# Fractional Ostrowski Type Inequalities via Generalized Mittag–Leffler Function

Authors: Xinghua You; Ghulam Farid; Kahkashan Maheen
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

If we study the theory of fractional differential equations then we notice the Mittag–Leffler function is very helpful in this theory. On the contrary, Ostrowski inequality is also very useful in numerical computations and error analysis of numerical quadrature rules. In this paper, Ostrowski inequalities with the help of generalized Mittag–Leffler function are established. In addition, bounds of fractional Hadamard inequalities are given as straightforward consequences of these inequalities.


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## Body

## 1. Introduction

Exponential function plays a vital role in the theory of integer order differential equations. The symbolEαz is well known as the Mittag–Leffler function and it is a generalization of exponential function. It occurs in the solutions of fractional differential equations such as exponential function which exists in the solutions of differential equations. Due to its importance, Mittag–Leffler function is generalized by many mathematicians: For example, Wiman [1], Prabhakar [2], Shukla and Prajapati [3], Salim [4], Salim and Faraj [5], and Rahman et al. [6]. Mittag–Leffler function is also used in the formation of fractional integral operators. These fractional integral operators provide generalizations of fractional differential equations and modeling of dynamic systems. Fractional integral operators also play a vital role in the advancement of classical mathematical inequalities. For example, Hadamard inequality, Ostrowski inequality, Gruss inequality, and many others have been presented for fractional integral and derivative operators, see [7–16]. The aim of this paper is to study well-known Ostrowski inequality for an integral operator which is directly associated with many fractional integral operators defined in near past.Recently, in [7], Andrić et al. defined the extended generalized Mittag–Leffler function Eμ,σ,lγ,δ,k,c.;p as follows.Definition 1.
Letμ,α,l,γ,c∈ℂ, ℜμ,ℜα,ℜl>0, and ℜc>ℜγ>0 with p≥0, δ>0, and 0<k≤δ+ℜμ. Then, the extended generalized Mittag–Leffler function Eμ,α,lγ,δ,k,ct;p is defined as(1)Eμ,α,lγ,δ,k,ct;p=∑n=0∞βpγ+nk,c−γβγ,c−γcnkΓμn+αtnlnδ,where βp is the generalized beta function defined as βpx,y=∫01tx−11−ty−1e−p/t1−tdt and cnk is the Pochhammer symbol given by cnk=Γc+nk/Γc.
The corresponding left- and right-sided generalized fractional integralsϵμ,α,l,ω,a+γ,δ,k,c and ϵμ,α,l,ω,b−γ,δ,k,c are defined as follows.Definition 2 (see [7]).
