Matrix inequalities via Bernstein functions (Q2120583)

The authors obtain a variety of matrix norm inequalities involving Bernstein functions, i.e., nonnegative, \(C^{\infty}\) functions defined on \((0, \infty)\), with completely monotone derivative. They prove that for \(f\) a Bernstein function, \(A,B\) positive definite matrices, \(X\) a complex matrix, all matrices of size \(n\times n\), we have \begin{align*} ||f^{(k)}((1-\nu)A+\nu B)|| &\le \max \{||f^{(k)}(A)||,||f^{(k)}(B)||\}, \quad 0\le \nu \le 1,\ k=0,1,2,\dots, \\ |||Df(A)||| &\le ||f' (A)||, \\ |||f(A)X-Xf(B)||| &\le \max \{||f'(A)||, ||f'(B)||\} \, |||AX-XB|||, \end{align*} where \(||.||\) is the operator norm, \(|||.|||\) any unitary invariant matrix norm, \(f^{(k)}\) stands for \(k\)-th derivative, and \(D\) denotes Fréchet derivative. Lévy-Khintchine representation of Bernstein functions is their principal tool.