Products of polynomials in uniform norms (Q2723460)

Hereafter, \(\|\cdot \|_E\) stands for the uniform norm on a compact set \(E\subset\mathbb{C}\). The article concerns with the inequalities NEWLINE\[NEWLINE\prod ^m _{k=1} \|p_k \|_E \leq C \|p_1 \cdot \dots \cdot p_m \|_E, NEWLINE\]NEWLINE where \(p_k, 1 \leq k \leq m, \) are polynomials in one complex variable \(z\). This inequality and its uses can be traced back to Kneser, Gel'fond, Mahler, etc. The main result is NEWLINENEWLINENEWLINETheorem 2.1. If the logarithmic capacity \(\text{cap} (E) > 0\) and \(\deg (p_1 \cdot \dots \cdot p_m) = n\), then the best constant in the inequality above is NEWLINE\[NEWLINEC=C(E)= \{\frac{\exp \{\int \log d _E (z) d\mu (z) \} }{\text{cap} (E)} \} ^n , NEWLINE\]NEWLINE where \(d_E (z)\) is the distance from the point \(z\) to the set \(E\) and \(\mu _E\) is the equilibrium measure of \(E\). The author also studies the asymptotic behavior of the constant \(C(E)\) and some applications of his results.