# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Proc. Amer. Math. Soc. 139 (2011), 4217-4223
DOI: https://doi.org/10.1090/S0002-9939-2011-10935-3
Published electronically: April 26, 2011

## Abstract:

According to a celebrated result by Löwner, a real-valued function $f$ is operator monotone if and only if its Löwner matrix, which is the matrix of divided differences $L_f=\left (\frac {f(x_i)-f(x_j)}{x_i-x_j}\right )_{i,j=1}^N$, is positive semidefinite for every integer $N>0$ and any choice of $x_1,x_2,\ldots ,x_N$. In this paper we answer a question of R. Bhatia, who asked for a characterisation of real-valued functions $g$ defined on $(0,+\infty )$ for which the matrix of divided sums $K_g=\left (\frac {g(x_i)+g(x_j)}{x_i+x_j}\right )_{i,j=1}^N$, which we call its anti-Löwner matrix, is positive semidefinite for every integer $N>0$ and any choice of $x_1,x_2,\ldots ,x_N\in (0,+\infty )$. Such functions, which we call anti-Löwner functions, have applications in the theory of Lyapunov-type equations.
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