Sandwich theorem for reciprocally strongly convex functions

Abstract

We introduce the notion of reciprocally strongly convex functions and we present some examples and properties of them. We also prove that two real functions f and g, defined on a real interval [a, b], satisfy for all x, y ∈ [a, b] and t ∈ [0, 1] iff there exists a reciprocally strongly convex function h : [a, b] → R such that f (x) ≤ h(x) ≤ g(x) for all x ∈ [a, b]. Finally, we obtain an approximate convexity result for reciprocally strongly convex functions; namely we prove a stability result of Hyers-Ulam type for this class of functions.

Hyers–Ulam Stability of 2D-Convex Mappings and Some Related New Hermite–Hadamard, Pachpatte, and Fejér Type Integral Inequalities Using Novel Fractional Integral Operators via Totally Interval-Order Relations with Open Problem

“…The topic of approximate convexity and its connection to other generalized convex mappings is rarely discussed, but a few recent advancements have been discussed by several authors. Using harmonically convex mappings, Bracamonte et al [44] discussed the sandwich theorem and Hyers-Ulam stability results. Forti [45] discussed Hyers-Ulam stability of functional equations with applications spanning varied disciplines.…”

“…In regard to the infinite version of the Hyers-Ulam stability theorem, Emanuele Casini and Pierluigi Papinia [47] provided an interesting counterexample. Bracamonte et al [48] defined an approximate convexity result for reciprocally strongly convex functions; specifically, they proved a Hyers-Ulam stability result for this class of functions. Flavia Corina [49] used set-valued mappings to explore convexity and its associated sandwich theorem, among other fascinating properties.…”

“…Inspiration from strong relevant literature concerning produced results, in particular publications [15,44,58], urges us to construct new and better versions of three well-known inequalities with applications. This article is structured as follows.…”

Strongly MφMψ -Convex Functions, The Hermite–Hadamard–Fejér Inequality and Related Results

“…Recently, some special cases of the class of functions defined above have been studied. For example, the definition and some properties of the reciprocally strongly convex functions (or strongly HA-convex) are given in [4], some properties of strongly p-convex functions (or strongly M p A-convex) are obtained in [15], the Hermite-Hadamard type inequalities for strongly GAconvex are studied in [14].…”

Post Quantum-Hermite-Hadamard Inequalities for Differentiable Convex Functions With a Critical Point

“…A large number of inequalities were obtained by means of convex functions see [9][10][11][12]. A classical inequality for convex functions is the Hermite-Hadamard inequality, and is given as follows:…”