Fractional inclusions of the Hermite\u2013Hadamard type for m-polynomial convex interval-valued functions

Abstract

The notion of m-polynomial convex interval-valued function Psi =[psi ^{-}, psi ^{+}] is hereby proposed. We point out a relationship that exists between Ψ and its component real-valued functions psi ^{-} and psi ^{+}. For this class of functions, we establish loads of new set inclusions of the Hermite–Hadamard type involving the ρ-Riemann–Liouville fractional integral operators. In particular, we prove, among other things, that if a set-valued function Ψ defined on a convex set S is m-polynomial convex, rho,epsilon >0 and zeta,eta in {mathbf{S}}, then mm+2−m−1Ψ(ζ+η2)⊇Γρ(ϵ+ρ)(η−ζ)ϵρ[ρJζ+ϵΨ(η)+ρJη−ϵΨ(ζ)]⊇Ψ(ζ)+Ψ(η)m∑p=1mSp(ϵ;ρ),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\frac{m}{m+2^{-m}-1}\\Psi \\biggl(\\frac{\\zeta +\\eta }{2} \\biggr)& \\supseteq \\frac{\\Gamma _{\\rho }(\\epsilon +\\rho )}{(\\eta -\\zeta )^{\\frac{\\epsilon }{\\rho }}} \\bigl[{_{\\rho }{\\mathcal{J}}}_{\\zeta ^{+}}^{\\epsilon } \\Psi (\\eta )+_{ \\rho }{\\mathcal{J}}_{\\eta ^{-}}^{\\epsilon }\\Psi (\\zeta ) \\bigr] \\\\ & \\supseteq \\frac{\\Psi (\\zeta )+\\Psi (\\eta )}{m}\\sum_{p=1}^{m}S_{p}( \\epsilon;\\rho ), \\end{aligned}$$ \\end{document} where Ψ is Lebesgue integrable on [zeta,eta ], S_{p}(epsilon;rho )=2-frac{epsilon }{epsilon +rho p}- frac{epsilon }{rho }mathcal{B} (frac{epsilon }{rho }, p+1 ) and mathcal{B} is the beta function. We extend, generalize, and complement existing results in the literature. By taking mgeq 2, we derive loads of new and interesting inclusions. We hope that the idea and results obtained herein will be a catalyst towards further investigation.