APPROXIMATE CONVEXITY WITH RESPECT TO INTEGRAL ARITHMETIC MEAN

Title & Authors
APPROXIMATE CONVEXITY WITH RESPECT TO INTEGRAL ARITHMETIC MEAN
Zoldak, Marek;

Abstract
Let ($\small{{\Omega}}$, $\small{\mathcal{S}}$, $\small{{\mu}}$) be a probabilistic measure space, $\small{{\varepsilon}{\in}\mathbb{R}}$, $\small{{\delta}{\geq}0}$, p > 0 be given numbers and let $\small{P{\subset}\mathbb{R}}$ be an open interval. We consider a class of functions $\small{f:P{\rightarrow}\mathbb{R}}$, satisfying the inequality $\small{f(EX){\leq}E(f{\circ}X)+{\varepsilon}E({\mid}X-EX{\mid}^p)+{\delta}}$ for each $\small{\mathcal{S}}$-measurable simple function $\small{X:{\Omega}{\rightarrow}P}$. We show that if additionally the set of values of $\small{{\mu}}$ is equal to [0, 1] then $\small{f:P{\rightarrow}\mathbb{R}}$ satisfies the above condition if and only if $\small{f(tx+(1-t)y){\leq}tf(x)+(1-t)f(y)+{\varepsilon}[(1-t)^pt+t^p(1-t)]{\mid}x-y{\mid}^p+{\delta}}$ for $\small{x,y{\in}P}$, $\small{t{\in}[0,1]}$. We also prove some basic properties of such functions, e.g. the existence of subdifferentials, Hermite-Hadamard inequality.
Keywords
Language
English
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