Sharp spectral constants for scaled $q$-numerical ranges

For $ n \geq 2$, $A\in M_n(\mathbb C)$ and $0<|q|\leq 1$, let $Ω_q(A)=q^{-1}W_q(A)$ be the scaled $q$-numerical range. We prove that for every $γ\geq 1$, \[ Ω_{η(γ)}(A) =\bigcup_{κ(S)\leqγ}W(S^{-1}AS), \qquad η(γ)=\frac{2}{γ+γ^{-1}}, \] where $κ(S)=\|S\|\,\|S^{-1}\|$. As a consequence, we prove the sharp inequality \[ \|p(A)\|\leq \max\!\left\{1,\frac{2|q|}{1+\sqrt{1-|q|^2}}\right\} \max_{z\inΩ_q(A)}|p(z)|, \] for all polynomials $p$.