# Scalar Crouzeix inequality OA-1-H0-to-H5

## Setting

You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.

## Problem

Prove 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 \(n,A,f\).

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.