# Inequalities Involving Conformable Approach for Exponentially Convex Functions and Their Applications

Authors: Jun-Feng Li; Saima Rashid; Jia-Bao Liu; Ahmet Ocak Akdemir; Farhat Safdar
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In the article, we present several Hermite–Hadamard-type inequalities for the exponentially convex functions via conformable integrals. As applications, we give new inequalities for certain bivariate means.


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## Body

## 1. Introduction

Recently, the fractional calculus has attracted the attention of several researchers [1–14]. The affect and motivation of the fractional calculus in both theoretical and applied science and engineering arose out substantially. Numerous studies are related with the discrete versions of the fractional calculus which benefit from countless applications in the theory of time scales, physics, different fields of engineering, chemistry, and so forth.In the past decades, the subject of fractional integrals has attracted the attention for mathematicians working on inequality theory and convexity. Fractional integral operators are sometimes the gateway to physical problems that cannot be expressed by classical integral, sometimes for the solution of problems expressed in fractional order. In recent years, a lot of new operator definitions have been given, and the properties and structures of these operators have been examined. Some of these operators are very close to classical operators in terms of their characteristics and definitions. In general, they have nonlocality property and defined with singular kernels.The derivatives in this calculus seemed complicated and lost some of the basic properties that usual derivatives have such as the product rule and the chain rule. However, the semigroup properties of these operators behave well in some cases. Recently, the authors in [15] defined a new well-behaved simple derivative called “conformable fractional derivative” which depends just on the basic limit definition of the derivative. It will define the derivative of higher order (i.e., order α>1) and also define the integral of order 0<α≤1 only. It will also prove the product rule and the mean value theorem and solve some (conformable) differential equations where the fractional exponential function eξα/α plays an important rule.Almost every mathematician knows the importance of convex sets and convex functions in many fields of mathematics, for example, in nonlinear programming and optimization theory. By using the concept of convexity, several integral inequalities have been introduced such as Jensen, Hermite–Hadamard, and Slater inequalities. But, the well-known one of them is the celebrated Hermite–Hadamard inequality.LetK⊆ℝ be an interval and χ:K⟶ℝ be a convex function. Then, the double inequality(1)χa+b2≤1b−a∫abχxdx≤χa+χb2,holds for all a,b∈K with a≠b.It is easy to see that ifχ is concave on K, then one has the reverse of inequality (1). Many upper and lower bounds for the mean value of a convex function can be derived by use of inequality (1). Recently, the generalizations, improvements, refinements, extensions, and variants for Hermite–Hadamard inequality (1) can be found in the literature [16–19].It is well known that convex functions have wide applications in pure and applied mathematics [20–35] and many other natural sciences [21–55]. The development in the history of convex function is the minimum of the differentiable convex functions that can be characterized by variational inequalities.Motivated and inspired by the previous study, a comprehensive investigation of exponentially convex functions for conformable integral in the present paper is new. The class of exponentially convex functions was introduced by Antczak [56] and Dragomir and Gomm [57]. Inspired by these facts, Awan et al. [58] defined a new class of convexity, namely, the exponentially convex function. The growth of research on big data analysis, time space management, and deep learning has recently increased interest in information theory involving exponentially convex functions. The smoothness of exponentially convex functions is exploited for statistical learning, sequential prediction, and stochastic optimization.Now, we recall and introduce some definitions for various convex functions.Definition 1.
A nonempty setK⊆ℝ is said to be convex if λx+1−λy∈K for all x,y∈K and λ∈0,1.Definition 2.
Letχ:K⟶ℝ be a real-valued function. Then, χ is said to be convex (concave) if the inequality holds(2)χ1−ξx+ξy≤1−ξχx+ξχy,which holds for all x,y∈K and ξ∈0,1. χ is said to be concave if −χ is convex.Definition 3.
A positive real-valued functionχ:K⊆ℝ⟶0,∞ is said to be exponentially convex if the inequality(3)eχ1−ξx+ξy≤1−ξeχx+ξeχy,holds for all x,y∈K and ξ∈0,1.
It is well known thatx∈K is the minimum of the differentiable exponentially convex functions χ if and only if it satisfies(4)χ′xeχx,y−x≥0,for all y∈K. Inequality (4) is known as the exponentially variational inequality.
