Comparative convexity
type: Item
instance of: scholarly article
title: Comparative convexity
zbMATH Open document ID: 0947.26009
author: F. W. McElroy
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-11-06T00:00:00Z
review text: The author discusses the usefulness of some concepts of the comparative convexity of two functions. He defines, for instance, \(f\) as \textit{absolutely} more convex than \(g\) on some interval \([a,b]\) if for all \(\lambda_1,\lambda_2\geq 0\), \(\lambda_1+ \lambda_2= 1\), and all \(x_1,x_2\in [a,b]\), \[ \lambda_1 f(x_1)+ \lambda_2f(x_2)- f(\lambda_1 x_1+ \lambda_2 x_2)\geq \lambda_1 g(x_1)+ \lambda_2g(x_2)- g(\lambda_1 x_1+ \lambda_2 x_2); \] and, for functions of the same sign, \(f\) as a \textit{relatively} more convex than \(g\), if for all \(\lambda_1,\lambda_2\geq 0\), \(\lambda_1+ \lambda_2=1\), and all \(x_1,x_2\in [a,b]\), \[ {\lambda_1 f(x_1)+ \lambda_2 f(x_2)\over |f(\lambda_1 x_1+ \lambda_2 x_2)|}\geq {\lambda_1 g(x_1)+ \lambda_2 g(x_2)\over|g(\lambda_1 x_1+ \lambda_2 x_2)|}. \] Several results, some of them written in the language of the random variables and the expectation operator, are presented. Relations to some earlier concepts are discussed.
Mathematics Subject Classification ID: 26A51
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 1271527
zbMATH Keywords: convex function
zbMATH Keywords: expectation operator
zbMATH Keywords: Jensen inequality
zbMATH Keywords: comparative convexity
zbMATH Keywords: random variables
reviewed by: Janusz Matkowski
MaRDI profile type: Publication
full work available at URL: jmaa.1998.6206
OpenAlex ID: W2912542613
cites work: Q4692666
cites work: Comparable means and generalized convexity
cites work: Risk Vulnerability and the Tempering Effect of Background Risk
cites work: Q4357532
cites work: Convex functions, partial orderings, and statistical applications
cites work: Risk Aversion in the Small and in the Large
DOI: 10.1006/JMAA.1998.6206
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description: scientific article; zbMATH DE number 1271527