3] for q-analogues of fractional calculus.In numerous pragmatic issues, convexity hypothesis has stayed as a significant device in formation of vital imbalances. In many practical problems, it is important to bound one quantity by another quantity. The classical inequalities such as Ostrowski’s inequalities are very useful for this purpose. Ostrowski type inequalities are well known to study the upper bounds for approximation of the integral average by the value of function and definition ofs-convex function. Some new Ostrowski type inequalities for Riemann-Liouville fractional integral are established. Fractional calculus has been a well-known topic since it was initiated in the seventeenth century and studied by many great mathematicians of the time. Some classical inequalities including Ostrowski’s inequality are examples of it.


## 2. Preliminary Results

In 1938, Ostrowski [4] established the following well-known and useful integral inequality:Theorem 1.
Supposeψ:I⟶ℝ is the function differentiable in open interval of I, where I⊆ℝ and let ξ,η∈I∘ with ξ<η. If ψ′θ≤M for all θ∈ξ,η, then the following inequality holds
(1)ψθ−1η−ξ∫ξηψudu≤Mη−ξ14+θ−ξ+η/22η−ξ2,for all θ∈ξ,η. The least value of constant on R.H.S of (1) is 1/4.Inequality (1) gives an approximate upper bound for the deviation of integral arithmetic mean 1/η−ξ∫ξηψθdθ to the function ψ· at point θ∈ξ,η. In recent years, this inequality is studied extensively by different researchers, and its different variants can be seen in number of research papers including [5–11]. Recently, in [12, 13], Ostrowski type inequalities are studied for q-integrals.The following notion ofs-convex function in the second sense is from [14]:A mappingψ:0,∞⟶ℝ is said to be s-convex in the second sense if
(2)ψuξ+1−uη≤usψξ+1−usψη,for all ξ,η∈0,∞, u∈0,1, and some static s∈0,1.The following Lemma is established by Alomari et al. (see [8]).Lemma 2.
Ifψ′∈Lξ,η, then we have the equality
(3)ψθ−1η−ξ∫ξηψudu=θ−ξ2η−ξ∫01uf′uθ+1−uξdu−η−θ2η−ξ∫01uf′uθ+1−uηdu,for each θ∈ξ,η.By using Lemma2, Alomari et al. in [8] proved the following results of Ostrowski type inequalities:Theorem 3.
Supposeψ′∈Lξ,η. If ψ′ in term of second sense is s-convex on ξ,η, for unique s∈0,1 and ψ′θ≤M,θ∈ξ,η, then
(4)ψθ−1η−ξ∫ξηψudu≤Mη−ξθ−ξ2+η−θ2s+1holds, for each θ∈ξ,η.Theorem 4.
Supposeψ′∈Lξ,η. If ψ′m is a s-convex in second sense in ξ,η for unique s∈0,1,m>1,n=m/m−1 and ψ′θ≤M,θ∈ξ,η, then
(5)ψθ−1η−ξ∫ξηψudu≤M1+n1/n2s+11/mθ−ξ2+η−θ2η−ξholds, for each θ∈ξ,η.Theorem 5.
Supposeψ′∈Lξ,η. If ψ′m is s-convex in second sense on ξ,η for static s∈0,1,m≥1 and ψ′θ≤M,θ∈ξ,η, then
(6)ψθ−1η−ξ∫ξηψudu≤M2s+11/mθ−ξ2+η−θ22η−ξholds, for each θ∈ξ,η.Further some existing results ons-convex functions can be seen in [11], and some results involving fractional operators can be found in [15–21]. In case of fractional integrals, see the following lemma from [1].Lemma 6.
Ifψ′∈Lξ,η, then for all θ∈ξ,η and β>0(7)θ−ξβ+η−θβη−ξψθ−Γβ+1η−ξJθ−βψξ+Jθ+βψη=θ−ξβ+1η−ξ∫01uβψ′uθ+1−uξdu−η−θβ+1η−ξ∫01uβψ′uθ+1−uηduholds, where Γβ=∫0∞e−uuβ−1du is Euler Gamma function.By using Lemma6, Set in [1] proved the following:Theorem 7.
Forψ′∈Lξ,η. If ψ′θ is s-convex in second sense on ξ,η for fix s∈0,1 and ψ′θ≤M,θ∈ξ,η, then the following fractional integrals inequality, for β>0(8)θ−ξβ+η−θβη−ξψθ−Γβ+1η−ξJθ−βψξ+Jθ+βψη≤Mη−ξ1+Γβ+1Γs+1Γβ+s+1θ−ξβ+1+η−θβ+1β+s+1holds.Theorem 8.
Supposeψ′∈Lξ,η. If ψ′m is s-convex in second sense on ξ,η for some fixed s∈0,1, n,m>1 and ψ′θ≤M,θ∈ξ,η, then
(9)θ−ξβ+η−θβη−ξψθ−Γβ+1η−ξJθ−βψξ+Jθ+βψη≤M1+nβ1/n2s+11/mθ−ξβ+1+η−θβ+1η−ξholds, where 1/n+1/m=1,β>0.Theorem 9.
Supposeψ′∈Lξ,η, and assume that ψ′m is s-convex in second sense on ξ,η for some fix s∈0,1, m≥1,ψ′θ≤M, and θ∈ξ,η. Then
(10)θ−ξβ+η−θβη−ξψθ−Γβ+1η−ξJθ−βψξ+Jθ+βψη≤M11+β1−1/m1β+s+11/m×1+Γβ+1Γs+1Γβ+s+11/mθ−ξβ+1+η−θβ+1η−ξholds for each β>0.Theorem 10.
Supposeψ′∈Lξ,η. If ψ′m is s-convex in second sense on ξ,η for some fixed s∈0,1 and n,m>1. Then
(11)θ−ξβ+η−θβη−ξψθ−Γβ+1η−ξJθ−βψξ+Jθ+βψη≤2s−1/m1+nβ1/nη−ξθ−ξβ+1ψ′θ+ξ2+η−θβ+1ψ′η+θ2holds, for 1/n+1/m=1 and β>0.Note that ifs=1, the definition of s-convexity reduces to classical convexity of functions defined on ℝ+.


