# Scalar Crouzeix inequality OA-1-H0-to-H3

## 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 H3: Let \(\Omega\subset\mathbb C\) be a bounded convex domain with \(C^2\) boundary, and let \(\gamma:[0,L]\to\partial\Omega\) traverse the boundary once counterclockwise at unit speed. Put \(\nu(s)=-\mathrm i\gamma'(s)\).
Let \(B\in M_n(\mathbb C)\) satisfy
\[
W(B)=\{x^*Bx:x\in\mathbb C^n,\ \|x\|_2=1\}\subset\Omega,
\]
and define
\[
R_s=(\gamma(s)I-B)^{-1},\qquad
D_B(s)=\frac1{2\pi}\left(\nu(s)R_s+\overline{\nu(s)}R_s^*\right).
\]
The map
\[
\Phi:C(\partial\Omega)\to M_n(\mathbb C),\qquad
\Phi(h)=\frac12\int_0^L h(\gamma(s))D_B(s)\,ds
\]
is positive and unital: \(h\ge0\) implies \(\Phi(h)\) is Hermitian positive semidefinite, and \(\Phi(1)=I\).

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.