The Riemann–Liouville fractional integral, conformable derivative, and conformable integral are very important in the theory of fractional calculus which are defined as follows.Definition 4.
Letα>0 and χ∈L1μ,ν. Then, the Riemann–Liouville integral Jμ+αχ and Jν−αχ of order α is defined by(5)Jμ+αχξ=1Γα∫μξξ−xα−1χxdx,ξ>μ,Jν−αχξ=1Γα∫ξνx−ξα−1χxdx,ξ<ν,respectively, where Γα=∫0∞e−ξξα−1dξ is the Euler gamma function [59–61].Definition 5.
Letα∈0,1 and χ:0,∞⟶ℝ be a real-valued function. Then, the α-order conformable derivative of f at ξ>0 is defined by(6)Dαχξ=limε⟶0χξ+εξ1−α−χξε.
Iflimξ⟶0+χαξ exists, then we define(7)Dαχ0=limξ⟶0+Dαχξ.
Additionally, ifχ is differentiable, then(8)Dαχξ=ξ1−αχ′ξ,where(9)χ′ξ=limε⟶0χξ+ε−χξε.
We writeχαξ for Dαχξ to denote the α-order conformable derivative of χ at ξ. Additionally, we say that χ is α-differentiable if the α-order conformable derivative of χ exists.Definition 6.
Letα∈0,1 and 0≤μ<ν. Then, the real-valued function χ:μ,ν⟶ℝ is said to be α integrable on μ,ν if the integral(10)∫abχxdαx=∫abχxxα−1dx,exists and is finite.Remark 1.
Letα∈0,1. Then, it is well known that(11)Iαμχξ=I1μtα−1χ=∫μξχxx1−αdx,where the integral is the classical Riemann improper integral.
Anderson [62] gave a variant of the Hermite–Hadamard inequality which is as follows.Theorem 1.
Letα∈0,1 and χ:I⟶ℝ be an α-differentiable function such that Dαχ is monotone. Then, the double inequality(12)χμ+ν2≤ανα−μα∫μvχxdx≤χμ+χν2,holds for all μ,ν∈I with μ≠ν.Note that, ifα=1, then inequality (12) leads to the classical Hermite–Hadamard inequality (1).The main purpose of the article is to present new Hermite–Hadamard-type inequalities for exponentially convex functions via the conformable integrals and find new inequalities for certain bivariate means.


## 2. Main Results

First of all, we provide an integral identity which can be proved easily by the readers.Lemma 1.
Letα∈0,1 and χ:μ,ν⟶0,∞ be an α-fractional exponentially differentiable function on μ,ν. The identity(13)να−xαeχν+xα−ναeχμνα−aα−ανα−μα∫μveχxdαx=x−μνα−μα∫011−ξμ+ξx2α−1−xα1−ξμ+ξxα−1×Dαeχ1−ξμ+ξxχα1−ξμ+ξxξ1−αdξ+ν−xνα−μα∫011−ξν+ξx2α−1−xα1−ξν+ξxα−1×Dαeχ1−ξν+ξxχα1−ξν+ξxξ1−αdξ,holds for any x∈μ,ν if Dαeχ∈Lαμ,ν.Remark 2.
Letα=1. Then, identity (13) reduces to(14)ν−xeχν+x−νeχμν−μ−1ν−μ∫μνeχxdx=x−μ2μ−ν∫01ξ−1eχ1−ξμ+ξxχ′1−ξμ+ξxdξ+ν−x2ν−μ∫01ξ−1eχ1−ξν+ξxχ′1−ξν+ξxdξ.Theorem 2.
Letμ,ν>0 with μ<ν, α∈0,1, and χ:μ,ν⟶ℝ be an α−differentiable exponentially function on μ,ν. Then, the inequality(15)να−xαeχν+xα−ναeχμνα−μα−ανα−μα∫μveχxdαx≤x−μνα−μαθ1α+ν−xνα−μαθ2α,holds, where(16)θ1α=xα−μα123eχμχ′μ+eχxχ′x+Δμ,x,θ2α=16012να+3xνα−1+xα−1ν−18xαeχνχ′ν+2να+3xνα−1+xα−1ν−8xαeχxχ′x+3να+2xνα−1+xα−1ν−7xαΔν,x,Δμ,x=eχμχ′x+eχxχ′μ,Δν,x=eχνχ′x+eχxχ′ν.Proof.
