# New Fractional Mercer–Ostrowski Type Inequalities with Respect to Monotone Function

Authors: Saad Ihsan Butt; Ammara Nosheen; Jamshed Nasir; Khuram Ali Khan; Rostin Matendo Mabela
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This research focuses on Ostrowski type inequality in the form of classical Mercer inequality viaψ-Riemann–Liouville fractional integral (F-I) operators. Using the ψ-Riemann–Liouville F-I operator, we first develop and demonstrate a new generalized lemma for differentiable functions. Based on this lemma, we derive some fractional Mercer–Ostrowski type inequalities by using the convexity theory. These new findings extend and recapture previous published results. Finally, we presented applications of our work via the known special functions of real numbers such as q-digamma functions and Bessel function.


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## Body

## 1. Introduction and Preliminaries

The well-known Ostrowski inequality, developed in 1938, established the following helpful and noteworthy integral inequality (see [1], page 468).Suppose a mappingλ:a,b⟶ℜ is continuous on a,b and differentiable on a,b. If λ′z≤M, for all z∈a,b, then, the following inequality holds:(1)λz−1b−a∫abλvdv≤Mb−a14+z−a+b/22b−a2.This finding is known as the Ostrowski inequality in the literature. Some generalizations, variations, and extensions of the Ostrowski inequality have been proposed in light of current findings and their related generalizations, variants, and extensions (see [2–4]).This inequality yields an upper bound for the approximation of the integral average1/b−a∫abλvdv by the value of λv at the point v∈a,b.In recent years, because of the widespread interest in the theory of inequalities, the theory of convex functions is now at the center of many studies. Convex functions are the topic of research in a number of disciplines due to their applicability in inequality theory [5–8] and defined as:Definition 1.
[5] A mapping λ:ℐ⊂ℜ⟶ℜ is called to be convex on ℐ, if(2)λ1−ζy1+ζy2≤1−ζλy1+ζλy2,holds for every y1,y2∈ℐ and ζ∈0,1.
Kian and Moslehian used the Jensen–Mercer inequality and demonstrated the Hermite–Hadamard Mercer inequality in [9] as:(3)λa+b−y1+y22≤1y2−y1∫y1y2λa+b−ζdζ≤λa+b−y1+λa+b−y22≤λa+λb−λy1+λy22,where λ is convex function on a,b.
The famous Jensen inequality, (see [5], Ch. 1) in the literature states that, if λ is convex function on the interval y1,y2, then(4)λ∑i=1nσiℓi≤∑i=1nσiλℓi,holds for all ℓi∈y1,y2 and all σi∈0,1,∑i=1nσi=1.
Jensen’s inequality was modified by Mercer (see [10]) as(5)λy1+y2−∑i=1nσiℓi≤λy1+λy2−∑i=1nσiλℓi,where λ is a convex function on y1,y2 holds for all ℓi∈y1,y2 and(6)σi∈0,1,∑i=1nσi=1.
Jensen and Hermite–Hadamard type inequalities are the most dynamic inequalities pertaining convex functions. Jensen and its related inequalities are well-known and significant inequalities in mathematical analysis due to its diverse applications and useability in applied and information sciences. Some recent discoveries can be found in [11, 12].
Jensen-type Mercer’s inequality is an effective inequality since it provides additional information with specific boundary constraints. The study of generalizations and improvements of Mercer’s variants of Hermite–Hadamard type inequalities considering the variety of fractional integral (F-I) operators have been of great interest for researchers in recent years, as evidenced by a large amount of research on it (see [13–15]).
The fractional calculus has been extensively studied by many researchers from the last few decades to generalize, improve, and extend several classic inequalities in order to obtain new variants in different dimensions. There are not only global derivatives in so called fractional calculus (for example: Riemann–Liouville and Caputo), but also local fractional derivatives (Khalil and Almeida, among others) (see [16–18]).
Yue [19], in 2013, discovered new Ostrowski inequalities for fractional integral (F-I) operators along with its associated fractional inequalities. Later in 2014, Aljinović [20] first developed Montgomery identity for fractional integrals of one function with respect to another function and then derived generalized fractional Ostrowski inequality from it. In the same article, he also presented the associated Ostrowski fractional inequalities for fractional integrals of functions with first derivatives in Lp spaces and computed sharp bounds. In the same year, Yildirim and Kirtay [21] used the generalized Riemann–Liouville F-I to establish new variants for Ostrowski inequalities. Some recent development about weighted Ostrowski fractional inequalities can be observed in [22].
Vanterler da Costa Sousa and Capelas de Oliveira in [23] recently introduced the ψ-Hilfer fractional derivative with respect to another function. Also, they investigate some Gronwall inequalities and Cauchy–type problem using the newly introduced ψ-Hilfer operator.Definition 2.
([24], p.3) Let y1,y2−∞≤y1<y2≤∞ be finite or infinite interval in ℜ and ϱ>0. Also, let ψ:y1,y2⟶ℜ+ be positive strictly increasing function possessing continuous derivative ψ′ on y1,y2. Then, left- and right-sided ψ-Riemann–Liouville F-I of a function λ with respect to another function ψ on y1,y2 can be given as(7)Iy1+ϱ:ψλℓ=1Γϱ∫y1ℓψ′ζψℓ−ψζϱ−1λζdζ,y1<ℓ,and(8)Iy2−ϱ:ψλℓ=1Γϱ∫ℓy2ψ′ζψζ−ψℓϱ−1λζdζ,ℓ<y2.Remark 1.
F-I operators elaborated in (7) and (8) yield several known F-I operators corresponding to various suitable selections of function ψ (see [25]), that are independently introduced by several authors with related results.(i)
By takingψy=y as an identity function in (7) and (8), we get Riemann–Liouville F-I operators [24].(ii)
Forψy=yϱ/ϱ, ϱ>0 in (7) and (8) produce Katugampola F-I operators defined by Chen and Katugampola in [26].(iii)
Forψy=yτ+s/τ+s, τ>0,s>0 in (7) and (8) produce generalized conformable F-I operators defined by Khan and Khan in [27].(iv)
Forψy=y−y1s/s, s>0, in (7), and ψy=−y2−ys/s, s>0, in (8), one can get conformable F-I operators presented by Jarad et al. in [28].
The main good articles about Hermite–Hadamard inequalities involvingψ-Riemann–Liouville F-I operators are in references [29–31]. Some recent results about Hermite–Jensen–Mercer inequalities for ψ-Riemann–Liouville F-I operators can be seen in [14, 15].
The striking motive of this study is to develop generalized fractional equality forψ-Riemann–Liouville F-I operators, which has a unique place among fractional integral operators, and to use this identity to generate some new Mercer–Ostrowski type inequalities for convex functions. The study also included applications of the findings, taking into account several specific circumstances of the primary conclusions.


