Abstract
The aim of our study is to establish, for convex functions on an interval, a midpoint version of the fractional HHF type inequality. The corresponding fractional integral has a symmetric weight function composed with an increasing function as integral kernel. We also consider a midpoint identity and establish some related inequalities based on this identity. Some special cases can be considered from our main results. These results confirm the generality of our attempt.
Highlights
Let J ⊂ R be an interval and let u : J → R be a continuous function
We have investigated a midpoint fractional HHF integral inequality by using the weighted fractional integrals with positive weighted symmetric function kernels, which is the midpoint version of (9)
The existing versions of HHF integral inequalities (7) and (8) have been successfully applied to other classes of convex functions, see [46,47,48]
Summary
Let J ⊂ R be an interval and let u : J → R be a continuous function. the function u is called convex if it satisfies. Various forms of fractional derivatives including RL, Hadamard, Caputo, Caputo–Hadamard, Riesz, ψ–RL, Prabhakar, and weighted versions [12,13,14,15,16] have been developed to date. Most of these versions are described in the RL sense based on the corresponding fractional integral. In 2013, the HH integral inequality (2) was generalized and reformulated by Sarikaya et al [25] in terms of RL fractional integrals. Their result is given by: c1 + c2
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