## 3. Preliminaries aboutq-Integrals and Related Inequalities

The following properties ofq-derivatives are recalled from [3].


### 3.1.q-Derivative

Forϕ∈Cξ,η, q-derivative of ϕ at θ∈ξ,η is given by
(12)Dξqϕθ=ϕθ−ϕqθ+1−qξ1−qθ−ξθ≠ξ.Forn≥1, we have the following relation:
(13)θ−ξqn=∏J=0nθ−qJξ,ξ−θqn=∏J=0nξ−qJθ.Respective derivatives are(14)Dqθ−ξqn=nθ−ξqn−1,Dqξ−θqn=−nξ−qθqn−1,ξ−qθqn=−1n+1Dqξ−θqn+1,where n=qn−1/q−1. The fractional calculus is a generalization of classical calculus concerned with operations of integration and differentiation of noninteger fractional order. The concept of fractional operators has been introduced almost simultaneously with the development of the classical ones. The first known reference can be found in the correspondence of G. W. Leibniz and Marquis de l’Hospital in 1695 where the question of meaning of the semiderivative has been raised. This question consequently attracted the interest of many well-known mathematicians, including Euler, Liouville, Laplace, Riemann, Grünwald, and Letnikov.


### 3.2. Fractional Integral from [8]

Letψ∈L1ξ,η. The Riemann-Liouville integrals Jξ+βψ and Jη−βψ of order β>0 for ξ≥0 are defined by
(15)Jξ+βψθ=1Γβ∫ξθθ−uβ−1ψudu,θ>ξ,Jη−βψθ=1Γβ∫θηu−θβ−1ψudu,θ<η.


