# Scalar Crouzeix inequality — H4

For every \(n\ge1\), every \(B\in M_n(\mathbb C)\) with pairwise distinct eigenvalues, and every \(f\) holomorphic on a neighborhood of
\[
W(B)=\{x^*Bx:x\in\mathbb C^n,\ \|x\|_2=1\},
\]
one has
\[
\|f(B)\|\le2\max_{z\in W(B)}|f(z)|.
\]
The norm is the Euclidean operator norm, and \(f(B)\) is defined by holomorphic functional calculus.