%0 Journal Article
%T Higher order extension of L£¿wner's theory: Operator $k$-tone functions
%A Uwe Franz
%A Fumio Hiai
%A ¨¦ric Ricard
%J Mathematics
%D 2011
%I arXiv
%R 10.1090/S0002-9947-2014-05942-4
%X The new notion of operator/matrix $k$-tone functions is introduced, which is a higher order extension of operator/matrix monotone and convex functions. Differential properties of matrix $k$-tone functions are shown. Characterizations, properties, and examples of operator $k$-tone functions are presented. In particular, integral representations of operator $k$-tone functions are given, generalizing familiar representations of operator monotone and convex functions.
%U http://arxiv.org/abs/1105.3881v4