On polynomials most deviating from zero on a domain boundary (Q1280315)

The author considers the polynomial \(R_n(z)\) determined by \[ \begin{gathered} \max\Big\{\min_{z\in\partial D}| P(z)|: P(z) = (z-a_1)\cdot\dots\cdot(z-a_n),\;\{a_k\}^n_{k=1}\subset D\Big\},\\ \min_{z\in\partial D}| R_n(z)| := m_n\qquad (n=1,2,\dots). \end{gathered} \] The main properties of \(R_n(z)\) are established as well as their concrete form for some domains. It is proved that there exists a \(\lim\limits_{n\to\infty} \tau_n^{\frac 1n} = \gamma(\overline D)\), where \(\gamma(\bar D)\) is the harmonic capacity of the clousure of the domain \(D\).