On an inequality for convex functions with some applications on fractional derivatives and fractional integrals

Abstract

Abstract. The main goal of the paper is to state and prove the new general inequality for convex and increasing functions. We introduce some new inequalities by involving some fractional integrals and fractional derivatives of Riemman-Liouville, Canavati, Hadamard and Erdelyi-Kóber type and apply our result to multidimensional setting to obtain new results involving mixed Riemman-Liouville fractional integrals.Mathematics subject classification (2010): Primary 26D10, Secondary 26D15.

Hardy-type inequalities within fractional derivatives without singular kernel

“…In 2010, Iqbal et al [7] obtained new fractional inequalities within fractional derivatives and integrals of Riemann-Liouville type. In 2011, 2013 and 2014, they proved some new inequalities involving Riemann-Liouville fractional integrals, Caputo fractional derivative and other fractional derivatives; see [11][12][13][14].…”

“…In Theorem 2.1, by replacing k 1 (x, t) by k 1 (x, t)g 2 (t) and g by g 1 g 2 , where the functions g j : Ω 2 → R are measurable for j = 1, 2, the following result is obtained (see [11]).…”

Untitled

“…This work is inspired by [6], [8], [9], [11] - [15]. Here we consider the Prabhakar function (also known as the three parameter Mittag-Leffler function, an entire function if z ∈ C), (see [5], p. 97; [4])…”