Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle

We consider the class $\mathcal P_n^*$ of algebraic polynomials of a complex variable with complex coefficients of degree at most $n$ with real constant terms. In this class 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 $\|P_n\|_{C(\Gamma_r)}\leq\mu(r,n)\|\operatorname{Re}P_n\|_{C(\Gamma_1)}$. We prove that $\mu(r,n)=r^n$ for $r^{n+2}-r^n-3r^2-4r+1\geq0$. In order to justify this result, we obtain the corresponding quadrature formula. We give an example which shows that the strict inequality $\mu(r,n)>r^n$ is valid for $r$ sufficiently close to 1.