Exact inequality between uniform norms of an algebraic polynomial and its real part on concentric circles in the complex plane

In the class $\mathcal P_n^*$ of algebraic polynomials of a complex variable of degree at most n with complex coefficients and a real constant term, we estimate the uniform norm of a polynomial $P_n\in\mathcal P_n^*$ on the circle $\Gamma_r=\{z\in\mathbb C\colon|z|=r\}$ of radius $r>1$ in terms of the norm of its real part on the unit circle $\Gamma_1$. More precisely, we study the best constant $\mu(r,n)$ in the inequality $\|P_n\|_{C(\Gamma_r)}\leq\mu(r,n)\|\operatorname{Re}P_n\|_{C(\Gamma_1)}$. Necessary and sufficient conditions for the equality $\mu(r,n)=r^n$ are found.