A characterisation of anti-Loewner functions

Abstract

In this paper we answer a question of R. Bhatia, by providing a characterisation of real-valued functions $g$ defined on $[0,+\infty]$ for which the matrix $(\frac{g(x_i)+g(x_j)}{x_i+x_j})_{i,j=1}^N$ is positive semidefinite, for every integer $N>0$ and any choice of $x_1,x_2,\ldots,x_N \in (0,+\infty)$.