## 2. New Mercer–Ostrowski Type Inequalities

Throughout this portion, Mercer–Ostrowski inequalities for theψ-Riemann–Liouville F-I operators are obtained for differentiable functions on a,b. As a result, we present a novel identity pertaining ψ-Riemann–Liouville F-I operators, that will serve as an auxiliary equality to produce subsequent inequalities.Lemma 1.
Considerλ:a,b⊂0,∞⟶ℜ be a differentiable function and λ′∈ℒ1a,b, with b>a. If ψ is a strictly increasing, positive monotone function on a,b with continuous derivative ψ′ on a,b, then, for all ϰ,y1,y2,ν∈a,b and ϱ>0, the following identity holds(9)Ξϱ,ψλy1,y2,ϰ,v=1y2−y1∫ψ−1ϰ+a−vψ−1ϰ+a−y1ψζ−ϰ+a−vϱψ′ζλ′°ψζdζ+1y2−y1∫ψ−1ϰ+b−y2ψ−1ϰ+b−vϰ+b−v−ψζϱψ′ζλ′°ψζdζ,where(10)Ξϱ,ψλy1,y2,ϰ,v=v−y1ϱy2−y1λϰ+a−y1+y2−vϱy2−y1λϰ+b−y2−Γϱ+1y2−y1Iψ−1ϰ+a−y1−ϱ:ψλ°ψψ−1ϰ+a−v+Iψ−1ϰ+b−y2+ϱ:ψλ°ψψ−1ϰ+b−v.Proof.
Let us start with(11)I1:=Γϱ+1y2−y1Iψ−1ϰ+a−y1−ϱ:ψλ°ψψ−1ϰ+a−v, (12)=ϱy2−y1∫ψ−1ϰ+a−vψ−1ϰ+a−y1ψζ−ϰ+a−vϱ−1ψ′ζλ°ψζdζ=1y2−y1∫ψ−1ϰ+a−vψ−1ϰ+a−y1λ°ψζdψζ−ϰ+a−vϱdζ=v−y1ϱy2−y1λϰ+a−y1−1y2−y1∫ψ−1ϰ+a−vψ−1ϰ+a−y1ψζ−ϰ+a−vϱψ′ζλ′°ψζdζ,and similarly, we get(13)I2:=Γϱ+1y2−y1Iψ−1ϰ+b−y2+ϱ:ψλ°ψψ−1ϰ+b−v=y2−vϱy2−y1λϰ+b−y2−1y2−y1∫ψ−1ϰ+b−y2ψ−1ϰ+b−vϰ+b−v−ψζϱψ′ζλ′°ψζdζ.
It follows from (11) and (13) that(14)v−y1ϱy2−y1λϰ+a−y1+y2−vϱy2−y1λϰ+b−y2−I1−I2=1y2−y1∫ψ−1ϰ+a−vψ−1ϰ+a−y1ψζ−ϰ+a−vϱψ′ζλ′°ψζdζ+1y2−y1∫ψ−1ϰ+b−y2ψ−1ϰ+b−vϰ+b−v−ψζϱψ′ζλ′°ψζdζ.
By simplifying, we get the required result.Remark 2.
Placing identity functionψy=y in (9), then, we get Lemma 2.1 given in [32].Remark 3.
If we setψy=y, y1=a,y2=b, and v=ϰ in (9), it reduces to Lemma 2 in [33].Remark 4.
Settingψy=y, y1=a,y2=b, v=ϰ, and ϱ=1 in (9), it recaptures Lemma 1 proved in [2].Theorem 1.
Under the assumptions of Lemma 1, ifλ′ is convex function on a,b, then for all ϱ>0, the following inequality is valid.(15)Ξϱ,ψλy1,y2,ϰ,v≤v−y1ϱ+1y2−y11ϱ+1λ′ϰ+λ′a−1ϱ+2λ′y1+1ϱ+1ϱ+2λ′v+y2−vϱ+1y2−y11ϱ+1λ′ϰ+λ′b−1ϱ+2λ′y2+1ϱ+1ϱ+2λ′v.Proof.
By means of (9),(16)Ξϱ,ψλy1,y2,ϰ,v=1y2−y1∫ψ−1ϰ+a−vψ−1ϰ+a−y1ψζ−ϰ+a−vϱψ′ζλ′°ψζdζ+1y2−y1∫ψ−1ϰ+b−y2ψ−1ϰ+b−vϰ+b−v−ψζϱψ′ζλ′°ψζdζ.
Change of variabless1=ψζ−ϰ+a−v/v−y1 and s2=ϰ+b−v−ψζ/y2−v and then ζ=s1=s2 into the resulting equality, we get(17)Ξϱ,ψλy1,y2,ϰ,v=v−y1ϱ+1y2−y1∫01ζϱλ′ϰ+a−ζy1+1−ζvdζ+y2−vϱ+1y2−y1∫01ζϱλ′ϰ+b−ζy2+1−ζvdζ.
Sinceλ′ is convex function on a,b, we obtain(18)Ξϱ,ψλy1,y2,ϰ,v≤v−y1ϱ+1y2−y1∫01ζϱλ′ϰ+a−ζy1+1−ζvdζ+y2−vϱ+1y2−y1∫01ζϱλ′ϰ+b−ζy2+1−ζvdζ≤v−y1ϱ+1y2−y1∫01ζϱλ′ϰ+λ′a−ζλ′y1+1−ζλ′vdζ+y2−vϱ+1y2−y1∫01ζϱλ′ϰ+λ′b−ζλ′y2+1−ζλ′vdζ≤v−y1ϱ+1y2−y11ϱ+1λ′ϰ+λ′a−1ϱ+2λ′y1+1ϱ+1ϱ+2λ′v+y2−vϱ+1y2−y11ϱ+1λ′ϰ+λ′b−1ϱ+2λ′y2+1ϱ+1ϱ+2λ′v.Remark 5.