### 3.3.q-Antiderivative

q-Antiderivative along with its properties can be studied in [22]. Suppose that ψ∈Cξ,η. Then q-definite integral for θ∈ξ,η is defined as
(16)∫ξθψuξdqu=1−qθ−ξ∑n=0∞qnψqnθ+1−qnξ,which gives
(17)∫ξ−θqβdqθ=−qξ−q−1θqβ+1β+1β≠−1,∫01uβ+sdqu=1β+s+1,∫01uβ1−usdqu=Γqβ+1Γqs+1Γqβ+s+2.Formula forq-integration by parts [23]: Let ψ,g∈Cξ,η, and θ∈ξ,η, and then
(18)∫cθψuξDqgudqu=ψθgθ−ψcgc−∫cθgqu+1−qξξDqψudqu.The followingq-integral inequality is from [23]:Theorem 11.
Supposeψ:ξ,η⟶ℝ is a q-differentiable mapping. If Dqψθ≤M for all θ∈ξ,η and 0<q<1, then
(19)ψθ−1η−ξ∫ξηψuξdqu≤M2q1+qθ−3q−1ξ+1+qη/4qη−ξ2+−q2+6q−18q1+qholds for all θ∈ξ,η.The least value of constant on R.H.S of (19) is −q2+6q−1/8q1+q.q-Hölder inequality [24]. Let ψ,Φ be q-integrable on ξ,η and 0<q<1 and 1/n+1/m=1 with m>1, and then we state as
(20)∫ξηψuΦudqu≤∫ξηψundqu1/n∫ξηΦumdqu1/m.The following inequalities are derived fromq-Hölder inequality:q-Minkowski’s inequality. Let ξ,η∈ℝ and n>1, for continuous functions ψ,Φ:ξ,η⟶ℝ, we have stated
(21)∫ξηψu+Φundqu1/n≤∫ξηψundqu1/n+∫ξηΦundqu1/n.q-Power mean inequality. Let 1/n+1/m=1 with n,m>1, and let ξ,η∈ℝ and for continuous functions ψ,g:ξ,η⟶ℝ, and then
(22)∫ξηψuΦudqu≤∫ξηψudqu1−1/m∫ξηψuΦumdqu1/m.Theorem 12 (see [25]).
Letψ:ξ,η⟶ℝ be a function and 0<q<1. Then
(23)∫01ψuη+1−uξdqu=1η−ξ∫ξηψtdqt.Example 1.
(24)∫ξθuu−ξdqu=1q+1θθ−ξ2−∫ξθu−ξ2dqu=1q+1θθ−ξ2−θ−ξ31+q+q2=θ−ξ21+qξ1+q+q2a1+q+q2.
Now, we have(25)∫ξθuu−ξdqu=∫0θ−ξt+ξtdqt=1q+1θθ−ξ2−∫0θ−ξqt2dqt=θ−ξ21+qξ1+q+q2a1+q+q2.Proposition 13.
For eachk,r∈ℕorℤq∈ℝ+ [2], we have
(26)k+rq=kq+qkrq.Exponential functions and Taylor series (q-analogues) from [26]:
(27)Eqθ=∑n=0∞qnn−1/2θnnq!=1+1−qθq∞.q-Gamma and q-Beta functions:For anyu>0(28)Γqu=∫0∞θu−1Eq−qθdqθis called q-Gamma Euler function and for any u,p>0(29)βqu,p=∫01θu−11−qθqp−1dqθis called q-Beta function.Relation betweenq-Gamma and q-Beta Function:
(30)βqθ,σ=ΓqθΓqσΓqθ+σ,(31)Γqm+1=mΓqmm>0.The following definition is introduced by Agarwal in [27] when ξ=0 and by Rajkovic et al. [28] for ξ≠0.q-Fractional integral from [29]. Let ψ∈L1ξ,η. The Riemann-Liouville q-integrals Jξ+βψ and Jη−βψ of order β>0 for ξ≥0 are defined by
(32)Jq,ξ+βψθ=1Γqβ∫ξθθ−quβ−1ψudqu,θ>ξ,Jq,η−βψθ=1Γqβ∫qθηu−qθβ−1ψudqu,θ<η,respectively, where Γqβ=∫0∞uβ−1Eq−qudqu. Here Jξ+0ψθ=Jη−0ψθ=ψθ.


## 3.1.q-Derivative

Forϕ∈Cξ,η, q-derivative of ϕ at θ∈ξ,η is given by
(12)Dξqϕθ=ϕθ−ϕqθ+1−qξ1−qθ−ξθ≠ξ.Forn≥1, we have the following relation:
(13)θ−ξqn=∏J=0nθ−qJξ,ξ−θqn=∏J=0nξ−qJθ.Respective derivatives are(14)Dqθ−ξqn=nθ−ξqn−1,Dqξ−θqn=−nξ−qθqn−1,ξ−qθqn=−1n+1Dqξ−θqn+1,where n=qn−1/q−1. The fractional calculus is a generalization of classical calculus concerned with operations of integration and differentiation of noninteger fractional order. The concept of fractional operators has been introduced almost simultaneously with the development of the classical ones. The first known reference can be found in the correspondence of G. W. Leibniz and Marquis de l’Hospital in 1695 where the question of meaning of the semiderivative has been raised. This question consequently attracted the interest of many well-known mathematicians, including Euler, Liouville, Laplace, Riemann, Grünwald, and Letnikov.


## 3.2. Fractional Integral from [8]

Letψ∈L1ξ,η. The Riemann-Liouville integrals Jξ+βψ and Jη−βψ of order β>0 for ξ≥0 are defined by
(15)Jξ+βψθ=1Γβ∫ξθθ−uβ−1ψudu,θ>ξ,Jη−βψθ=1Γβ∫θηu−θβ−1ψudu,θ<η.


