Positive definiteness of certain functions and comparison of power difference and Heinz means (Q782510)

A function \(f\) on \(\mathbb{R}\) is said to be positive definite if, for every set \(\{t_1,\cdots,t_n\}\) of real numbers, the matrix \([f(t_i-t_j)]_{1\leq i,j\leq n}\) is positive semi-definite. A~non-negative function \(f\) on \(\mathbb{R}\) is said to be infinitely divisible if \(f^r\) is positive definite for every \(r>0\). The positive definiteness and infinite divisibility of the functions \begin{align*} \frac{\cosh(at)\cosh(bt)}{\cosh t}\quad ,\quad \frac{\cosh t }{\cosh(at)\cosh(bt)}\quad ,\quad \frac{\cosh(at)\sinh(bt)}{\sinh t} \end{align*} have been investigated in [\textit{R. Bhatia} and the author, Linear Algebra Appl. 424, No. 1, 36--54 (2007; Zbl 1124.15015)] and [the author, Positive definiteness of functions with applications to operator norm inequalities. Providence, RI: American Mathematical Society (AMS) (2011; Zbl 1227.47005)]. In the present article, the author studies these properties for the function \[f(t)=\frac{\sinh t}{\cosh(at)\sinh(bt)}\qquad (a,b>0)\] and obtains a characterization of \(a,b>0\) for which \(f(t)\) is infinite divisible and positive definite. Some applications to norm inequalities for power difference and Heinz means are given as well.