It follows from Lemma1 and the convexity of the functions xα−1, −xα, and eχ′ that(17)να−xαeχν+xα−ναeχμνα−μα−ανα−μα∫μveχxdαx=x−μνα−μα∫01xα−1−tμ+xtαeχ1−ξμ+xξχ′1−ξμ+xξdξ+ν−xνα−μα∫01∫011−ξν+xξα−xαeχ1−ξν+xξχ′1−ξν+xξdξ=x−μνα−μα∫01xα−1−ξμ+xtαeχ1−ξμ+xξχ′1−ξμ+xξdξ+ν−xνα−μα∫011−ξν+xtα+1−1−xαeχ1−ξν+xξeχ1−ξν+xξχ′1−ξν+xξdξ≤x−μνα−μα∫01xα−1−ξμα+ξxαeχ1−ξμ+xξχ′1−ξμ+ξxdξ+b−xνα−μα∫011−ξν+ξxα−11−ξν+ξx−xαeχ1−ξν+xξχ′1−ξν+ξxdξ≤x−μνα−μα∫01xα−1−ξμα+ξxαeχ1−ξμ+xξχ′1−ξμ+ξxdξ+ν−xνα−μα∫011−ξνα−1+ξxα−11−ξν+ξx−xαeχ1−ξν+xξχ′1−ξν+ξxdξ≤x−μνα−μα∫01xα−1−ξμα+ξxα1−ξeχμ+ξeχx1−ξχ′μ+ξχ′xdξ+ν−xνα−μα∫011−ξνα−1+ξxα−11−ξν+ξx−xα×1−ξeχν+ξeχx1−ξχ′ν+ξχ′xdξ≤x−μνα−μα∫01xα−1−ξμα+ξxα×1−ξ2eχμχ′μ+ξ2efxχ′x+ξ1−ξeχμχ′x+eχxχ′μdξ+ν−xνα−μα∫011−ξνα−1+ξxα−11−ξν+ξx−xα×1−ξ2eχνχ′ν+ξ2eχxχ′x+ξ1−ξeχνχ′x+eχxχ′νdξ≤x−μνα−μα∫01xα−1−ξμα+ξxα×1−ξ2eχμχ′μ+ξ2eχxχ′x+ξ1−ξΔμ,xdξ+ν−xνα−μα∫011−ξνα−1+ξxα−11−ξν+ξx−xα×1−ξ2eχνχ′ν+ξ2eχxχ′x+ξ1−ξΔν,xdξ=x−μνα−μαθ1α+ν−xνα−μαθ2α.Remark 3.
Letα=1. Then, inequality (15) leads to(18)ν−xeχν+x−νeχμν−μ−1ν−μ∫μveχxdx≤x−μ2ν−μ3eχμχ′μ+eχxχ′x+Δx,μ12+ν−x2ν−μ3eχνχ′ν+eχxχ′x+Δx,ν12.Theorem 3.
Letμ,μ>0 with μ<ν, α∈0,1, p,q>1 such that 1/p+1/q=1, and χ:μ,ν⟶ℝ be an α-differentiable function on μ,ν. Then, the inequality(19)να−xαeχν+xα−ναeχμνα−μα−ανα−μα∫μveχxdαx≤x−μνα−μαφ1α,p1/p2eχμχ′μq+eχxχ′xq+Δ1x,μ61/q+ν−xνα−μαφ2α,p1/p2eχνχ′νq+eχxχ′xq+Δ2x,ν61/qholds if Dαeχ∈Lαμ,ν and eχ′q is convex on μ,ν, where(20)φ1α,p=∫01xα−1−ξμα+ξxαpdξ,(21)φ2α,p=∫011−tν+ξxα−11−tν+ξx−xαpdξ,(22)Δ1x,μ=eχμχ′xq+eχxχ′μq,Δ2x,ν=eχνχ′xq+eχxχ′νq.Proof.