## 3.3.q-Antiderivative

q-Antiderivative along with its properties can be studied in [22]. Suppose that ψ∈Cξ,η. Then q-definite integral for θ∈ξ,η is defined as
(16)∫ξθψuξdqu=1−qθ−ξ∑n=0∞qnψqnθ+1−qnξ,which gives
(17)∫ξ−θqβdqθ=−qξ−q−1θqβ+1β+1β≠−1,∫01uβ+sdqu=1β+s+1,∫01uβ1−usdqu=Γqβ+1Γqs+1Γqβ+s+2.Formula forq-integration by parts [23]: Let ψ,g∈Cξ,η, and θ∈ξ,η, and then
(18)∫cθψuξDqgudqu=ψθgθ−ψcgc−∫cθgqu+1−qξξDqψudqu.The followingq-integral inequality is from [23]:Theorem 11.
Supposeψ:ξ,η⟶ℝ is a q-differentiable mapping. If Dqψθ≤M for all θ∈ξ,η and 0<q<1, then
(19)ψθ−1η−ξ∫ξηψuξdqu≤M2q1+qθ−3q−1ξ+1+qη/4qη−ξ2+−q2+6q−18q1+qholds for all θ∈ξ,η.The least value of constant on R.H.S of (19) is −q2+6q−1/8q1+q.q-Hölder inequality [24]. Let ψ,Φ be q-integrable on ξ,η and 0<q<1 and 1/n+1/m=1 with m>1, and then we state as
(20)∫ξηψuΦudqu≤∫ξηψundqu1/n∫ξηΦumdqu1/m.The following inequalities are derived fromq-Hölder inequality:q-Minkowski’s inequality. Let ξ,η∈ℝ and n>1, for continuous functions ψ,Φ:ξ,η⟶ℝ, we have stated
(21)∫ξηψu+Φundqu1/n≤∫ξηψundqu1/n+∫ξηΦundqu1/n.q-Power mean inequality. Let 1/n+1/m=1 with n,m>1, and let ξ,η∈ℝ and for continuous functions ψ,g:ξ,η⟶ℝ, and then
(22)∫ξηψuΦudqu≤∫ξηψudqu1−1/m∫ξηψuΦumdqu1/m.Theorem 12 (see [25]).
Letψ:ξ,η⟶ℝ be a function and 0<q<1. Then
(23)∫01ψuη+1−uξdqu=1η−ξ∫ξηψtdqt.Example 1.
(24)∫ξθuu−ξdqu=1q+1θθ−ξ2−∫ξθu−ξ2dqu=1q+1θθ−ξ2−θ−ξ31+q+q2=θ−ξ21+qξ1+q+q2a1+q+q2.
Now, we have(25)∫ξθuu−ξdqu=∫0θ−ξt+ξtdqt=1q+1θθ−ξ2−∫0θ−ξqt2dqt=θ−ξ21+qξ1+q+q2a1+q+q2.Proposition 13.
For eachk,r∈ℕorℤq∈ℝ+ [2], we have
(26)k+rq=kq+qkrq.Exponential functions and Taylor series (q-analogues) from [26]:
(27)Eqθ=∑n=0∞qnn−1/2θnnq!=1+1−qθq∞.q-Gamma and q-Beta functions:For anyu>0(28)Γqu=∫0∞θu−1Eq−qθdqθis called q-Gamma Euler function and for any u,p>0(29)βqu,p=∫01θu−11−qθqp−1dqθis called q-Beta function.Relation betweenq-Gamma and q-Beta Function:
(30)βqθ,σ=ΓqθΓqσΓqθ+σ,(31)Γqm+1=mΓqmm>0.The following definition is introduced by Agarwal in [27] when ξ=0 and by Rajkovic et al. [28] for ξ≠0.q-Fractional integral from [29]. Let ψ∈L1ξ,η. The Riemann-Liouville q-integrals Jξ+βψ and Jη−βψ of order β>0 for ξ≥0 are defined by
(32)Jq,ξ+βψθ=1Γqβ∫ξθθ−quβ−1ψudqu,θ>ξ,Jq,η−βψθ=1Γqβ∫qθηu−qθβ−1ψudqu,θ<η,respectively, where Γqβ=∫0∞uβ−1Eq−qudqu. Here Jξ+0ψθ=Jη−0ψθ=ψθ.