If we setψy=y in Theorem 1, one can get above inequality for Riemann–Liouville F-I operators given in Theorem 2.1 [32].Remark 6.
If we setψy=y, y1=a,y2=b, and v=ϰ in Theorem 1, it reduces to Theorem 7 in [33] that yields the same results with s=1.Corollary 1.
If we setψy=y, y1=a,y2=b, and v=ϰ with ϱ=1 in Theorem 1, we get the following inequality(19)λϰ−1b−a∫abλζdζ≤ϰ−a23b−a12λ′a+λ′ϰ+b−ϰ23b−a12λ′b+λ′ϰ.Corollary 2.
If we setψy=y and ϱ=1 in Theorem 1, we get the following Mercer–Ostrowski inequality:(20)v−y1y2−y1λϰ+a−y1+y2−vy2−y1λϰ+b−y2−1y2−y1∫ϰ+a−vϰ+a−y1λζdζ+∫ϰ+b−y2ϰ+b−vλζdζ≤v−y12y2−y112λ′ϰ+λ′a−13λ′y1+16λ′v+y2−v2y2−y112λ′ϰ+λ′b−13λ′y2+16λ′v.Corollary 3.
The following Mercer–Ostrowski inequality can be found in Theorem1 with λ′≤M(21)Ξϱ,ψλy1,y2,ϰ,v≤My2−y1ϱ+1v−y1ϱ+1+y2−vϱ+1.Proof.
The result can be obtained by usingλ′ϰ+a−ζy1+1−ζv≤M and λ′ϰ+b−ζy2+1−ζv≤M.Remark 7.
If we setψy=y, y1=a,y2=b and v=ϰ in Corollary 3, it reduces to Corollary 1 in [33].Remark 8.
If we setψy=y, y1=a,y2=b and v=ϰ and ϱ=1 in Corollary 3, it reduces to Theorem 2 in [2] that yields the same result with s=1.Theorem 2.
We assume that all the conditions of Lemma1 hold. If λ′q is convex function on a,b, then, for all ϱ>0, the following inequality(22)Ξϱ,ψλy1,y2,ϰ,v≤v−y1ϱ+1y2−y11ϱp+11/pλ′ϰq+λ′aq−12λ′y1q+λ′vq1/q+y2−vϱ+1y2−y11ϱp+11/pλ′ϰq+λ′bq−12λ′y2q+λ′vq1/q,holds, where p,q>1 are conjugate exponents.Proof.
Applying classical Hölder integral inequality and the convexity ofλ′q on the right side of (9), we get(23)Ξϱ,ψλy1,y2,ϰ,v≤v−y1ϱ+1y2−y1∫01ζϱλ′ϰ+a−ζy1+1−ζvdζ+y2−vϱ+1y2−y1∫01ζϱλ′ϰ+b−ζy2+1−ζvdζ.≤v−y1ϱ+1y2−y1∫01ζϱpdζ1/p∫01λ′ϰ+a−ζy1+1−ζvqdζ1/q+y2−vϱ+1y2−y1∫01ζϱpdζ1/p∫01λ′ϰ+b−ζy2+1−ζvqdζ1/q≤v−y1ϱ+1y2−y1∫01ζϱpdζ1/p×∫01λ′ϰq+λ′aq−ζλ′y1q+1−ζλ′vqdζ1/q+y2−vϱ+1y2−y1∫01ζϱpdζ1/p×∫01λ′ϰq+λ′bq−ζλ′y2q+1−ζλ′vqdζ1/q≤v−y1ϱ+1y2−y11ϱp+11/pλ′ϰq+λ′aq−12λ′y1q+λ′vq1/q+y2−vϱ+1y2−y11ϱp+11/pλ′ϰq+λ′bq−12λ′y2q+λ′vq1/q.
That finish the proof.Remark 9.
If we setψy=y in Theorem 2, it reduces to Theorem 2.2 in [32].Remark 10.
If we setψy=y, y1=a,y2=b, and v=ϰ in Theorem 2, it reduces to Theorem 8 in [33] that yields the same results with s=1.Corollary 4.
If we setψy=y, y1=a,y2=b, and v=ϰ with ϱ=1 in Theorem 2, we have the following inequality:(24)λϰ−1b−a∫abλζdζ≤121/qb−a1p+11/p×ϰ−a2λ′aq+λ′ϰq1/q+b−ϰ2λ′bq+λ′ϰq1/q.Corollary 5.
If we setψy=y and ϱ=1 in Theorem 2, we lead to following Mercer–Ostrowski inequality:(25)|v−y1y2−y1λϰ+a−y1+y2−vy2−y1λϰ+b−y2−1y2−y1∫ϰ+a−vϰ+a−y1λζdζ+∫ϰ+b−y2ϰ+b−vλζdζ|≤v−y12y2−y11p+11/pλ′ϰq+λ′aq−12λ′y1q+λ′vq1/q+y2−v2y2−y11p+11/pλ′ϰq+λ′bq−12λ′y2q+λ′vq1/q.Corollary 6.