From (13) and (20)–(22), we clearly see that(23)να−xαeχν+xα−ναeχμνα−μα−ανα−μα∫μveχxdαx≤x−μνα−μα∫01xα−1−ξμα+ξxαeχ1−ξμ+xξχ′1−ξμ+ξxdξ+ν−xνα−μα∫011−ξν+ξxα−11−ξν+ξx−xαeχ1−ξν+xtχ′1−ξν+ξxdξ.
Making use of Hölder’s inequality, one has(24)∫01xα−1−ξμα+ξxαeχ1−ξμ+xξχ′1−ξμ+ξxdξ≤∫01xα−1−ξμα+ξxαpdξ1/peχ1−ξμ+xξχ′1−ξμ+ξxqdξ1/q≤φ1α,p1/p∫011−ξeχμq+ξeχxq1−ξχ′μq+ξχ′xqdξ1/q=φ1α,p1/p∫011−ξ2eχμχ′μq+ξ2eχxχ′xq+ξ1−ξΔ1x,μdξ1/q=φ1α,p1/p2eχμχ′μq+eχxχ′xq+Δ1x,μ61/q.
Similarly, we have(25)∫011−ξν+ξxα−11−ξν+ξx−xαeχ1−ξν+xξχ′1−ξν+ξxdξ≤∫011−ξν+ξxα−11−ξν+ξx−xαdξ1/p∫01eχ1−ξν+xξχ′1−ξν+ξxqdξ1/q≤φ2α,p1/p2eχνχ′νq+eχxχ′xq+Δ2x,ν61/q.
Therefore, inequality (19) follows from (24) and (25).Remark 4.
Letα=1. Then, inequality (19) becomes(26)ν−xeχν+x−νeχμν−μ−1ν−μ∫μveχxdx≤1p+11/px−μ2ν−μ2eχμχ′μq+eχxχ′xq+Δ1x,μ61/q+ν−x2ν−μ2eχνχ′νq+eχxχ′xq+Δ2x,ν61/q.Theorem 4.
Letμ,ν>0 with μ<ν, α∈0,1, p,q>1 such that 1/p+1/q=1, and χ:μ,ν⟶ℝ be an α-differentiable function on μ,ν. Then, the inequality(27)να−xαeχν+xα−ναeχμνα−μα−ανα−μα∫μveχxdαx≤x−μνα−μαψ1α1−1/qψ2αeχμχ′μq+ψ3αeχxχ′xq+ψ4αΔ1x,μ1/q+ν−xνα−μαϑ1α1−1/qϑ2αeχνχ′νq+ϑ3αeχxχ′xq+ϑ4αΔ2x,ν1/q,holds if Dαeχ∈Lαμ,ν and eχ′q is convex on μ,ν, where(28)ψ1α=xα−μα2,ψ2α=xα−μα4,ψ3α=ψ4α=xα−μα12,ϑ1α=να−1x+2ν+xα−1ν−4x6,ϑ2α=12να+3xα−1ν+xνα−1−18xα60,ϑ3α=2να+3xα−1ν+xνα−1−8xα60,ϑ4α=3να+2xα−1ν+xνα−1−7xα60,and Δ1x,μ and Δ2x,ν are given in Theorem 3.Proof.
It follows from Lemma1 that(29)να−xαeχν+xα−ναeχμνα−μα−ανα−μα∫μνeχxdαx≤x−μνα−μα∫01xα−1−ξμα+ξxαeχ1−ξμ+xξχ′1−ξμ+ξxdξ+ν−xνα−μα∫011−ξνα−1+ξxα−11−ξν+ξx−xαeχ1−ξν+xξχ′1−ξν+ξxdξ.
Making use of power mean inequality, we get(30)∫01xα−1−ξμα+ξxαeχ1−ξμ+xξχ′1−ξμ+ξxdξ≤∫01xα−1−ξμα+ξxαdξ1−1/q×∫01xα−1−ξμα+ξxαeχ1−ξμ+xξχ′1−ξμ+ξxqdξ1/q.
