Harmonically Convex Fuzzy-Interval-Valued Functions and Fuzzy-Interval Riemann–Liouville Fractional Integral Inequalities

Abstract

It is well known that the concept of convexity establishes strong relationship with integral inequality for single-valued and interval-valued function. The single-valued function and interval-valued function both are special cases of fuzzy interval-valued function. The aim of this paper is to introduce a new class of convex fuzzy interval-valued functions, which is called harmonically convex fuzzy interval-valued functions (harmonically convex fuzzy-IVFs) by means of fuzzy order relation and to investigate this new class via fuzzy-interval Riemann-Liouville fractional operator. With the help of fuzzy order relation and fuzzyinterval Riemann-Liouville fractional, we derive some integrals inequalities of Hermite-Hadamard (H-H) type and Hermite-Hadamard-Fejér (H-H Fejér) type as well as some product inequities for harmonically convex fuzzy-IVFs. Our results represent a significant improvement and refinement of the known results. We hope that these interesting outcomes may open a new direction for fuzzy optimization, modeling and interval-valued function.

An extensive note on various fractional-order type operators and some of their effects to certain holomorphic functions

“…Our paper is one of numerous applications of those operators described by fractional order calculus, which has been expressed as Tremblay operator (for certain functions with complex variable) in the recent passed years. For the details of those fractional-order (type) operators and some of their (comprehensive) applications, one may refer to the main works in [6], [21], [22], [23], [24] and [25], and also see the earlier papers in [2], [3], [8], [27], [11], [13], [14], [15], [16], [17], [18], [19] and [24] as numerous different investigations.…”

Riemann–Liouville Fractional Integral Inequalities for Generalized Pre-Invex Functions of Interval-Valued Settings Based upon Pseudo Order Relation

“…Khan et al also presented the new class of convex fuzzy mappings known as (χ 1 , χ 2 )-convex fuzzy-interval-valued functions ((χ 1 , χ 2 )-convex F-I-V•F) and obtained the new version of H-H inequalities for (χ 1 , χ 2 )-convex F-I-V•Fs. Moreover, Khan et al introduced new notions of generalized convex F-I-V•Fs, and derived new fractional H-H type and H-H type inequalities for convex F-I-V•Fs [27][28][29][30][31][32]. For more analysis and applications of F-I-V•Fs, see [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50] and the references therein.…”

Fractional Calculus - Theory and Applications

“…Due to the vast applications of convexity and fractional HH-inequality in mathematical analysis and optimization, many authors have discussed the applications, refinements, generalizations, and extensions, see [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] and the references therein.…”