Abstract
In this paper, we derive a new extension of Hermite-Hadamard’s inequality via k-Riemann-Liouville fractional integrals. Two new k-fractional integral identities are also derived. Then, using these identities as an auxiliary result, we obtain some new k-fractional bounds which involve k-Appell’s hypergeometric functions. These bounds can be viewed as new k-fractional estimations of trapezoidal and mid-point type inequalities. These results are obtained for the functions which have the harmonic convexity property. We also discuss some special cases which can be deduced from the main results of the paper.
Highlights
1 Introduction and preliminaries Convexity theory has played a pivotal role through its numerous applications in different fields of pure and applied sciences
In the past few years several new generalizations and extensions of classical convexity have been proposed in the literature, see [ – ]
Definition . ([ ]) A set ⊂ R+ is said to be a harmonically convex set if xy ∈, ∀x, y ∈, t ∈ [, ]
Summary
Introduction and preliminariesConvexity theory has played a pivotal role through its numerous applications in different fields of pure and applied sciences. Shi et al [ ] introduced the notion of harmonic convex sets as follows. Iscan [ ] introduced the class of harmonic convex functions. The Riemann-Liouville integrals Jaα+ f and Jbα– f of order α > with a ≥ are defined by
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