Approximate convexity with respect to integral arithmetic mean
Bull. Korean Math. Soc. 2014 Vol. 51, No. 6, 1829-1839
Printed November 30, 2014
Marek \.Zo\l{}dak
Prof. St. Pigonia 1
Abstract : Let $(\Omega, {\mathcal S}, \mu)$ be a probabilistic measure space, $\varepsilon \in \R$, $\delta \geq 0$, $p>0$ be given numbers and let $P \subset \R$ be an open interval. We consider a class of functions $f: P \rightarrow \R$, satisfying the inequality $$f(EX) \leq E(f \circ X)+\varepsilon E(|X-EX|^p)+\delta$$ for each ${\mathcal S}$-measurable simple function $X: \Omega \rightarrow P$. We show that if additionally the set of values of $\mu$ is equal to $[0,1]$ then $f: P \rightarrow \R$ satisfies the above condition if and only if $$f(tx+(1-t)y) \leq tf(x)+(1-t)f(y)+\varepsilon \left[(1-t)^p t+t^p (1-t)\right] |x-y|^p +\delta$$ for $x,y \in P$, $t \in [0,1]$. We also prove some basic properties of such functions, e.g. the existence of subdifferentials, Hermite-Hadamard inequality.
Keywords : approximate convexity, Jensen integral inequality, Hermite-Hadamard inequality
MSC numbers : 26A51, 26B25
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