Let the functionλ′ in Theorem 2 is assumed to be bounded that is λ′≤M, then the following result holds:(26)Ξϱ,ψλy1,y2,ϰ,v≤My2−y11ϱp+11pv−y1ϱ+1+y2−vϱ+1.Proof.
The result can be demonstrated by usingλ′ϰ+a−ζy1+1−ζv≤M and λ′ϰ+b−ζy2+1−ζv≤M.Remark 11.
If we setψy=y, y1=a,y2=b, and v=ϰ in Corollary 6, it reduces to Corollary 2 in [33].Remark 12.
If we setψy=y, y1=a,y2=b, v=ϰ and ϱ=1 in Corollary 6, it reduces to Theorem 3 in [2] that yields the same result with s=1.Theorem 3.
We assume that all the conditions of Lemma1 hold. If λ′q is convex function on a,b,q≥1, then for all ϱ>0, the following inequality(27)Ξϱ,ψλy1,y2,ϰ,v≤v−y1ϱ+1y2−y11ϱ+11−1/q×1ϱ+1λ′ϰq+λ′aq−1ϱ+2λ′y1q+1ϱ+1ϱ+2λ′vq1/q+y2−vϱ+1y2−y11ϱ+11−1/q×1ϱ+1λ′ϰq+λ′bq−1ϱ+2λ′y2q+1ϱ+1ϱ+2λ′vq1/q,is valid.Proof.
Applying power-mean integral inequality and the convexity ofλ′q on the right side of (9), we get(28)Ξϱ,ψλy1,y2,ϰ,v≤v−y1ϱ+1y2−y1∫01ζϱλ′ϰ+a−ζy1+1−ζvdζ+y2−vϱ+1y2−y1∫01ζϱλ′ϰ+b−ζy2+1−ζvdζ≤v−y1ϱ+1y2−y1∫01ζϱdζ1−1/q∫01ζϱλ′ϰ+a−ζy1+1−ζvqdζ1/q+y2−vϱ+1y2−y1∫01ζϱdζ1−1/q∫01ζϱλ′ϰ+b−ζy2+1−ζvqdζ1/q≤v−y1ϱ+1y2−y1∫01ζϱdζ1−1/q×∫01ζϱλ′ϰq+λ′aq−ζλ′y1q+1−ζλ′vqdζ1/q+y2−vϱ+1y2−y1∫01ζϱdζ1−1/q×∫01ζϱλ′ϰq+λ′bq−ζλ′y2q+1−ζλ′vqdζ1/q≤v−y1ϱ+1y2−y11ϱ+11−1/q×1ϱ+1λ′ϰq+λ′aq−1ϱ+2λ′y1q+1ϱ+1ϱ+2λ′vq1/q+y2−vϱ+1y2−y11ϱ+11−1/q×1ϱ+1λ′ϰq+λ′bq−1ϱ+2λ′y2q+1ϱ+1ϱ+2λ′vq1/q.
Which ends the proof.Remark 13.
If we setψy=y in Theorem 3, it reduces to Theorem 2.3 in [32].Remark 14.
If we setψy=y, y1=a,y2=b, and v=ϰ in Theorem 3, it reduces to Theorem 9 in [33] that yields the same results with s=1.Corollary 7.
If we setψy=y, y1=a,y2=b, and v=ϰ with ϱ=1 in Theorem 3, we have the following inequality:(29)λϰ−1b−a∫abλζdζ≤1b−a121−1/q×13ϰ−a212λ′aq+λ′ϰq1/q+b−ϰ212λ′bq+λ′ϰq1/q.Corollary 8.
If we chooseψy=y and ϱ=1 in Theorem 3, we have the following Mercer–Ostrowski inequality:(30)v−y1y2−y1λϰ+a−y1+y2−vy2−y1λϰ+b−y2−1y2−y1∫ϰ+a−vϰ+a−y1λζdζ+∫ϰ+b−y2ϰ+b−vλζdζ≤v−y12y2−y1121−1/q12λ′xq+λ′aq−13λ′y1q+16λ′vq1/q+y2−v2y2−y1121−1/q12λ′ϰq+λ′bq−13λ′y2q+16λ′vq1/q.Corollary 9.
Assuming thatλ′≤M, in Theorem 3, the following Mercer–Ostrowski inequality holds:(31)Ξϱ,ψλy1,y2,ϰ,v≤My2−y1ϱ+1v−y1ϱ+1+y2−vϱ+1.Proof.
Under the assumed conditions, we haveλ′ϰ+a−ζy1+1−ζv≤M and λ′ϰ+b−ζy2+1−ζv≤M. Thus, inequality (27) in Theorem 3 leads to inequality (31).Remark 15.
If we setψy=y, y1=a,y2=b, and v=ϰ in Corollary 9, it reduces to Corollary 3 in [33].Remark 16.
If we setψy=y, y1=a,y2=b, v=ϰ, and ϱ=1 in Corollary 9, it reduces to Theorem 4 in [2] that yields the same result with s=1.Theorem 4.
