A uniform estimate for the value of a conjugate polynomial in the plane. (Q1896991)

Let \(P_n (z)= \sum_{k=0}^n c_k z^k\), \(z= \rho e^{ix}\), be an arbitrary complex polynomial of degree at most \(n\). Norming its real part \(T_n (\rho, x)\) by \(\max_x |T_n (1,x) |= |T_n |\leq 1\), the author posed the problem of investigating the magnitude \(W(\rho, n)\) of its imaginary part \(\widetilde {T}_n (\rho, x)\): \(W(\rho, n):= \sup\{ |\widetilde {T}_n (\rho) |\): \(|T_n|\leq 1\}\). An exact value of the magnitude of \(W(\rho, n)\) has been known only for \(\rho=1\), \(n\geq 1\) [\textit{L. V. Taikov}, Mat. Zametki 48, No. 4, 110--114 (1990; Zbl 0746.42009)]. The aim of the paper is to find an exact value of the magnitude \(W(\rho,n)\) for \(0<\rho \neq 1\) and \(n\geq 1\). Main result is the following: \[ W(\rho, n)= (2\pi)^{-1} (1+ \rho^{n+1}) \ln \bigl( 1- 2\rho \cos \pi (n+1)^{-1}+ \rho^2 \bigr)^{-1} (1+ \rho)^2+ R, \] where \(|R|\leq 3\rho\), for \(0< \rho< 1\), \(2\rho\leq 1- \rho^2\) and \(n\geq 1\); \[ W(\rho, n)= \rho^n, \] for \(2\rho\leq (\rho^2 - 1) \rho^n\) and \(n\geq 1\).