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A result by Crouzeix and Palencia states that the spectral norm of a matrixfunction $f(A)$ is bounded by $K = 1+\sqrt{2}$ times the maximum of $f$ on$W(A)$, the numerical range of $A$. The purpose of this work is to point outthat this result extends to a certain notion of bivariate matrix functions; thespectral norm of $f\{A,B\}$ is bounded by $K^2$ times the maximum of $f$ on$W(A)\times W(B)$. As a special case, it follows that the spectral norm of theFr\'echet derivative of $f(A)$ is bounded by $K^2$ times the maximum of$f^\prime$ on $W(A)$. An application to the convergence analysis of certainKrylov subspace methods and the extension to functions in more than twovariables are discussed.

A bivariate extension of the Crouzeix-Palencia result with an application to Fr\'echet derivatives of matrix functions