Ohlin and Levin-Stečkin-type results for strongly convex functions
type: Item
instance of: scholarly article
title: Ohlin and Levin-Stečkin-type results for strongly convex functions
zbMATH Open document ID: 1466.26010
author: Teresa Rajba
author: Kazimierz Nikodem
published in: Annales Mathematicae Silesianae
publication date: 2021-05-03T00:00:00Z
review text: Let \(X\), \(Y\) be two real valued random variables such that \(E[X]=E[Y]\). It is said that \(X\) is dominated by \(Y\) in convex stochastic ordering sense if \(E[f(X)]\leq E[f(Y)]\) for every convex function \(f:R\to R\). Ohlin's lemma gives sufficient conditions for \(X\) to be dominated by \(Y\) in such ordering. More precisely, if the distribution functions \(F_{X}\), \(F_{Y}\) cross one time, i.e., there exists a real number \(t_0\) such that \(F_{X}(t)\leq F_{Y}(t)\) if \(t<t_0\), and \(F_{X}(t)\geq F_{Y}(t)\) if \(t>t_0\), then \(X\) is dominated by \(Y\) in convex stochastic ordering sense. Necessary and sufficient conditions in this context were given by \textit{V. I. Levin} and \textit{S. B. Stechkin} [Transl., Ser. 2, Am. Math. Soc. 14, 1--29 (1960; Zbl 0100.04503)]. A function \(f:I\to R\) defined on an interval \(I\subset R\) is called strongly convex with modulus \(c>0\) if \(f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-ct(1-t)(x-y)^2\), for all \(x,y\in I\) and \(t\in (0,1)\). In this paper the authors prove counterparts of the Ohlin and Levin-Steckin theorems for strongly convex functions. As applications they prove some known inequalities related with strongly convex functions in an alternative and unified way.
reviewed by: Ioan Raşa
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 39B62
zbMATH DE Number: 7342910
zbMATH Keywords: Ohlin's lemma
zbMATH Keywords: stochastic ordering
zbMATH Keywords: Levin-Stečkin theorem
zbMATH Keywords: strongly convex functions
zbMATH Keywords: Hermite-Hadamard inequalities
MaRDI profile type: Publication
full work available at URL: amsil-2020-0017
OpenAlex ID: W3044396786
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DOI: 10.2478/AMSIL-2020-0017
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description: scientific article; zbMATH DE number 7342910