On the Generalized Hermite–Hadamard Inequalities via the Tempered Fractional Integrals

Pshtiwan Othman Mohammed,Dumitru Baleanu,Mehmet Zeki Sarikaya

doi:10.3390/sym12040595

Pshtiwan Othman Mohammed, Dumitru Baleanu + Show 1 more

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https://doi.org/10.3390/sym12040595

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Abstract

Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if the method is not convergent or stable. Thus, there is a present day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Convexity play a concrete role in the field of inequalities due to the behaviour of its definition. There is a strong relationship between convexity and symmetry. Which ever one we work on, we can apply to the other one due to the strong correlation produced between them especially in recent few years. In this article, we first introduced the notion of λ -incomplete gamma function. Using the new notation, we established a few inequalities of the Hermite–Hadamard (HH) type involved the tempered fractional integrals for the convex functions which cover the previously published result such as Riemann integrals, Riemann–Liouville fractional integrals. Finally, three example are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.

Highlights

Let h : J ⊆ R → R be a convex function and o3, o4 ∈ J with o3 < o4
We followed the Sarikaya et al [3] and Sarikaya and Yildirim [6] technique to establish a few inequalities of Hermite–Hadamard type which involved the tempered fractional integrals and the notion of λ-incomplete gamma function for convex functions
We introduced an extension of the well known incomplete gamma function, namely the λ-incomplete gamma function to connect with the model of tempered fractional integrals

Summary

Introduction

Let h : J ⊆ R → R be a convex function and o3 , o4 ∈ J with o3 < o4. the well known inequalities, namely, the Hermite–Hadamard inequalities [1], defined by ho3 + o4. Let us recall the above definition of the Riemann-Liouville fractional integrals (left and right) which are defined by [15,17]:. In [3], Sarikaya et al established the following trapezoidal type equality and inequality for Riemann-Liouville integral, respectively: Lemma 3. We recall that several researchers the Riemann-Liouville fractionals integrals and provided important generalizations of Hermite–Hadamard type inequalities utilising these type of integrals for various type of convex functions, see, for instance, [3,19,20]. We followed the Sarikaya et al [3] and Sarikaya and Yildirim [6] technique to establish a few inequalities of Hermite–Hadamard type (including both trapezoidal and midpoint type) which involved the tempered fractional integrals and the notion of λ-incomplete gamma function for convex functions.

Hermite–Hadamard Inequalities Involving Beta Function

Examples

Conclusions