Letω,μ,α,l,γ,c∈ℂ, ℜμ,ℜα,ℜl>0, ℜc>ℜγ>0 with p≥0, δ>0, and 0<k≤δ+ℜμ. Let ψ1∈L1a,b and x∈a,b. Then, the generalized fractional integrals ϵμ,α,l,ω,a+γ,δ,k,cψ1 and ϵμ,α,l,ω,b−γ,δ,k,cψ1 are defined as(2)ϵμ,α,l,ω,a+γ,δ,k,cψ1x;p=∫axx−tα−1Eμ,α,lγ,δ,k,cωx−tμ;pψ1tdt,(3)ϵμ,α,l,ω,b−γ,δ,k,cψ1x;p=∫xbt−xα−1Eμ,α,lγ,δ,k,cωt−xμ;pψ1tdt.
Recently, Farid defined a unified integral operator in [17] (also see [18]). This unifies several kinds of fractional and conformable integrals in a compact formula and is given as follows.Definition 3.
Letψ1,ψ2:a,b⟶ℝ, 0<a<b, be the functions such that ψ1 be positive and ψ1∈L1a,b, and ψ2 be differentiable and strictly increasing. Also, let ϕ/x be an increasing function on a,∞ and α,l,γ,c∈ℂ, ℜα,ℜl>0, ℜc>ℜγ>0, p,μ,δ≥0, and 0<k≤δ+μ. Then, for x∈a,b the left and right integral operators are defined by(4)Fμ,α,l,a+ϕ,γ,δ,k,cgψ1x,ω;p=∫axKxyEμ,α,lγ,δ,k,c,g;ϕψ1ydψ2y,(5)Fμ,α,l,b−ϕ,γ,δ,k,cgψ1x,ω;p=∫xbKyxEμ,α,lγ,δ,k,c,g;ϕψ1ydψ2y,where KxyEμ,α,lγ,δ,k,c,g;ϕ=ϕψ2x−ψ2y/ψ2x−ψ2yEμ,α,lγ,δ,k,cωψ2x−ψ2yμ;p.
The following definition can be deduced from Definition3 (see [16]).Definition 4.
Letψ1,ψ2:a,b⟶ℝ, 0<a<b, be the functions such that ψ1 be positive and ψ1∈L1a,b, and ψ2 be differentiable and strictly increasing and α,l,γ,c∈ℝ+,c>γ, p,μ,δ≥0 and 0<k≤δ+μ. Then, for x∈a,b the left and right integral operators are defined by(6)ϒμ,α,l,ω,a+γ,δ,k,cψ2ψ1x;p=∫axψ2x−ψ2tα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2tμ;p×ψ1tψ2′tdt,(7)ϒμ,α,l,ω,b−γ,δ,k,cψ2ψ1x;p=∫xbψ2t−ψ2xα−1Eμ,α,lγ,δ,k,cωψ2t−ψ2xμ;p×ψ1tψ2′tdt.
It can be noted that(8)ϒμ,α,l,ω,a+γ,δ,k,cψ21x;p=ψ2x−ψ2tα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2tμ;p,(9)ϒμ,α,l,ω,b−γ,δ,k,cψ21x;p=ψ2t−ψ2xα−1Eμ,α,lγ,δ,k,cωψ2t−ψ2xμ;p.
In the following, we state the Ostrowski inequality which is proved by Ostrowski [19] in 1938.Theorem 1.
Letψ1:I⟶ℝ, where I is an interval in ℝ, be a mapping differentiable in Io, the interior of I and a,b∈Io, a<b. If ψ1′t≤M for all t∈a,b, then for x∈a,b we have(10)ψ1x−1b−a∫abψ1tdt≤14+x−a+b/22b−a2b−aM.The Ostrowski inequality has been studied by many researchers to obtain its refinements, generalizations, and extensions. Also, their applications are analyzed for establishing the bounds of relations among special means and for estimations of numerical quadrature rules. For recent developments of Ostrowski inequality, we refer the reader to [8, 9, 11, 20–26] and references therein.In Section2, fractional version of Ostrowski inequalities with the help of Mittag–Leffler function has been established. The presented results may be useful in the study of fractional integral operators and their applications. Also, the error bounds of fractional Hadamard inequalities are presented in Section 3.


