# Scalar Crouzeix inequality — H5

For every \(n\ge1\), every \(A\in M_n(\mathbb C)\), and every \(f\) holomorphic on a neighborhood of
\[
W(A)=\{x^*Ax:x\in\mathbb C^n,\ \|x\|_2=1\},
\]
one has
\[
\|f(A)\|\le2\max_{z\in W(A)}|f(z)|.
\]
The norm is the Euclidean operator norm, \(f(A)\) is defined by holomorphic functional calculus, and the constant \(2\) is optimal uniformly over \(n,A,f\).