## 4. Ostrowski Type Inequalities viaq-Fractional Integrals

In order to prove our main results, we have to prove the following lemma with the help of ([30], Lemma 1), which can be seen in [31]).Lemma 14.
Suppose thatψ:ξ,η⟶ℝ is q-differentiable mapping. If Dqψ∈Lξ,η, then for all θ∈ξ,η and β>0, we have
(33)θ−ξβ+1η−ξ∫01uβDqψuθ+1−uξdqu−η−θβ+1η−ξ∫01uβDqψuθ+1−uηdqu=θ−ββ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βψξ+Jq,θ+βψη,where Γqβ=∫0∞uβ−1Eq−qudqu.Proof.
By using the formula ofq-integration by parts
(34)∫01uβDqψuθ+1−uξdqu=uβψuθ+1−uξθ−ξ01−∫01βuβ−1ψquθ+1−quξθ−ξdqu=ψθθ−ξ−βθ−ξ∫ξqθ−qξ+ξu−qξqθ−ξβ−1ψuqθ−ξdqu=ψθθ−ξ+βqβθ−ξβ+1∫ξqθ−qξ+ξu−ξβ−1ψudqu,where
(35)∫ξqθ−qξ+ξu−ξβ−1ψudqu=qθ−qξ+ξ−ξ1−q∑n=0∞qnqnqθ−qξ+ξ+1−qnξ−ξβ−1×ψqnqθ−qξ+ξ+1−qnξ=ψθθ−ξ−βqθ−ξ1−qqβθ−ξβ+1q∑n=1∞qnqnθ−qnξβ−1ψqnθ+1−qnξ+βθ−ξθ−ξβ−11−qqβθ−ξβ+1ψθ−βθ−ξθ−ξβ−11−qqβθ−ξβ+1ψθ=ψθθ−ξ−βθ−ξ1−qqqθ−ξβ+1∑n=1∞qnqnθ−qnξβ−1ψqnθ+1−qnξ+βθ−ξθ−ξβ−11−qqβθ−ξβ+1ψθ+β1−qqβθ−ξψθ=ψθθ−ξ−βθ−ξ1−qqβθ−ξβ+1∑n=0∞qnqnθ−qnξβ−1ψqnθ+1−qnξ+β1−qqβθ−ξψθ.(36)∫01uβDqψuθ+1−uξdqu=ψθθ−ξ−βΓqβqβθ−ξβ+1Γqβ∫ξqθu−qξβ−1ψudqu+β1−qqβθ−ξψθ.
Multiply both sides of (36) by θ−ξβ+1/η−ξ(37)θ−ξβη−ξ∫01uβDqψuθ+1−uξdqu=θ−ξβη−ξψθ−Γqβ+1qβη−ξJq,θ−βξ+β1−qθ−ξβqβη−ξψθ.
Similar, calculation gives(38)∫01uβDqψuθ+1−uηdqu=ψθθ−η+βΓqβqβη−θβ+1Γqβ∫ηθη−quβ−1ψudqu−β1−qqβη−θψθ.
Multiply both sides of (38) by η−θβ+1/η−ξ to get
(39)η−θβ+1η−ξ∫01uβDqψuθ+1−uηdqu=ψθθ−η+βΓqβqβη−θβ+1Γqβ∫ηθη−quβ−1ψudqu−β1−qqβη−θψθ.
By combining (37) and (39), we obtain the following desired result.
(40)θ−ξβ+1η−ξ∫01uβDqψuθ+1−uξdqu−η−θβ+1η−ξ∫01uβDqψuθ+1−uηdqu=θ−ξβη−ξψθ−Γqβ+1qβη−ξJq,θ−βξ+β1−qθ−ξβqβη−ξψθ+η−θβη−ξψθ−Γqβ+1qβη−ξJq,θ+βη+β1−qη−θβqβη−ξψθ=θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βξ+Jq,θ+βη.Remark 15.
(a)
Takingq=1, Lemma 14 becomes Lemma 6(b)
Takingβ=1, Lemma 14 reduces to ([32], Lemma 3.1)(41)1qψθ−1η−ξ∫ξηψuadqu=θ−ξ2η−ξ∫01uDqψuθ+1−uξ0dqu−η−θ2η−ζ∫01uDqψuθ+1−uη0dqu.By using Lemma14, we established some Ostrowski type q-fractional integral inequalities.Theorem 16.
Supposeψ:ξ,η⊂ℝ+⟶ℝ is a q-differentiable mapping in such a way that Dqψ∈Lξ,η. If Dqψ is s−convex in second sense on ξ,η for some static s,q∈0,1 and Dqψθ≤M,θ∈ξ,η, subsequently, the following integral inequality for q-fractional integrals is valid.
(42)θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βψξ+Jq,θ+βψη≤Mη−ξ1+Γqβ+1Γqs+1Γqβ+s+1θ−ξβ+1+η−θβ+1β+s+1.Proof.
Consider Lemma14, and since Dqψ is s−convex mapping on ξ,η, we can write
(43)θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βξ+Jq,θ+βη≤θ−ξβ+1η−ξ∫01uβDqψuθ+1−uξdqu+η−θβ+1η−ξ∫01uβDqψuθ+1−uηdqu≤θ−ξβ+1η−ξ∫01uβ+sDqψθdqu+uβ1−usDqψξdqu+η−θβ+1η−ξ∫01uβ+sDqψθdqu+uβ1−usDqψηdqu≤θ−ξβ+1Mη−ξ∫01uβ+s+uβ1−usdqu+η−θβ+1Mη−ξ∫01uβ+s+uβ1−usdqu≤θ−ξβ+1Mη−ξ1β+s+1+Γqβ+1Γqs+1Γqβ+s+2+η−θβ+1Mη−ξ1β+s+1+Γqβ+1Γqs+1Γqβ+s+2=Mη−ξ1β+s+1+Γqβ+1Γqs+1Γqβ+s+2θ−ξβ+1+η−θβ+1=Mη−ξ1+Γqβ+1Γqs+1Γqβ+s+1θ−ξβ+1+η−θβ+1β+s+1,where we have used the fact that
