Abstract
In this paper, firstly we have established a new generalization of Hermite–Hadamard inequality via p-convex function and fractional integral operators which generalize the Riemann–Liouville fractional integral operators introduced by Raina, Lun and Agarwal. Secondly, we proved a new identity involving this generalized fractional integral operators. Then, by using this identity, a new generalization of Hermite–Hadamard type inequalities for fractional integral are obtained.
Highlights
In mathematical literature, the Hermite–Hadamard inequality, named after Charles Hermite (1822–1901) and Jacques Hadamard (1865–1963) and sometimes called Hadamard’s inequality, states that if a function f : [a, b] −→ R is convex, the following chain of inequalities holds: f a+b 2≤1 b−a b f (x)dx ≤ f (a) + f (b) . a (1)The inequality (1) is one of the most famous result for convex functions
A number of paper have been written on this inequality providing new proofs note worthy extensions, generalizations, refinements and new inequalities connected with the Hermite–Hadamard inequality
Since f is p-convex function on [a, b], we have for all x, y ∈ [a, b]
Summary
The inequality (1) is one of the most famous result for convex functions. In [8], Zhang and Wan gave definition of p-convex function as follows. The Hermite–Hadamard inequality for p-convex functions is as the following (see [11,12]) For some result related to p-convex functions and its generalizations, we refer the reader to see [8,9,11,12,13,14,15,16,17,18].
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