We assume that all the assumptions of Lemma1 holds. If λ′q is convex function on a,b, for all ϱ>0, the following inequality(32)Ξϱ,ψλy1,y2,ϰ,v≤v−y1ϱ+1y2−y11ϱp+1p+1qλ′ϰq+λ′aq−12λ′y1q+λ′vq+y2−vϱ+1y2−y11ϱp+1p+1qλ′ϰq+λ′bq−12λ′y2q+λ′vq,holds where p,q>1 are conjugate exponents.Proof.
From (9), we obtain(33)Ξϱ,ψλy1,y2,x,v≤v−y1ϱ+1y2−y1∫01ζϱλ′ϰ+a−ζy1+1−ζvdζ+y2−vϱ+1y2−y1∫01ζϱλ′ϰ+b−ζy2+1−ζvdζ.
Utilizing Young’s inequality as(34)uv<1pup+1qvq.Ξϱ,ψλy1,y2,ϰ,v≤v−y1ϱ+1y2−y11p∫01ζϱpdζ+1q∫01λ′ϰ+a−ζy1+1−ζvqdζ+y2−vϱ+1y2−y11p∫01ζϱpdζ+1q∫01λ′ϰ+b−ζy2+1−ζvqdζ.
By the convexity ofλ′q, we have(35)Ξϱ,ψλy1,y2,ϰ,v≤v−y1ϱ+1y2−y1×1p∫01ζϱpdζ+1q∫01λ′ϰq+λ′aq−ζλ′y1q+1−ζλ′vqdζ+y2−vϱ+1y2−y1×1p∫01ζϱpdζ+1q∫01λ′ϰq+λ′bq−ζλ′y2q+1−ζλ′vqdζ≤v−y1ϱ+1y2−y11ϱp+1p+1qλ′ϰq+λ′aq−12λ′y1q+λ′vq+y2−vϱ+1y2−y11ϱp+1p+1qλ′ϰq+λ′bq−12λ′y2q+λ′vq,and the proof is done.Remark 17.
In Theorem4, putting ψy=y, then, one can get Mercer–Ostrowski inequality pertaining Riemann–Liouville F-I operators given in [32].Corollary 10.
If we chooseψy=y and ϱ=1 in Theorem 4, the following Mercer–Ostrowski inequality holds:(36)v−y1y2−y1λϰ+a−y1+y2−vy2−y1λϰ+b−y2−1y2−y1∫ϰ+a−vϰ+a−y1λζdζ+∫ϰ+b−y2ϰ+b−vλζdζ≤v−y12y2−y11p+1p+1qλ′ϰq+λ′aq−12λ′y1q+λ′vq+y2−v2y2−y11p+1p+1qλ′ϰq+λ′bq−12λ′y2q+λ′vq.Corollary 11.
The following Mercer–Ostrowski inequality can be obtained from Theorem4 by assuming λ′≤M:(37)Ξϱ,ψλy1,y2,ϰ,v≤1y2−y11ϱp+1p+1qMqv−y1ϱ+1+y2−vϱ+1.Theorem 5.
We assume that all the assumptions of Lemma1 hold. If λ′q is concave function on a,b,q>1, for all ϱ>0, the following inequality holds:(38)Ξϱ,ψλy1,y2,ϰ,v≤1ϱp+11p×v−y1ϱ+1y2−y1λ′ϰ+a−y1+v2+y2−vϱ+1y2−y1λ′ϰ+b−y2+v2,where p,q>1 are conjugate exponents.Proof.
From (9), by using the Hölder’s inequality, we obtain(39)Ξϱ,ψλy1,y2,ϰ,v≤v−y1ϱ+1y2−y1∫01ζϱλ′ϰ+a−ζy1+1−ζvdζ+y2−vϱ+1y2−y1∫01ζϱλ′ϰ+b−ζy2+1−ζvdζ≤v−y1ϱ+1y2−y1∫01ζϱpdζ1/p∫01λ′ϰ+a−ζy1+1−ζvqdζ1/q+y2−vϱ+1y2−y1∫01ζϱpdζ1/p∫01λ′ϰ+b−ζy2+1−ζvqdζ1/q.
Sinceλ′q is concave mapping, therefore from (3), we obtain(40)∫01λ′ϰ+a−ζy1+1−ζvqdζ≤λ′ϰ+a−y1+v2q,and(41)∫01λ′ϰ+b−ζy2+1−ζvqdζ≤λ′ϰ+b−y2+v2q.
By placing the inequalities (40) and (41) in (39), leads to (38).Remark 18.
If we setψy=y in Theorem 5, it reduces to Theorem 2.5 in [32].Remark 19.
If we setψy=y, y1=a,y2=b, v=ϰ, and ϱ=1 in Theorem 5, it reduces to Theorem 5 in [2] that yields the same result with s=1.Corollary 12.
If we chooseψy=y and ϱ=1 in Theorem 5, we deduce the Mercer–Ostrowski inequality as(42)v−y1y2−y1λϰ+a−y1+y2−vy2−y1λϰ+b−y2−1y2−y1∫ϰ+a−vϰ+a−y1λζdζ+∫ϰ+b−y2ϰ+b−vλζdζ≤1p+11pv−y12y2−y1λ′ϰ+a−y1+v2+y2−v2y2−y1λ′ϰ+b−y2+v2.