Similarly, one has(31)∫011−ξνα−1+ξxα−11−ξν+ξx−xα1eχ1−ξν+xξχ′1−ξν+ξxdξ≤∫011−ξνα−1+ξxα−11−ξν+ξx−xαdξ1/q×∫011−ξνα−1+ξxα−11−ξb+ξx−xαeχ1−ξν+xξχ′1−ξν+ξxqdξ1/q.
From the convexity ofeχ′q, we get(32)∫01xα−1−ξμα+ξxαeχ1−ξμ+xξχ′1−ξμ+ξxqdξ≤∫01xα−1−ξμα+ξxα1−ξ2eχμχ′μq+ξ2eχxχ′xq+ξ1−ξΔ1x,μdξ=ψ2αeχμχ′μq+ψ3αeχxχ′xq+ψ4αΔ1x,μ,(33)∫011−ξνα−1+ξxα−11−ξν+tx−xαeχ1−ξν+xξχ′1−ξν+ξxqdξ≤∫011−ξνα−1+ξxα−11−ξν+ξx−xα×1−ξ2eχνχ′νq+ξ2eχxχ′xq+ξ1−ξΔ2x,νdξ=ϑ2αeχμχ′μq+ϑ3αeχxχ′xq+ϑ4αΔ2x,ν,where we have used the facts that(34)ψ1α=∫01xα−1−ξμα+ξxαdξ,ϑ1α=∫011−ξνα−1+ξxα−11−ξν+ξx−xαdξ.
Therefore, Theorem4 follows from (30)–(33).Remark 5.
Letα=1. Then, inequality (27) leads to(35)ν−xeχν+x−νeχμν−μ−1ν−μ∫μveχxdx≤121−1/qx−μ2ν−μ3eχμχ′μq+eχxχ′xq+Δ1x,μ121/q+ν−x2ν−μ3eχνχ′νq+eχxχ′xq+Δ2x,ν121/q.Theorem 5.
Letμ,ν>0 with μ<ν, α∈0,1, p,q>1 such that 1/p+1/q=1, and χ:μ,ν⟶ℝ be an α-differentiable function on μ,ν. Then, the inequality(36)να−xαeχν+xα−ναeχμνα−μα−ανα−μα∫μueχxdαx≤x−μνα−μαℵ1α1−1/qℵ2αeχμχ′μq+ℵ3αeχxχ′xq+ℵ4αΔ1x,μ1/q+ν−xνα−μαℑ1α1−1/qℑ2αeχνχ′νq+ℑ3αeχxχ′xq+ℑ4αΔ2x,ν1/q,holds if Dαeχ∈Lαμ,ν and eχ′q is convex on μ,ν, where(37)ℵ1α=xα−xα+1−μα+1α+1x−μ,ℵ2α=13α+1α+2α+3x−μ3×xαα+1α+2α+3x−μ3+3μα+1α+2α+3x−μ2+6μα+2α+3x−μ−6xα+3−μα+3,ℵ3α=13α+1α+2α+3x−μ3×xαα+1α+2α+3x−μ3−3xα+1α+2α+3x−μ2+6xα+2α+3x−μ−6xα+3−μα+3,ℵ4α=16α+1α+2α+3x−μ3×xαα+1α+2α+3x−μ3−6xα+2−να+2+12xα+3−να+3,ℑ1α=xα+1−να+1α+1ν−x−xα,ℑ2α=13α+1α+2α+3x−ν3×6xα+3−να+3−6να+2α+3x−ν−3να+1α+2α+3x−ν−xαα+1α+2α+3x−ν3,ℑ3α=13α+1α+2α+3x−ν3×3xα+1α+2α+3x−ν2−6α+3x−νxα+2−να+2+6xα+3−να+3−xαα+1α+2α+3x−ν3,ℑ4α=16α+1α+2α+3x−ν3×6xα+2+να+2α+3x−ν−12xα+3−να+3−xαα+1α+2α+3x−ν3.Proof.
