Sharp spectral constants for scaled $q$-numerical ranges

For $n\ge 2$, $A\in M_n(\mathbb{C})$ and $0<\mathrm{\mid }q\mathrm{\mid }\le 1$, let ${\mathrm{\Omega }}_q(A)=q^{-1}{W}_q(A)$ be the scaled $q$-numerical range. We prove that for every $\gamma \ge 1$, $${\mathrm{\Omega }}_{\eta (\gamma )}(A)=\bigcup _{\kappa (S)\le \gamma }W(S^{-1}AS),\eta (\gamma )=\frac{2}{\gamma +{\gamma }^{-1}},$$ where $\kappa (S)=\mathrm{\parallel }S\mathrm{\parallel }\mathrm{\parallel }{S}^{-1}\mathrm{\parallel }$. As a consequence, we prove the sharp inequality $$\mathrm{\parallel }p(A)\mathrm{\parallel }\le \max \text{\hspace{0.17em}⁣}\{1,\frac{2\mathrm{\mid }q\mathrm{\mid }}{1+\sqrt{1-\mathrm{\mid }q{\mathrm{\mid }}^2}}\}\underset{z\in {\mathrm{\Omega }}_q(A)}{\max }\mathrm{\mid }p(z)\mathrm{\mid },$$ for all polynomials $p$.