A bivariate extension of the Crouzeix–Palencia result with an application to Fréchet derivatives of matrix functions

A result by Crouzeix and Palencia states that the spectral norm of a matrix function (Formula presented.) is bounded by (Formula presented.) times the maximum of f on (Formula presented.), the numerical range of A. The purpose of this work is to point out that this result extends to a certain notion of bivariate matrix functions; the spectral norm of (Formula presented.) is bounded by (Formula presented.) times the maximum of f on (Formula presented.). As a special case, it follows that the spectral norm of the Fréchet derivative of (Formula presented.) is bounded by (Formula presented.) times the maximum of (Formula presented.) on (Formula presented.). An application to the convergence analysis of certain Krylov subspace methods and the extension to functions in more than two variables are discussed.