(44)∫01uβ+sdqu=1β+s+1,∫01uβ1−usdqu=Γqβ+1Γqs+1Γqβ+s+2.
Therefore by applying the reduction formulaΓqm+1=mΓqmn>0 for Euler gamma function, it completes the proof.Corollary 17.
Suppose thatψ:ξ,η⊂ℝ+⟶ℝ, is q-differentiable function in such a way that Dqψ∈Lξ,η. If Dqψ is convex on ξ,η for some fixed s,q∈0,1 and Dqψθ≤M,θ∈ξ,η, then we have the following q-fractional integral inequality
(45)θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βψξ+Jq,θ+βψη≤Mη−ξθ−ξβ+1+η−θβ+1β+1is valid for β>0.Proof.
Takings=1 in (42), the required result follows.Remark 18.
(a)
q-analogue of Theorem 3, (4) is followed by taking β=1 and q=1 in (42)(b)
Takingq=1 in (42), it follows the inequality (8) of Theorem 7Theorem 19.
Suppose thatψ:ξ,η⊂ℝ+⟶ℝ is q-differentiable function in such a way that Dqψ∈Lξ,η. If Dqψm is s-convex in second sense on ξ,η for some static s,q∈0,1, n,m>1 and Dqψθ≤M,θ∈ξ,η, then
(46)θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βψξ+Jq,θ+βψη≤M1+nβ1/n1+q1−1−q−1s+1s+11/mθ−ξβ+1+η−θβ+1η−ξ,where 1/n+1/m=1, and β>0.Proof.
From Lemma14,
(47)θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βξ+Jq,θ+βη≤θ−ξβ+1η−ξ∫01uβDqψuθ+1−uξdqu+η−θβ+1η−ξ∫01uβDqψuθ+1−uηdqu.
By virtue of Hölder’s inequality, we have(48)θ−ξβ+1η−ξ∫01uβDqψuθ+1−uξdqu+η−θβ+1η−ξ∫01uβDqψuθ+1−uηdqu≤θ−ξβ+1η−ξ∫01uβndqu1/n∫01Dqψuθ+1−uξmdqu1/m+η−θβ+1η−ξ∫01uβndqu1/n∫01Dqψuθ+1−uηmdqu1/m.
SinceDqψm is s−convex in the second sense on ξ,η and Dqψθ is bounded by number M,
(49)∫01uβndqu=1βn+1(50)∫01Dqψuθ+1−uξmdqu≤∫01usDqψθmdqu+∫011−usDqψξmdqu≤∫01usdqu+∫011−usdqu=Mmus+1s+101+−q1−q−1us+1s+101=Mm1s+1−q1−q−1s+1s+1+qs+1=Mm1+q−q1−q−1s+1s+1.
Similarly, we have(51)∫01Dqψuθ+1−uηmdqu≤∫01usDqψθmdqu+∫011−usDqψηmdqu,(52)∫01Dqψuθ+1−uηmdqu≤Mm1+q1−1−q−1s+1s+1.
Substitute (49), (50), and (52) in (48) to get,
(53)θ−ξβ+1η−ξ1βn+11/nMm1+q−q1−q−1s+1s+11/m+η−θβ+1η−ξ1βn+11/n1+q−q1−q−1s+1s+11/m=Mβn+11/n1+q1−1−q−1s+1s+11/mθ−ββ+1+η−θβ+1η−ξ.
Hence, this completes the proof.Corollary 20.
SupposeDqψ∈Lξ,η. If Dqψm is convex on ξ,η, for some static s,q∈0,1, m,n>1, and Dqψθ≤M,θ∈ξ,η, then subsequently, we have
(54)θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βψξ+Jq,θ+βψη≤M1+nβ1/nθ−ξβ+1+η−θβ+1η−ξ,where 1/n+1/m=1 and β>0.Proof.
Takings=1 in (46), the required result follows.Remark 21.
(a)
Takingβ=1 and q=1 in (46), it follows the q-analogue of (5), Theorem 4(b)
Takingq=1 in (46), it follows the inequality (9) of Theorem 8Theorem 22.
