Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities : Open Mathematics

摘要

The connection between the functional inequalities $$fleft( {rac{{x + y}}{2}} ight) leqslant rac{{fleft( x ight) + fleft( y ight)}}{2} + lpha _J left( {x - y} ight), x,y in D,$$ and $$int_0^1 {fleft( {tx + left( {1 - t} ight)y} ight)ho left( t ight)dt leqslant lambda fleft( x ight) + left( {1 - lambda } ight)fleft( y ight) + lpha _{m H} left( {x - y} ight),} x,y in D,$$ is investigated, where D is a convex subset of a linear space, f: D → ?, α H;α J: D-D → ? are even functions, λ ∈ [0; 1], and ρ: [0; 1] →?+ is an integrable nonnegative function with ∫01 ρ(t) dt = 1.