## 2. Main Results

First, we establish the following lemma for extended generalized Mittag–Leffler function.Lemma 1.
Ifω,μ,α,l,γ,c∈C,ℜμ,ℜα,ℜl>0,ℜc>ℜγ>0 with p≥0,δ>0 and 0<k<δ+ℜμ, then(11)ddtψ2tα−1Eμ,α,lγ,δ,k,cωψ2tμ;p=ψ2′tψ2tα−2Eμ,α−1,lγ,δ,k,cωψ2tμ;p.Proof.
We have(12)ddtψ2tα−1Eμ,α,lγ,δ,k,cωψ2tμ;p=∑n=0∞βpγ+nk,c−γβγ,c−γcnkΓμn+αωnμn+α−1ψ2tμn+α−2ψ2′tlnδ=∑n=0∞βpγ+nk,c−γβγ,c−γcnkΓμn+α−1ωnψ2tμn+α−2ψ2′tlnδ.
After simple computation, one can obtain (11).
Next, we give the generalized fractional Ostrowski type inequality containing extended generalized Mittag–Leffler function.Theorem 2.
Letψ1:I⟶ℝ, where I is an interval in ℝ, be a mapping differentiable in Io, the interior of I and a,b∈Io, a<b. If ψ1 is an integrable function, ψ1′ψ2t≤M for all t∈a,b and ψ2:a,b⟶ℝ be an increasing and positive function on a,b, having continuous derivative ψ2′ on a,b, then for α,β≥1, the following inequality for fractional integrals (6) and (7) holds:(13)ψ1ψ2xψ2b−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p+ψ2x−ψ2aα−1×Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ϒμ,α−1,l,ω,a+γ,δ,k,cψ2ψ1∘ψ2x;p+ϒμ,β−1,l,ω,b−γ,δ,k,cψ2ψ1∘ψ2x;p≤Mψ2x−ψ2aα×Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p+ψ2b−ψ2xβEμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p−ψ2ϒμ,α−1,l,ω,a+γ,δ,k,c1x;p+ϒμ,β−1,l,ω,b−γ,δ,k,cψ21x;p.Proof.
Letx∈a,b, t∈a,x, and α≥1. Then, the following inequality holds for the monotonically increasing function ψ2 and the Mittag–Leffler function (1):(14)ψ2x−ψ2tα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2tμ;pψ2′t≤ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;pψ2′t.
From (14) and given condition of boundedness of ψ1′, one can have the following integral inequalities:(15)∫axM−ψ1′ψ2tψ2x−ψ2tα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2tμ;pψ2′tdt≤ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p∫axM−ψ1′ψ2tψ2′tdt,(16)∫axM+ψ1′ψ2tψ2x−ψ2tα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2tμ;pψ2′tdt≤ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,c0ωψ2x−ψ2aμ;p∫axM+ψ1′ψ2tψ2′tdt.
First, we consider inequality (15) as follows:(17)M∫axψ2x−ψ2tα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2tμ;pψ2′tdt−∫axψ2x−ψ2tα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2tμ;pψ1′ψ2tψ2′tdt≤ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p∫axM−ψ1′ψ2tψ2′tdt.
Therefore, (17) takes the following form after integrating by parts and using derivative property (11) and a simple computation:(18)ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;pψ1ψ2x−ϒψ2μ,α−1,l,ω,a+γ,δ,k,cψ1∘ψ2x;p≤Mψ2x−ψ2aαEμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ϒψ2μ,α−1,l,ω,a+γ,δ,k,c1x;p.
Similarly, adopting the same pattern from (16), one can obtain(19)ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;pψ1ψ2x−ϒψ2μ,α−1,l,ω,a+γ,δ,k,cψ1∘ψ2x;p≥−Mψ2x−ψ2aαEμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ϒψ2μ,α−1,l,ω,a+γ,δ,k,c1x;p.
From (18) and (19), the following inequality is obtained:(20)ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;pψ1ψ2x−ψ2ϒμ,α−1,l,ω,a+γ,δ,k,cψ1∘ψ2x;p≤Mψ2x−ψ2aαEμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ψ2ϒμ,α−1,l,ω,a+γ,δ,k,c1x;p.
Now, on the contrary, we letx∈a,b, t∈x,b, and β≥1. Then, the following inequality holds for Mittag–Leffler function:(21)ψ2t−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2t−ψ2xμ;pψ2′t≤ψ2b−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;pψ2′t.
From (21) and the condition of boundedness of ψ1′, one can have the following integral inequalities:(22)∫xbM−ψ1′ψ2tψ2t−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2t−ψ2xμ;pψ2′tdt≤ψ2b−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p∫xbM−ψ1′ψ2tψ2′tdt,(23)∫xbM+ψ1′ψ2tψ2t−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2t−ψ2xμ;pψ2′tdt≤ψ2b−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p∫xbM+ψ1′ψ2tψ2′tdt.