Supposeψ:ξ,η⊂ℝ+⟶ℝ is q-differentiable mapping in such a way that Dqψ∈Lξ,η. If Dqψm is s−convex in second sense on ξ,η for some static s,q∈0,1, m≥1, and Dqψθ≤M,θ∈ξ,η, then for β>0, we have
(55)θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βξ+Jq,θ+βη≤M1β+11−1/m1β+s+11/m1+Γqβ+1Γqs+1Γqβ+s+11/mθ−ξβ+1+η−θβ+1η−ξ.Proof.
From Lemma14,
(56)θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βψξ+Jq,θ+βψη≤θ−ξβ+1η−ξ∫01uβDqψuθ+1−uξdqu+η−θβ+1η−ξ∫01uβDqψuθ+1−uηdqu.
Now applying familiar power mean inequality, we have(57)θ−ξβ+1η−ξ∫01uβDqψuθ+1−uξdqu+η−θβ+1η−ξ∫01uβDqψuθ+1−uηdqu≤θ−ξβ+1η−ξ∫01uβdqu1−1/m∫01uβDqψuθ+1−uξmdqu1/m+η−θβ+1η−ξ∫01uβ1−1/mdqu∫01Dqψuθ+1−uηmdqu1/m.
SinceDqψm is s-convex in the second sense on ξ,η and Dqψθ≤M, we get
(58)∫01uβDqψuθ+1−uξmdqu≤∫01uβ+sDqψθmdqu+∫01uβ1−usDqψξmdqu≤Mm∫01uβ+sdqu+Mm∫01uβ1−usdqu=Mm∫01uβ+sdqu+∫01uβ1−usdqu,∫01uβ+sdqu=1β+s+1,∫01uβdqu=1β+1,∫01uβ1−usdqu=Γqβ+1Γqs+1Γqβ+s+2=Γqβ+1Γqs+1β+s+1Γqβ+s+1,∫01uβDqψuθ+1−uξmdqu≤Mmβ+s+11+Γqβ+1Γqs+1Γqβ+s+1,∫01uβDqψuθ+1−uηmdqu≤∫01uβ+sDqψθmdqu+∫01uβ1−usDqψηmdqu≤Mmβ+s+11+Γqβ+1Γqs+1Γqβ+s+1.
By using (30), we have
(59)∫01uβDqψuθ+1−uηmdqu≤θ−ξβ+1η−ξ1β+11−1/mMmβ+s+11+Γqβ+1Γqs+1Γqβ+s+11/m+η−θβ+1η−ξ1β+11−1/mMmβ+s+11+Γqβ+1Γqs+1Γqβ+s+11/m=M1β+11−1/m1β+s+11/m1+Γqβ+1Γqs+1Γqβ+s+11/mθ−ξβ+1+η−θβ+1η−ξ,which completes the proof.Corollary 23.
SupposeDqψ∈Lξ,η. If Dqψm is convex on ξ,η and for some static s,q∈0,1, m≥1, and Dqψθ≤M,θ∈ξ,η, then for β>0, we have
(60)θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βψξ+Jq,θ+βψη≤M1β+1θ−ξβ+1+η−θβ+1η−ξ.Proof.
Takings=1 in (55), the required result follows.Remark 24.
(a)
Takingβ=1 and q=1, in (55), it follows the q-analogue of (6), Theorem 5(b)
Takingq=1, in (55), it follows formula (10) of Theorem 9Theorem 25.
Suppose thatψ:ξ,η⊂ℝ+⟶ℝ is q-differentiable mapping and Dqψ∈Lξ,η. If Dqψm is s-convex in second sense on ξ,η for some static s∈0,1 and m,n>1, therefore, the following integral inequality for q-fractional integrals is valid.
(61)θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βψξ+Jq,θ+βψη≤2s−1/m1+nβ1/nη−ξθ−ξβ+1Dqψθ+ξ2+η−θβ+1Dqψη+θ2,where 1/n+1/m=1, and β>0.Proof.
From Lemma14 and keeping the familiar Hölder inequality in use, it follows
(62)θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βξ+Jq,θ+βη≤θ−ξβ+1η−ξ∫01uβDqψuθ+1−uξdqu+η−θβ+1η−ξ∫01uβDqψuθ+1−uηdqu≤θ−ξβ+1η−ξ∫01uβn1/ndqu∫01Dqψuθ+1−uξmdqu1/m+η−θβ+1η−ξ∫01uβn1/ndqu∫01Dqψuθ+1−uηmdqu1/m.
SinceDqψm is s−concave, we have
(63)∫01Dqψuθ+1−uξmdu≤2s−1Dqψθ+ξ2m,∫01Dqψuθ+1−uηmdu≤2s−1Dqψη+θ2m.
Therefore,(64)θ−ξβ+1η−ξ∫01uβDqψuθ+1−uξdqu+η−θβ+1η−ξ∫01uβDqψuθ+1−uηdqu≤θ−ξβ+1η−ξ1βn+11/n2s−1Dqψθ+ξ2m1/m+η−θβ+1η−ξ1βn+11/n2s−1Dqψη+θ2m1/m=2s−1/mβn+11/nη−ξθ−ξβ+1Dqψθ+ξ2+η−θβ+1Dqψη+θ2,which completes the proof.Remark 26.
Takingq=1 in (61), it follows the inequality (11) of Theorem 10.


