Fuzzy integral inequalities on coordinates of convex fuzzy interval-valued functions.

In this study, we introduce and study new fuzzy-interval integral is known as fuzzy-interval double integral, where the integrand is fuzzy-interval-valued functions (FIVFs). Also, some fundamental properties are also investigated. Moreover, we present a new class of convex fuzzy-interval-valued functions is known as coordinated convex fuzzy-interval-valued functions (coordinated convex FIVFs) through fuzzy order relation (FOR). The FOR (≼) and fuzzy inclusion relation (⊇) are two different concepts. With the help of fuzzy-interval double integral and FOR, we have proved that coordinated convex fuzzy-IVF establish a strong relationship between Hermite-Hadamard (HH-) and Hermite-Hadamard-Fejér (HH-Fejér) inequalities. With the support of this relation, we also derive some related HH-inequalities for the product of coordinated convex FIVFs. Some special cases are also discussed. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.

Hermite-Hadamard inequalities and their applications.

New Hermite-Hadamard type inequalities are established. Some corresponding examples are also discussed in detail.

Fractional Hermite-Hadamard inequalities containing generalized Mittag-Leffler function.

The objective of this paper is to establish some new refinements of fractional Hermite-Hadamard inequalities via a harmonically convex function with a kernel containing the generalized Mittag-Leffler function.

Generalized geometrically convex functions and inequalities.

In this paper, we introduce and study a new class of generalized functions, called generalized geometrically convex functions. We establish several basic inequalities related to generalized geometrically convex functions. We also derive several new inequalities of the Hermite-Hadamard type for generalized geometrically convex functions. Several special cases are discussed, which can be deduced from our main results.

On bounds involving k-Appell's hypergeometric functions.

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.