Bivariate log-convexity of the more extended means and its applications

Abstract

In this paper, the bivariate log-convexity of the two-parameter homogeneous function in parameter pair is vestigated. From this the bivariate log-convexity of the more extended means with respect to a parameter pair is solved. It follows that Stolarsky means, Gini means, two-parameter identric (exponential) means and two-parameter Heronian means are all bivariate log-concave on $[0{,\infty )}^2$ and log-convex on with respect to parameters. Lastly, some classical and new inequalities for means are given.

12 pages