Abstract
In this paper, a new identity for the generalized fractional integral is defined. Using this identity we studied a new integral inequality for functions whose first derivatives in absolute value are convex. The new generalized Hermite-Hadamard inequality for generalized convex function on fractal sets involving Katugampola type fractional integral is established. This fractional integral generalizes Riemann-Liouville and Hadamard’s integral, which possess a symmetric property. We derive trapezoid and mid-point type inequalities connected to this generalized Hermite-Hadamard inequality.
Highlights
The emergence of convexity theory, in the field of mathematical analysis, has been considered as the remarkable development
Many new classes of inequalities that are related to the convex functions have been derived and applied to other field of studies, see [1,2]
A very new study was carried out on the improvement of Hermite-Hadamard type inequalities via generalized convex functions on fractal set, see [21], and we provide the definition of this concept as Definition 2
Summary
The emergence of convexity theory, in the field of mathematical analysis, has been considered as the remarkable development. The aim of this paper is to generalize the Hermite-Hadamard inequality for generalized convex functions on fractal sets via Katugampola fractional integrals. This can be the generalization of the work of Chen and Katugampola [25], who proposed the inequality stated in Theorem 3 Another objective of this study is to define a new identity for generalized fractional integrals, through which generalized Hermite-Hadamard type inequalities for convex function are derived. The following theorem generalizes the result obtained by [25] of the Hermite-Hadamard inequality involving the Katugampola fractional integrals for generalized convex function on fractal sets. We derive the mid-point type inequalities via generalized convex functions on the fractal set for the Katugampola fractional integral. The trapezoid-type inequalities via generalized convex function on fractal sets for Katugampola fractional integrals can be derived using Lemma 1.
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