From Definition3 and Lemma 1, we get(38)να−xαeχν+xα−ναeχμνα−μα−ανα−μα∫μveχxdαx=x−μνα−μα∫011−ξμ+ξx2α−1−xα1−ξμ+ξxα−1Dαeχ1−ξμ+ξxχα1−ξμ+ξxdξ+ν−xνα−aα∫011−ξν+ξx2α−1−xα1−ξν+ξxα−1Dαeχ1−ξν+ξxχα1−ξν+ξxdξ≤x−μνα−μα∫01xα−1−ξμ+ξxαeχ1−ξμ+ξxχ′1−ξμ+ξxdξ+ν−xνα−μα∫011−ξν+ξxα−xαeχ1−ξν+ξxχ′1−ξν+ξxdξ.
Making use of power mean inequality, we get(39)∫01xα−1−ξμ+ξxαeχ1−ξμ+ξxχ′1−ξμ+ξxdξ≤∫01xα−1−ξμ+ξxα1−1/q×∫01xα−1−ξμ+ξxαeχ1−ξμ+ξxχ′1−ξμ+ξxqdξ1/q.
Similarly, we have(40)∫011−ξν+ξxα−xαeχ1−ξμ+ξxχ′1−ξν+ξxdξ≤∫011−ξν+ξxα−xα1−1/q×∫011−ξν+ξxα−xαeχ1−ξν+ξxχ′1−ξν+ξxqdξ1/q.
It follows from the convexity ofeχ′q that(41)∫01xα−1−ξμ+ξxαeχ1−ξμ+ξxχ′1−ξμ+ξxqdξ≤∫01xα−1−ξμ+ξxα1−ξ2eχμχ′μq+ξ2eχxχ′xq+ξ1−ξΔ1x,μdξ=ℵ2αeχμχ′μq+ℵ3αeχxχ′xq+ℵ4αΔ1x,μ,(42)∫011−ξν+ξxα−xαeχ1−ξν+ξxχ′1−ξν+ξxqdξ≤∫011−ξν+ξxα−xα1−ξ2eχνχ′νq+ξ2eχxχ′xq+ξ1−ξΔ2x,νdξ=ℑ2αeχνχ′νq+ℑ3αeχxχ′xq+ℑ4αΔ2x,ν,where we have used the facts that(43)ℵ1α=∫01xα−1−ξμ+ξxαdξ=xα−xα+1−μα+1α+1x−μ,ℑ1α=∫011−ξν+ξxα−xαdξ=xα+1−να+1α+1ν−x−xα.
Therefore, inequality (36) follows from (38)–(42).Theorem 6.
Letμ,ν>0 with μ<ν, α∈0,1, p,q>1 such that 1/p+1/q=1, and χ:μ,ν⟶ℝ be an α−differentiable function on μ,ν. Then, the inequality(44)να−xαeχν+xα−ναeχμνα−μα−ανα−μα∫μveχxdαx≤x−μνα−μαψ1αeχQ1α/ψ1αχ′Q1αψ1α+ν−xνα−aαϑ1αeχQ2α/ϑ1αχ′Q2αϑ1α,holds if Dαeχ∈Lαμ,ν and eχ′q is concave on μ,ν, where(45)ψ1α=xα−μα2,ϑ1α=να−1x+2ν+xα−1ν−4x6,Q1α=xα−μα2μ+x6,Q2α=3να+1+xα+1+ν+xνα−1x+xα−1ν−6xαν+2x12.Proof.
