On the order of vanishing at 1 of a polynomial (Q1592143)

Let \(P(z)=\sum_{k=0}^na_kz^k\) be a polynomial with complex coefficients and assume that \(\max\{|a_k|\}=\max\{|a_0|,|a_n|\}\). The author shows that the order of a possible zero of \(f\) at \(z=1\) cannot exceed \(31\sqrt n/15+2.5\). For sufficiently large \(n\) this bound can be replaced by \(21\sqrt n/13\).