## 3. Some Applications

### 3.1. Applications to Means

For two real numbers0<κ1<κ2, consider the following two important means:The arithmetic mean:(43)Aκ1,κ2=κ1+κ22.The generalized logarithmic-mean:(44)Lmκ1,κ2=κ2m+1−κ1m+1m+1κ2−κ11/m;m∈ℜ∖−1,0.Proposition 1.
Supposea,b>0, then, we have the following inequality(45)v−y1y2−y12Aϰ,a−y1n+y2−vy2−y12Aϰ,b−y2n−1y2−y1v−y1Lmmϰ+a−y1,ϰ+a−v+v−y2Lmmϰ+b−v,ϰ+a−y2≤nv−y12y2−y1121−1/qAϰn−1q,an−1q−13y1n−1q+16vn−1q1/q+ny2−v2y2−y1121−1/qAϰn−1q,bn−1q−13y2n−1q+16vn−1q1/q.Proof.
The result can be obtained immediately by taking into account Corollary8 along with the convex function λϰ=ϰn,ϰ>0.Proposition 2.
Supposea,b>0, then, we have the following inequality:(46)v−y1y2−y12Aϰ,a−y1n+y2−vy2−y12Aϰ,b−y2n−1y2−y1v−y1Lmmϰ+a−y1,ϰ+a−v+v−y2Lmmϰ+b−v,ϰ+a−y2≤v−y12y2−y11p+1p+nq2Aϰn−1q,an−1q−Ay1n−1q,vn−1q+y2−v2y2−y11p+1p+nq2Aϰn−1q,bn−1q−Ay2n−1q,vn−1q.Proof.
The proof is the direct consequence of Corollary10 by considering the convex function λϰ=ϰn,ϰ>0.


