Abstract
Abstract The key purpose of this study is to suggest a new fractional extension of Hermite–Hadamard, Hermite–Hadamard–Fejér and Pachpatte-type inequalities for harmonically convex functions with exponential in the kernel. Taking into account the new operator, we derived some generalizations that capture novel results under investigation with the aid of the fractional operators. We presented, in general, two different techniques that can be used to solve some new generalizations of increasing functions with the assumption of convexity by employing more general fractional integral operators having exponential in the kernel have yielded intriguing results. The results achieved by the use of the suggested scheme unfold that the used computational outcomes are very accurate, flexible, effective and simple to perform to examine the future research in circuit theory and complex waveforms.
Highlights
Introduction and preliminariesThe Hermite–Hadamard inequality is a well-known, paramount and extensively used inequality in the applied literature of mathematical inequalities [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]
We propose an innovative class of functional variants for harmonically convex functions and several other generalizations for the convexity theory as novel fractional operators with the exponential kernel are new and effectively applicable
The key procedure of the new adaption in extended form with an exponential kernel to the more general fractional integral operator is helpful in deriving several generalizations for the convexity theory
Summary
The Hermite–Hadamard inequality is a well-known, paramount and extensively used inequality in the applied literature of mathematical inequalities [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. We use the fractional integral operator for the integrable functions to establish Hermite–Hadamard, Hermite–Hadamard– Fejér and Pachpatte-type integral inequalities for harmonically convex functions. We define the one-sided definition of a more general fractional integral operator having exponential in their kernel as follows. We derive the Hermite–Hadamard inequality for harmonically convex functions in the frame of a new fractional integral operator as follows. For the proof of the second inequality in (2.1), we first note that if is a harmonically convex function, for ζ ∈ [0, 1], it yields (η1) + (1 − ζ ) Multiplying on both sides of (2.4) by e−θζ and integrating the inequality with respect to ζ from 0 to 1, one obtains. Which is proposed by Iscan in [34]
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