Comparative convexity (Q1284040)

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.