Convexity According to Means

Generalized Steffensen Means

Generalizing Jensen and Bregman divergences with comparative convexity and the statistical Bhattacharyya distances with comparable means

Comparative convexity is a generalization of convexity relying on abstract notions of means. We define the (skew) Jensen divergence and the Jensen diversity from the viewpoint of comparative convexity, and show how to obtain the generalized Bregman divergences as limit cases of skewed Jensen divergences. In particular, we report explicit formula of these generalized Bregman divergences when con...

Schur Power Convexity of the Daróczy Means

In this paper, the Schur convexity is generalized to Schur f -convexity, which contains the Schur geometrical convexity, harmonic convexity and so on. When f : R+ →R is defined by f (x) = (xm−1)/m if m = 0 and f (x) = lnx if m = 0 , the necessary and sufficient conditions for f -convexity (is called Schur m -power convexity) of Daróczy means are given, which improve, generalize and unify Shi et...

Schur–harmonic Convexity for Differences of Some Special Means in Two Variables

In the paper, the authors find Schur-harmonic convexity of linear combinations of differences between some means such as the arithmetic, geometric, harmonic, and root-square means, and establish some inequalities related to these means and differences. Mathematics subject classification (2010): Primary 26E60; Secondary 26B25, 26D20.

Convexity of the Zero-balanced Gaussian Hypergeometric Functions with Respect to Hölder Means

In this note we investigate the convexity of zero-balanced Gaussian hypergeometric functions and general power series with respect to Hölder means.