### 3.2. q-Digamma Function

Theφq-digamma function, which is described as the logarithmic derivative of the q-gamma function, is an essential function related to the q-gamma function (see [34]) given as(47)φq=−ln1−q+lnq∑k=0∞qk+ζ1−qk+ζ=−ln1−q+lnq∑k=0∞qkζ1−qkζ.Forq>1 and ζ>0, q-digamma function φq can be given as(48)φq=−lnq−1+lnqζ−12−∑k=0∞q−k+ζ1−q−k+ζ=−lnq−1+lnqζ−12−∑k=0∞q−kζ1−q−kζ.Proposition 3.
Assume thataandb are the real numbers such that 0<a<b, q>1, 0<q<1, and q−1=1−p−1. Then, the following inequality is valid:(49)v−y1y2−y1φqϰ+a−y1+y2−vy2−y1φqϰ+b−y2−1y2−y1∫ϰ+a−vϰ+a−y1φqζdζ+∫ϰ+b−y2ϰ+b−vφqζdζ≤v−y12y2−y1121−1/qAφq′ϰq,φq′aq−13φq′y1q+16φq′vq1/q+y2−v2y2−y1121−1/qAφq′ϰq,φq′bq−13φq′y2q+16φq′vq1/q.Proof.
The statement can be obtained by using Corollary8 by considering λζ⟶φqζ. Since, φq′ζ is a completely monotone function on 0,∞ for all ζ>0, consequently, λ′ζ:=φq′ζ is convex on the same interval 0,∞, (see [34]).


### 3.3. Bounds Involving Modified Bessel Function

We know the first type of modified Bessel functionBτ1, which has the series interpretation (see [35], p.77)(50)Bτ1ϰ=∑n≥0ϰ/2τ1+2nn!Γτ1+n+1,where ϰ∈ℜ and τ1>−1, while the second kind modified Bessel function ℏτ1 (see [35], p.78) is usually defined as(51)ϕτ1ϰ=π2B−τ1ϰ−Bτ1ϰsinτ1π.Consider the functionΨτ1:ℜ⟶[1,∞) defined by(52)Ψτ1ϰ=2τ1Γτ1+1ϰ−τ1Bτ1ϰ.The first- and second-order derivative formula ofΨτ1ϰ is given as [35]:(53)Ψτ1′ϰ=ϰ2τ1+1Ψτ1+1ϰ,(54)Ψτ1′′ϰ=ϰ2Ψτ1+2ϰ4τ1+1τ1+2+Ψτ1+1ϰ2τ1+1.Proposition 4.
Suppose thatτ1>−1 and 0<a<b,q>1. Then, we have(55)v−y1y2−y1.ϰ+a−y12τ1+1.Ψτ1+1ϰ+a−y1+y2−vy2−y1.ϰ+b−y22τ1+1.Ψτ1+1ϰ+b−y2−1y2−y1Ψτ1ϰ+a−y1−Ψτ1ϰ+a−v+Ψτ1ϰ+b−v−Ψτ1ϰ+b−y2≤v−y12y2−y11p+1p+1qϰ2Ψτ1+2ϰ4τ1+1τ1+2+Ψτ1+1ϰ2τ1+1q+a2Ψτ1+2a4τ1+1τ1+2+Ψτ1+1a2τ1+1q−12y12Ψτ1+2y14τ1+1τ1+2+Ψτ1+1y12τ1+1q+v2Ψτ1+2v4τ1+1τ1+2+Ψτ1+1v2τ1+1q+y2−v2y2−y11p+1p+1qϰ2Ψτ1+2ϰ4τ1+1τ1+2+Ψτ1+1ϰ2τ1+1q+b2Ψτ1+2b4τ1+1τ1+2+Ψτ1+1b2τ1+1q−12y22Ψτ1+2y24τ1+1τ1+2+Ψτ1+1y22τ1+1q+v2Ψτ1+2v4τ1+1τ1+2+Ψτ1+1v2τ1+1q.Proof.
Substituting the mappingλ⟶Ψτ1′ to the inequality in Corollary 10. Note that all assumptions of Corollary 10 are satisfied (see [34]). Therefore, using the identities (53) and (54) gives required result.


## 3.1. Applications to Means

For two real numbers0<κ1<κ2, consider the following two important means:The arithmetic mean:(43)Aκ1,κ2=κ1+κ22.The generalized logarithmic-mean:(44)Lmκ1,κ2=κ2m+1−κ1m+1m+1κ2−κ11/m;m∈ℜ∖−1,0.Proposition 1.
Supposea,b>0, then, we have the following inequality(45)v−y1y2−y12Aϰ,a−y1n+y2−vy2−y12Aϰ,b−y2n−1y2−y1v−y1Lmmϰ+a−y1,ϰ+a−v+v−y2Lmmϰ+b−v,ϰ+a−y2≤nv−y12y2−y1121−1/qAϰn−1q,an−1q−13y1n−1q+16vn−1q1/q+ny2−v2y2−y1121−1/qAϰn−1q,bn−1q−13y2n−1q+16vn−1q1/q.Proof.
The result can be obtained immediately by taking into account Corollary8 along with the convex function λϰ=ϰn,ϰ>0.Proposition 2.
Supposea,b>0, then, we have the following inequality:(46)v−y1y2−y12Aϰ,a−y1n+y2−vy2−y12Aϰ,b−y2n−1y2−y1v−y1Lmmϰ+a−y1,ϰ+a−v+v−y2Lmmϰ+b−v,ϰ+a−y2≤v−y12y2−y11p+1p+nq2Aϰn−1q,an−1q−Ay1n−1q,vn−1q+y2−v2y2−y11p+1p+nq2Aϰn−1q,bn−1q−Ay2n−1q,vn−1q.Proof.
The proof is the direct consequence of Corollary10 by considering the convex function λϰ=ϰn,ϰ>0.


