Strong monotonicity for various means (Q403304)

A scalar mean is a symmetric and homogeneous continuous non-negative function \(M : [0, \infty)^2 \to \mathbb R^{\geq 0}\) such thatNEWLINENEWLINE(i) \(M (s, t) = M (t, s)\),NEWLINENEWLINE(ii) \(M (rs, rt) = rM (s, t)\) (\(r > 0\)),NEWLINENEWLINE(iii) \(M (s, t)\) is non-decreasing in \(s\) and \(t\),NEWLINENEWLINE(iv) \(\min(s, t) \leq M (s, t) \leq \max(s, t)\).NEWLINENEWLINEAssume that \(\mu\) is a probability measure on \(\mathbb R\). Then \(\mu\) is called infinitely divisible if \(\mu = \mu_1 \ast \dots \ast \mu_n\), where \(n \in \mathbb N\), \(\ast\) is the convolution product and \(\mu_n\) are probability measures.NEWLINENEWLINE In the paper under review, the author studies norm comparison for operator means with the power difference means, binomial means and Stolarsky means. For instance, the ratio \(B_\alpha (e^{2t} , 1)/B_\beta (e^{2t} , 1)\) is infinitely divisible for \(\beta \geq \alpha\), where NEWLINE\[NEWLINEB_\alpha (s, t) =\left(\frac{s^\alpha + t^\alpha} {2}\right)^\frac1\alphaNEWLINE\]NEWLINE is the binomial mean. In continuation, the author proves that, for Hilbert space operators \(H, K, X\) with \(H, K \geq 0\) and unitarily invariant norm \(||| \cdot |||\), the function NEWLINE\[NEWLINE\alpha \in \mathbb R \mapsto |||B_\alpha (H, K)X|||NEWLINE\]NEWLINE is increasing.