Following the same procedure as we did for (15) and (16), one can obtain from (22) and (23) the following modulus inequality:(24)ψ2b−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;pψ1ψ2x−ψ2ϒμ,β−1,l,ω,b−γ,δ,k,cψ1∘ψ2x;p≤Mψ2b−ψ2xβEμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p−ψ2ϒμ,β−1,l,ω,b−γ,δ,k,c1x;p.
Inequalities (20) and (24) give (13) which is the required inequality.
In the following, we give direct consequences of above theorem.Corollary 1.
If we putα=β in (13), then we get the following fractional integral inequality:(25)ψ1ψ2xψ2b−ψ2xα−1Eμ,α,lγ,δ,k,cωψ2b−ψ2xμ;p+ψ2x−ψ2aα−1×Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ψ2ϒμ,α−1,l,ω,a+γ,δ,k,cψ1∘ψ2x;p+ψ2ϒμ,α−1,l,ω,b−γ,δ,k,cψ1∘ψ2x;p≤Mψ2x−ψ2aα×Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p+ψ2b−ψ2xαEμ,α,lγ,δ,k,cωψ2b−ψ2xμ;p−ψ2ϒμ,α−1,l,ω,a+γ,δ,k,c1x;p+ψ2ϒμ,α−1,l,ω,b−γ,δ,k,c1x;p.Remark 1.
(i)
If we putψ2x=x in (13), then we obtain Theorem 5 in [9](ii)
If we putω=p=0 and ψ2x=x in (13), then we obtain Theorem 1 in [6](iii)
If we putα=β=1, ψ2x=x, and ω=p=0 in (13), then we obtain Ostrowski inequality (10)(iv)
If we putψ2x=x in (25), then we obtain Corollary 1 in [9]
The next result is a general form of fractional Ostrowski inequality containing generalized Mittag–Leffler function.Theorem 3.
Letψ1:I⟶ℝ, where I is an interval in ℝ, be a mapping differentiable in Io, the interior of I and a,b∈Io, a<b. If ψ1 is integrable function and m<ψ1′ψ2t≤M for all t∈a,b and ψ2:a,b⟶ℝ be an increasing and positive function on a,b, having continuous derivative ψ2′ on a,b, then, for α,β≥1, the following inequalities for fractional integrals (6) and (7) hold:(26)ψ1ψ2xψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ψ2b−ψ2xβ−1×Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p−ψ2ϒμ,α−1,l,ω,a+γ,δ,k,cψ1∘ψ2x;p−ψ2ϒμ,β−1,l,ω,b−γ,δ,k,cψ1∘ψ2x;p≤Mψ2x−ψ2aα×Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ψ2ϒμ,α−1,l,ω,a+γ,δ,k,c1x;p−mψ2b−ψ2xβ×Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p−ψ2ϒμ,β−1,l,ω,b−γ,δ,k,c1x;p,(27)ψ1ψ2xψ2b−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p−ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ψ2ϒμ,α−1,l,ω,a+γ,δ,k,cψ1∘ψ2x;p−ψ2ϒμ,β−1,l,ω,b−γ,δ,k,cψ1∘ψ2x;p≤Mψ2b−ψ2xβ×Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p−ψ2ϒμ,β−1,l,ω,b−γ,δ,k,c1x;p−mψ2x−ψ2aαEμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ψ2ϒμ,α−1,l,ω,a+γ,δ,k,c1x;p.Proof.
The proof is similar to the proof of Theorem2, just after comparing conditions on derivative of ψ1, so we left it for the reader.
Some comments on the abovementioned result are given as follows.Remark 2.
(i)
If we putω=p=0 and ψ2x=x in (26) and (27), then we obtain Theorem 1 in [6](ii)
If we putm=−M in Theorem 3, then with some rearrangements we obtain Theorem 2(iii)
If we putψ2x=x in (26) and (27), then we obtain Theorem 6 in [9]
In the following, we have established a result related to fractional Ostrowski inequality containing generalized Mittag–Leffler function.Theorem 4.
Letψ1:I⟶ℝ, where I is an interval in ℝ, be a mapping differentiable in Io, the interior of I and a,b∈Io, a<b. If ψ1 is integrable function, ψ1′ψ2t≤M for all t∈a,band and ψ2:a,b⟶ℝ be an increasing and positive function on a,b, then, for α,β≥1, the following inequality for fractional integrals (6) and (7) holds:(28)ψ1ψ2bψ2b−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p+ψ1ψ2aψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ψ2Υμ,β−1,l,ω,x+γ,δ,k,cψ1°ψ2b;p−ψ2Υμ,α−1,l,ω,x−γ,δ,k,cψ1°ψ2a;p≤Mψ2a−ψ2xα×Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p+ψ2b−ψ2xβEμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p−ψ2ϒμ,α−1,l,ω,x−γ,δ,k,c1a;p+ψ2ϒμ,β−1,l,ω,x+γ,δ,k,c1b;p.Proof.
