Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions

Abstract

The purpose of this study is to introduce the new class of Hermite–Hadamard inequality for LR-convex interval-valued functions known as LR-interval Hermite–Hadamard inequality, by means of pseudo-order relation ( ≤p ). This order relation is defined on interval space. We have proved that if the interval-valued function is LR-convex then the inclusion relation “ ⊆ ” coincident to pseudo-order relation “ ≤p ” under some suitable conditions. Moreover, the interval Hermite–Hadamard–Fejér inequality is also derived for LR-convex interval-valued functions. These inequalities also generalize some new and known results. 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.

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

“…If ϕ(ω, µ) = ω − µ, then from Theorem 3, we get the following new result in fractional calculus, see [42].…”

“…+ D(µ + ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − D(µ) ≤ I α µ + S * D(µ + ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − S * D(µ) * 2µ+ϕ(ω,µ) 2 I α µ + D(µ + ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − D(µ) ≤ I α µ + S * D(µ + ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − S * D(µ) ,from which, we have:+ D(µ + ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − D(µ) ≤ p I α µ + S * D(µ + ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − S * D(µ), I α µ + S * D(µ + ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − S * D(µ) + D(µ + ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − D(µ) ≤ p I α µ + SD(µ + ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − SD(µ)This completes the proof. If D(z) = 1, then from (26) and (31), we get Theorem 3.If χ(ς) = ς, then from (26) and (31), we achieve the following coming inequality, see[42]:S 2µ+ϕ(ω,µ) 2 I α µ + D(µ + ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − D(µ) ≤ p I α µ + SD(µ + {ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − SD(µ) ≤ p S(µ)+S(µ+ϕ(ω,µ)) 2 I α µ + D(µ + ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − D(µ) ≤ p S(µ)+S(ω) 2 I α µ + D(µ + ϕ(ω, µ)) + I α µ+ϕ(ω,µ) − D(µ)…”

“…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.…”

Advances in Optimization and Nonlinear Analysis

“…The purpose of the next paper Khan et al [7] published in this Special Issue is to introduce a new class of Hermite-Hadamard inequalities for LR-convex interval-valued functions, by means of a pseudo-order relation. This order relation is defined on interval space.…”

Post-Quantum Midpoint-Type Inequalities Associated with Twice-Differentiable Functions

“…In the case of concave mappings, the above inequality is satisfied in reverse order. For more recent refinements of Inequality (1), one can consult [2,3].…”