We clearly see thateχ′ is concave due to eχ′q is concave for q>1. From Definition 3, Lemma 1, the concavity of eχ′, and Jensen’s integral inequality, we get(46)να−xαeχν+xα−ναeχμνα−μα−ανα−μα∫μveχxdαx≤x−μνα−μα∫01xα−1−ξμα+ξxαeχ1−ξμ+ξxχ′1−ξμ+ξxdξ+ν−xνα−μα∫011−ξνα−1+ξxα−11−ξν+ξx−xαeχ1−ξν+ξxχ′1−ξν+ξxdξ,(47)∫01xα−1−ξμ+ξxαeχ1−ξμ+ξxχ′1−ξμ+ξxdξ≤∫01xα−1−ξμα+ξxαdξeχ∫01xα−1−ξμα+ξxα1−ξμ+ξxdξ/∫01xα−1−ξμα+ξxαdξ×χ′∫01xα−1−ξμα+ξxα1−ξμ+ξxdξ∫01xα−1−ξμα+ξxαdξ=ψ1αeχQ1α/ψ1αχ′Q1αψ1α,(48)∫011−ξνα−1+ξxα−11−ξν+ξx−xαeχ1−ξν+ξxχ′1−ξν+ξxdξ≤∫011−ξνα−1+ξxα−11−ξν+ξx−xαdξ×eχ∫011−ξνα−1+ξxα−11−ξν+ξx−xα1−ξν+xξdξ∫011−ξνα−1+ξxα−11−ξν+ξx−xαdξ×χ′∫011−ξνα−1+ξxα−11−ξν+ξx−xα1−ξν+xξdξ∫011−ξνα−1+ξxα−1α1−ξν+ξx−xαdξ=ϑ1αeχQ2α/ϑ1αχ′Q2αϑ1α,where we have used the identities(49)ψ1α=∫01xα−1−ξμα+ξxαdξ=xα−μα2,ϑ1α=∫011−ξνα−1+ξxα−11−ξν+ξx−xαdξ=να−1x+2ν+xα−1ν−4x6,Q1α=∫01xα−1−ξμα+ξxα1−ξμ+ξxdξ=xα−μα2μ+x6,Q2α=∫011−ξνα−1+tξxα−11−ξν+ξx−xα1−ξν+xξdξ=3να+1+xα+1+ν+xνα−1x+xα−1ν−6xαν+2x12.
Therefore, inequality (44) follows from (46)–(48).


## 3. Applications to Special Means

It is very important to give applications in terms of efficiency and usefulness of the obtained results. At the same time, the accuracy of the findings will be confirmed by the applications.A bivariate functionΩ:0,∞×0,∞⟶0,∞ is said to be a mean if mina,b≤Ωa,b≤maxa,b for all a,b∈0,∞. Recently, the bivariate means have been the subject of intensive research [63–75]; in particular, many remarkable inequalities and properties for the bivariate means and their related special functions can be found in the literature [76–85].In this section, we use the results obtained in Section2 to derive some inequalities for the weighted arithmetic mean(50)Aμ,ν;w1,w2=w1μ+w2νw1+w2,μ,ν>0and α,s-th generalized logarithmic mean(51)Lα,sμ,ν=ανα+s−μα+sα+sνα−μα1/sfor μ,ν>0 with μ≠ν, s∈ℝ\0 and α∈−1,0∪0,1.Letχx=xlogx. Then, Theorems 2–4 and (50) and (51) lead to Theorems 7–9 immediately.Theorem 7.
Letμ,ν>0 with μ<ν. Then, the inequality(52)να−μαAμs,νs;xα−μα,να−xα−Lα,ssμ,ν≤sx−μνα−μα3μs−1+xs−1+μsx−1+xsμ−1+sν−xνα−μα12να+3xνα−1+νxα−1−18xανs−1+2να+3xνα−1+νxα−1−8xαxs−1+3να+2xνα−1+νxα−1−7xανsx−1+ν−1xs,holds for all α∈0,1, s>1, and x∈μ,ν.Theorem 8.
Letμ,ν>0 with μ<ν. Then, the inequality(53)να−μαAμs,νs;xα−μα,να−xα−Lα,ssμ,ν≤sx−μνα−μαφ1α,p1/p2μqs−1+xqs−1+μsqxqs−1+μsq−1xqs61/q+sν−xνα−μαφ2α,p1/p2νqs−1+xqs−1+νsqxqs−1+νsq−1xqs61/q,holds for all α∈0,1, s>1 and x∈μ,ν.Theorem 9.
Letμ,ν>0 with μ<ν. Then, the inequality(54)να−μαAμs,νs;xα−μα,να−xα−Lα,ssμ,ν≤sx−μνα−μαψ1α,p1−1/qψ2αμqs−1+ψ3αxqs−1+ψ4αμsqxqs−1+μsq−1xqs1/q+sν−xνα−μαϑ1α,p1−1/qϑ2ανqs−1+ϑ3αxqs−1+ϑ4ανsqxqs−1+νsq−1xqs1/qholds for all α∈0,1, s>1, and x∈μ,ν, where ψiα and ϑiα for i=1,2,3 are defined in Theorem 4.



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*Source: 6517068-2020-01-27.xml*