Letx∈a,b, t∈a,x, and α≥1. Then, the following inequality holds true for Mittag–Leffler function:(29)ψ2t−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2t−ψ2aμ;pψ2′t≤ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;pψ2′t.
From (29) and given condition of boundedness on ψ1′, one can have the following integral inequalities:(30)∫axM−ψ1′ψ2tψ2t−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2t−ψ2aμ;pψ2′tdt≤ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p∫axM−ψ1′ψ2tψ2′tdt,(31)∫axM+ψ1′ψ2tψ2t−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2t−ψ2tμ;pψ2′tdt≤ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p∫axM+ψ1′ψ2tψ2′tdt.
First, we consider inequality (30) as follows:(32)M∫axψ2t−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2t−ψ2aμ;pψ2′tdt−∫axψ2t−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2t−ψ2aμ;pψ2′tψ1′ψ2tdt≤ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p∫axM−ψ1′ψ2tψ2′tdt.
Therefore, (32) takes the following form after integrating by parts and using derivative property (11) and a simple computation(33)ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;pψ1ψ2a−ψ2ϒμ,α−1,l,ω,x−γ,δ,k,cψ1∘ψ2a;p≤Mψ2x−ψ2aα×Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ψ2ϒμ,α−1,l,ω,x−γ,δ,k,c1a;p.
Similarly, adopting the same pattern from (31), one can obtain(34)ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;pψ1ψ2a−ψ2ϒμ,α−1,l,ω,x−γ,δ,k,cψ1∘ψ2a;p≥−Mψ2x−ψ2aα×Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ψ2ϒμ,α−1,l,ω,x−γ,δ,k,c1a;p.
From (33) and (34), the following inequality is obtained:(35)ψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;pψ1ψ2a−ψ2ϒμ,α−1,l,ω,x−γ,δ,k,cψ1∘ψ2a;p≤Mψ2x−ψ2aα×Eμ,α,lγ,δ,k,cωψ2x−ψ2aαμ;p−ψ2Υμ,α−1,l,ω,x−γ,δ,k,c1a;p.
Now, on the contrary, we letx∈a,b, t∈x,b, and β≥1. Then, the following inequality holds for Mittag–Leffler function:(36)ψ2b−ψ2tβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2tμ;pψ2′t≤ψ2b−ψ2xβ−1×Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;pψ2′t.
From (36) and given condition of boundedness of ψ1′, one can have the following integral inequalities:(37)∫xbM−ψ1′ψ2tψ2b−ψ2tβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2tμ;pψ2′tdt≤ψ2b−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p∫xbM−ψ1′ψ2tψ2′tdt,(38)∫xbM+ψ1′ψ2tψ2b−ψ2tβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2tμ;pψ2′tdt≤ψ2b−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p∫xbM+ψ1′ψ2tψ2′tdt.
Following the same procedure as we did for (30) and (31), one can obtain from (37) and (38) the following modulus inequality:(39)ψ2b−ψ2xβ−1Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;pψ1ψ2b−ψ2ϒμ,β−1,l,ω,x+γ,δ,k,cψ1∘ψ2b;p≤Mψ2b−ψ2xβ×Eμ,β,lγ,δ,k,cωψ2b−ψ2xμ;p−ψ2ϒμ,β−1,l,ω,x+γ,δ,k,c1b;p.
Inequalities (35) and (39) give (28) which is required inequality.
Some direct consequences of the above theorem are given below.Corollary 2.
If we putα=β in (28), then we get the following fractional integral inequality:(40)ψ1ψ2bψ2b−ψ2xα−1Eμ,α,lγ,δ,k,cωψ2b−ψ2xμ;p+ψ1ψ2aψ2x−ψ2aα−1Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p−ψ2ϒμ,α−1,l,ω,x+γ,δ,k,cψ1∘ψ2b;p+ψ2ϒμ,α−1,l,ω,x−γ,δ,k,cψ1∘ψ2a;p≤Mψ2x−ψ2aα×Eμ,α,lγ,δ,k,cωψ2x−ψ2aμ;p+ψ2b−ψ2xαEμ,α,lγ,δ,k,cωψ2b−ψ2xμ;p−ψ2ϒμ,α−1,l,ω,x−γ,δ,k,c1a;p+ψ2ϒμ,α−1,l,ω,x+γ,δ,k,c1b;p.Remark 3.
(i)
If we putψ2x=x in (28), then we obtain Theorem 5 in [9](ii)
If we putω=p=0 and ψ2x=x in (28), then we obtain Theorem 1 in [6](iii)
If we putψ2x=x in (40), then we obtain Corollary 2 in [9]


