Abstract
In this paper, we establish inequalities of Hermite–Hadamard type for harmonically convex functions using a generalized fractional integral. The results of our paper are an extension of previously obtained results (İşcan in Hacet. J. Math. Stat. 43(6):935–942, 2014 and İşcan and Wu in Appl. Math. Comput. 238:237–244, 2014). We also discuss some special cases for our main results and obtain new inequalities of Hermite–Hadamard type.
Highlights
1 Introduction The Hermite–Hadamard inequality introduced by Hermite and Hadamard, see [4], and [17, p. 137], is one of the best-established inequalities in the theory of convex analysis with a nice geometrical interpretation and several applications
It is worth mentioning that the Hermite–Hadamard inequality may be regarded as a refinement of the concept of convexity and it follows from Jensen’s inequality
If f is harmonically convex function on [a, b], the following double inequality holds for the fractional integrals: f
Summary
The Hermite–Hadamard inequality introduced by Hermite and Hadamard, see [4], and [17, p. 137], is one of the best-established inequalities in the theory of convex analysis with a nice geometrical interpretation and several applications. If |f |q is harmonically convex function on [a, b] for q ≥ 1, the following inequality holds: f (a) + f (b) ab b–a b f (x) a x2 dx ab(b – 2 a) İşcan [10] established the following identity and inequalities of Hermite–Hadamard type for harmonically convex functions via Riemann–Liouville fractional integrals.
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