Ohlin and Levin-Stečkin-type results for strongly convex functions
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.
scientific article; zbMATH DE number 7342910
scientific article