## 3. Applications

In this section, we just describe some applications of Theorem4 and leave such applications of other results for the reader. By applying Theorem 4 at end points of the interval a,b and adding the resulting inequalities, one obtains the error bounds of compact form of the fractional Hadamard inequality.Theorem 5.
Under the assumptions of Theorem4, the following estimation of Hadamard inequality can be obtained:(41)ψ1ψ2b+ψ1ψ2b2ψ2ϒμ,α−1,l,ω,a+γ,δ,k,c1b;p−12ψ2ϒμ,α−1,l,ω,b−γ,δ,k,cψ1∘ψ2a;p+ψ2ϒμ,α−1,l,ω,a+γ,δ,k,cψ1∘ψ2b;p≤M2ψ2b−ψ2aαEμ,α,lγ,δ,k,cωψ2b−ψ2aμ;p−M2ψ2ϒμ,α−1,l,ω,b−γ,δ,k,c1a;p+ψ2ϒμ,α−1,l,ω,a+γ,δ,k,c1b;p.Proof.
By puttingx=a, α=β, and x=b in (40) then adding the resulting inequalities, we obtain(42)ψ1ψ2b+ψ1ψ2bψ2b−ψ2aaα−1Eμ,α,lγ,δ,k,cωψ2b−ψ2aμ;p−ψ2ϒμ,α−1,l,ω,b−γ,δ,k,cψ1∘ψ2a;p+ψ2ϒμ,α−1,l,ω,a+γ,δ,k,cψ1∘ψ2b;p≤Mψ2b−ψ2aαEμ,α,lγ,δ,k,cωψ2b−ψ2aμ;p−ψ2ϒμ,α−1,l,ω,b−γ,δ,k,c1a;p+ψ2ϒμ,α−1,l,ω,a+γ,δ,k,c1b;p.
Multiplying both sides of the above inequality by1/2 and using (8) and (9), inequality (41) can be obtained.Remark 4.
If in (41) α is replaced by α+1 and ω by ω′=ω/gb−gaμ, then we get an error bound of the Hadamard inequality given in Theorem 1 in [20].Theorem 6.
Under the assumptions of Theorem4, the following inequality can be obtained:(43)ψ1ψ2bψ2ϒμ,α−1,l,ω,b−γ,δ,k,c1a+b2;p+ψ1ψ2aψ2ϒμ,α−1,l,ω,a+γ,δ,k,c1a+b2;p−ψ2ϒμ,α−1,l,ω,a+b/2+γ,δ,k,cψ1∘ψ2b;p+ψ2ϒμ,α−1,l,ω,a+b/2−γ,δ,k,cψ1∘ψ2a;p≤Mψ2a+b2−ψ2aαEμ,α,lγ,δ,k,cωψ2a+b2−ψ2aμ;p+ψ2b−ψ2a+b2αEμ,α,lγ,δ,k,cωψ2b−ψ2a+b2μ;p−ψ2ϒμ,α−1,l,ω,a+b/2−γ,δ,k,c1a;p+ψ2ϒμ,α−1,l,ω,a+b/2+γ,δ,k,c1b;p.Proof.
By puttingx=a+b/2 for α=β in (40), (43) can be obtained.


## 4. Concluding Remarks

We have established generalized fractional integral inequalities of Ostrowski type. By applying boundedness of a differentiable function and using properties of an extended generalized Mittag–Leffler function different generalized versions of Ostrowski type inequalities are analyzed. Also, some deductions from results of this paper are connected with already published results. Furthermore, all the results can be calculated for fractional integral operators defined in [2, 3, 5, 6, 27], and we left it for the reader.



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