Strong monotonicity for various means

Abstract

Point-wise monotonicity (in parameters) for various one-parameter families of scalar means such as power difference means, binomial means and Stolarsky means is well known, but norm comparison for corresponding operator means requires monotonicity in the sense of positive definiteness. Among other things we obtain monotonicity in the sense of infinite divisibility, which is much stronger than that in the sense of positive definiteness. These strong monotonicity results are proved based on explicit computations for measures in relevant Lévy–Khintchine (or actually Kolmogorov) formulas.