## 3.2. q-Digamma Function

Theφq-digamma function, which is described as the logarithmic derivative of the q-gamma function, is an essential function related to the q-gamma function (see [34]) given as(47)φq=−ln1−q+lnq∑k=0∞qk+ζ1−qk+ζ=−ln1−q+lnq∑k=0∞qkζ1−qkζ.Forq>1 and ζ>0, q-digamma function φq can be given as(48)φq=−lnq−1+lnqζ−12−∑k=0∞q−k+ζ1−q−k+ζ=−lnq−1+lnqζ−12−∑k=0∞q−kζ1−q−kζ.Proposition 3.
Assume thataandb are the real numbers such that 0<a<b, q>1, 0<q<1, and q−1=1−p−1. Then, the following inequality is valid:(49)v−y1y2−y1φqϰ+a−y1+y2−vy2−y1φqϰ+b−y2−1y2−y1∫ϰ+a−vϰ+a−y1φqζdζ+∫ϰ+b−y2ϰ+b−vφqζdζ≤v−y12y2−y1121−1/qAφq′ϰq,φq′aq−13φq′y1q+16φq′vq1/q+y2−v2y2−y1121−1/qAφq′ϰq,φq′bq−13φq′y2q+16φq′vq1/q.Proof.
The statement can be obtained by using Corollary8 by considering λζ⟶φqζ. Since, φq′ζ is a completely monotone function on 0,∞ for all ζ>0, consequently, λ′ζ:=φq′ζ is convex on the same interval 0,∞, (see [34]).


## 3.3. Bounds Involving Modified Bessel Function

We know the first type of modified Bessel functionBτ1, which has the series interpretation (see [35], p.77)(50)Bτ1ϰ=∑n≥0ϰ/2τ1+2nn!Γτ1+n+1,where ϰ∈ℜ and τ1>−1, while the second kind modified Bessel function ℏτ1 (see [35], p.78) is usually defined as(51)ϕτ1ϰ=π2B−τ1ϰ−Bτ1ϰsinτ1π.Consider the functionΨτ1:ℜ⟶[1,∞) defined by(52)Ψτ1ϰ=2τ1Γτ1+1ϰ−τ1Bτ1ϰ.The first- and second-order derivative formula ofΨτ1ϰ is given as [35]:(53)Ψτ1′ϰ=ϰ2τ1+1Ψτ1+1ϰ,(54)Ψτ1′′ϰ=ϰ2Ψτ1+2ϰ4τ1+1τ1+2+Ψτ1+1ϰ2τ1+1.Proposition 4.
Suppose thatτ1>−1 and 0<a<b,q>1. Then, we have(55)v−y1y2−y1.ϰ+a−y12τ1+1.Ψτ1+1ϰ+a−y1+y2−vy2−y1.ϰ+b−y22τ1+1.Ψτ1+1ϰ+b−y2−1y2−y1Ψτ1ϰ+a−y1−Ψτ1ϰ+a−v+Ψτ1ϰ+b−v−Ψτ1ϰ+b−y2≤v−y12y2−y11p+1p+1qϰ2Ψτ1+2ϰ4τ1+1τ1+2+Ψτ1+1ϰ2τ1+1q+a2Ψτ1+2a4τ1+1τ1+2+Ψτ1+1a2τ1+1q−12y12Ψτ1+2y14τ1+1τ1+2+Ψτ1+1y12τ1+1q+v2Ψτ1+2v4τ1+1τ1+2+Ψτ1+1v2τ1+1q+y2−v2y2−y11p+1p+1qϰ2Ψτ1+2ϰ4τ1+1τ1+2+Ψτ1+1ϰ2τ1+1q+b2Ψτ1+2b4τ1+1τ1+2+Ψτ1+1b2τ1+1q−12y22Ψτ1+2y24τ1+1τ1+2+Ψτ1+1y22τ1+1q+v2Ψτ1+2v4τ1+1τ1+2+Ψτ1+1v2τ1+1q.Proof.
Substituting the mappingλ⟶Ψτ1′ to the inequality in Corollary 10. Note that all assumptions of Corollary 10 are satisfied (see [34]). Therefore, using the identities (53) and (54) gives required result.


## 4. Conclusions

The objective of this study is to introduce the idea of new generalized fractional variants of Ostrowski inequality by employing Jensen–Mercer inequality for differentiable convex functions. The obtained results are intersting and generalized in a sense that by substituting identity functionψy=y and special value of ϱ=1, we get connected to previously established results in the literature. Also, one can get variety of fractional Mercer–Ostrowski inequalities for different F-I operators by considering particular values of function ψ as mentioned in Remark 1. In addition, another motivating aspect of the study is that we try to give applications of means, q-digamma function, and Bessel function for Mercer–Ostrowski inequality in the similar passion as considered for Hermite–Hadamard type inequalities given in [8, 30]. Based on this study, researchers may contribute to the development of such results for twice differentiable functions.



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