Multiplicity of zeros of polynomials (Q2037068)

The paper grew out of the known result of \textit{P. Erdős} and \textit{P. Túran} [Ann. Math. (2) 41, 162--173 (1940; Zbl 0023.02201)] on zero distributions and bounds for their multiplicities of monic polynomials with all their zeros in \([-1,1]\). Theorem 1.1. Let \(K\) be a compact set consisting of pairwise disjoint \(C^{1+\alpha}\)-smooth Jordan curves or arcs lying exterior to each other. Given a monic polynomial \(P_n\) of degree at most \(n\) with a zero \(a\in K\) of multiplicity \(m=m(a)\), the following lower bound holds \[ \|P_n\|_k\ge e^{c\frac{m^2}{n}} (\mathrm{cap}\,K)^n, \qquad c>0,\] where \(\mathrm{cap}\,K\) is the logarithmic capacity of \(K\), \(\|\cdot\|_K\) the supremum norm on~\(K\). In the case when \(K\) is an analytic Jordan curve or arc, the result turns out to be sharp.