On the growth of polynomials not vanishing in the unit disc (Q2769225)

The reviewed paper contains the lower and upper bound (the last one is sharp) for the quantity NEWLINE\[NEWLINE\left|\frac{f(re^{i\gamma})}{f(Re^{i\gamma})}\right|,\qquad 0\leq r<R<1,\quad\gamma\in\mathbb RNEWLINE\]NEWLINE with the polynomial NEWLINE\[NEWLINEf(z)=\sum_{k=1}^nc_kz^k\neq 0\quad\text{in}\quad|z|<1.NEWLINE\]NEWLINE Some interesting corollaries are given as well. The results depend on the following interesting NEWLINENEWLINENEWLINELemma 1: Let \(f(z)=\sum_{k=0}^nc_kz^k\neq 0\) for \(|z|<1\). Then \(zf'(z)-nf(z)\neq 0\) for \(|z|<1\) and \(|f'(z)|\leq|zf'(z)-nf(z)|\) for \(|z|=1\), so that NEWLINE\[NEWLINE\varphi(z):=\frac{f'(z)}{zf'(z)-nf(z)}NEWLINE\]NEWLINE is analytic on the closed unit disk. Furthermore, \(|\varphi(z)|\leq 1\) for \(|z|\leq 1\).