New fuzzy-interval inequalities in fuzzy-interval fractional calculus by means of fuzzy order relation

Abstract

<abstract> <p>It is well-known that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval analysis and fuzzy-interval analysis, the inclusion relation (⊆) and fuzzy order relation $\left(\preccurlyeq \right)$ both are two different concepts, respectively. In this article, with the help of fuzzy order relation, we introduce fractional Hermite-Hadamard inequality (<italic>HH</italic>-inequality) for <italic>h</italic>-convex fuzzy-interval-valued functions (<italic>h</italic>-convex-IVFs). Moreover, we also establish a strong relationship between <italic>h</italic>-convex fuzzy-IVFs and Hermite-Hadamard Fejér inequality (<italic>HH</italic>-Fejér inequality) via fuzzy Riemann Liouville fractional integral operator. It is also shown that our results include a wide class of new and known inequalities for <italic>h</italic>-convex fuzz-IVFs and their variant forms as special cases. Nontrivial examples are presented to illustrate the validity of the concept suggested in this review. This paper's techniques and approaches may serve as a springboard for further research in this field.</p> </abstract>

Integral Inequalities for Generalized Harmonically Convex Functions in Fuzzy-Interval-Valued Settings

“…If h(ξ) = ξ, then from Theorems 10 and 11, we obtain results for harmonically convex F-I-V-Fs, see [28].…”

“…, where s ∈ (0, 1), then Theorem 4 reduces to the result for the harmonically s-convex fuzzy-interval-valued function, see [28]:…”

“…Kamran et al [26] developed the H•H inequality by means of the notion of interval-valued generalized p-convex functions. Khan et al introduced new classes of convex and generalized convex F-I-V-F, and derived fractional H•H type and H•H type inequalities for convex F-I-V-F [27], h-convex F-I-V-F [28], (h 1 , h 2 )-connvex F-I-V-F [29], (h 1 , h 2 )-preinvex F-I-V-F [30], log-h-convex F-I-V-Fs [31], logs-convex F-I-V-Fs in the second sense [32], and the references therein. We refer the readers for further analysis to literature on the applications and properties of fuzzy-interval, and inequalities and generalized convex fuzzy mappings, see and the references therein.…”

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.…”

Some New Versions of Hermite–Hadamard Integral Inequalities in Fuzzy Fractional Calculus for Generalized Pre-Invex Functions via Fuzzy-Interval-Valued Settings

“…Khan et al [24][25][26][27] extended the class of convex F•Ms and defined h-convex and (h 1 , h 2 )-convex F-IV-Fs using fuzzy partial order relation. Moreover, they introduced H.H, Hermite-Hadamard-Fejér (H.H Fejér), fractional H.H and H.H Fejér for h-convex and (h 1 , h 2 )-convex F•I•V•Fs via fuzzy Riemannian and fuzzy Riemann-Liouville fractional integrals.…”