Operator monotonicity of functions related to the Stolarsky mean and exp${f(x)}$ (The research of geometric structures in quantum information based on Operator Theory and related topics)

Abstract

The weighted power mean is one of the most famous 2-parameter operator mean, and its representing function is P_{s, $alpha$}[(1-$alpha$)+$alpha$x^{s}]^{frac{1}{$theta$}}(sin [-1, 1], $alpha$ in [0, 1 In [6] we constructed a 2-parameter family of operator monotone function F_{r, s}(x)(r, sin[-1, 1]) by integration of the function P_{s. $alpha$}(x) of $alpha$in[0, 1]. We shall extend its range of parameters r and s. We also consider operator monotonicity of exp{f(x)} for a non-constant continuous function f(x) defined on (0, infty).