## 5. Applications and Examples

Example 2.
Letψu=1−u and q∈0,1, and we fixed θ=1/2; ξ=0, η=1; β=1; q=1/2, and s=1, and then we get verification of Theorem 16.
(65)1/2−01+1−1/211−01/21+11−1/21/2ψ12−Γ1/221/21J1/2,1/2−1ψ0+J1/2,1/2+1ψ1≤11−01+Γ1/22Γ1/22Γ1/231/2−02+1−1/223,0.5+0.520.5+0.512−21Γ1/21∫01/21−udqu≤1+230.51.75,1−23≤1.66670.2857,0.3333≤0.4762.For anyq∈0,1 and s∈0,1 result holds.Example 3.
Letψu=1−u and q∈0,1, and we fixed θ=1/2;ξ=0,η=1;β=1;q=1/2;m=n=1/2, and s=1, and then we get verification of Theorem 19.
(66)1/2−01+1−1/211−01/21+11−1/21/2ψ12−Γ1/221/21J1/2,1/2−1ψ0+J1/2,1/2+1ψ1≤11+1/21/21+1/21−1−1/1/21/2220.51/2−02+1−1/221−0,0.5+0.520.5+0.512−21Γ1/21∫01/21−udqu+1Γ1/21∫1/211−udqu≤1−q1−q3/20.51+.51−0q+10.50.5,1−23≤0.51−0.35360.53/21+1/20.50.5,0.3333≤0.77350.50.5,0.3333≤0.4398.For anyq∈0,1 and s∈0,1 result holds.Example 4.
Letψu=1−u and q∈0,1, and we fixed θ=1/2;ξ=0,η=1;β=1;q=1/2;m=n=1/2, and s=1, and then we get verification of Theorem 22.
(67)1/2−01+1−1/211−01/21+11−1/21/2ψ12−Γ1/221/21J1/2,1/2−1ψ0+J1/2,1/2+1ψ1≤1121/2131/21+1Γ1/231/21/2−02+1−1/221−0,0.5+0.520.5+0.512−21Γ1/21∫01/21−udqu+1Γ1/21∫1/211−udqu≤11/2+10.511+1/2+1/220.51+231/20.5,1−23≤230.511.750.51.66670.50.5,0.3333≤0.66670.50.54140.51.66670.50.5,0.3333≤0.3984.For anyq∈0,1 and s∈0,1 result holds.Example 5.
Letψu=1−u and q∈0,1, and we fixed θ=1/2;ξ=0,η=1;β=1;q=1/2; and s=1, and then we get verification of Theorem 25.
(68)1/2−01+1−1/211−01/21+11−1/21/2ψ12−Γ1/221/21J1/2,1/2−1ψ0+J1/2,1/2+1ψ1≤11/2+10.5122−1+122−1,0.5+0.520.5+0.512−21Γ1/21∫01/21−udqu+1Γ1/21∫1/211−udqu≤1−q1−q3/20.5122−1+122−1,1−23≤0.51−0.35360.50.5,0.3333≤0.4398.Result holds for several example such as everyq∈0,1 and ψu=1−un,ψs=un,ψu=u1−un,ψu=u−ξ,ψu=u−ξn,ψs=ξ−u, and ψu=ξ−unn∈ℝ+ have good approximation. At the end, we compare Theorems 16, 19, and 22. We have seen that Theorem 19 has good approximation than Theorem 16, and Theorem 22 has better approximation than Theorems 16 and 19.
(69)θ−ξβ+η−θβη−ξqβ+β1−qqβψθ−Γqβ+1qβη−ξJq,θ−βψξ+Jq,θ+βψη≤M1β+11−1/m1β+s+11/m1+Γqβ+1Γqs+1Γqβ+s+11/mθ−ξβ+1+η−θβ+1η−ξ,≤M1+nβ1/n1+q1−1−q−1s+1s+11/mθ−ξβ+1+η−θβ+1η−ξ≤Mη−ξ1+Γqβ+1Γqs+1Γqβ+s+1θ−ξβ+1+η−θβ+1β+s+1.


## 6. Conclusion

The major goal of this paper is to prove fractional quantum integral identities in order to establish some new quantum Ostrowski type inequalities involvingq-fractional integrable inequalities. By the virtue of discrete fractional q-calculus, Ostrowski type inequalities are generalized for q-fractional integrals, which provide a method to study some properties of q-fractional integrals via other classes of integral inequalities. Similar method can be applied to other inequalities, like Sipmson’s and Newton’s type inequalities.



---

*Source: 9742133-2022-10-13.xml*