On growth of polynomials
The main result of this paper may be stated as follows. Let \(p(z)=a_{0}+ \sum_{\nu=s}^{n}a_{\nu}z^{\nu}\) be a polynomial of degree \(n\geq 2\) and set \(M(p,R)=\max_{|z|=R}|p(z)|\) for \(R\geq 1\) and \(|p |=M(p,1)\). If \(p(z)\) has no zeros in \(|z|<K\), \(K\geq 1\), then \[ M(p,R)\leq \left(\frac{R^n+K^s}{1+K^s}\right)|p |- \left(\frac{R^n-1}{1+K^s}\right)m-|a_1|\left(\frac{R^n-1}{n}-\frac{R^{n-2}-1} {n-2}\right), \] if \(n\geq 3\), and \[ M(p,R)\leq \left(\frac{R^n+K^s}{1+K^s}\right)|p |- \left(\frac{R^n-1}{1+K^s}\right)m-|a_1|\frac{(R-1)^n}{2}, \] if \(n=2\), where \(m=\min_{|z|=K}|p(z)|\). These inequalities generalize and sharpen the classical inequalities of S.~Bernstein and Ankeny and Rivlin as well as several other related inequalities due to the author ([Complex Variables, Theory Appl. 40, 249-259 (2000; Zbl 1020.30002)]) and to Rahman and Schmeisser ([Séminaire de Mathématiques Supérieures, Séminaire Scientifique OTAN (NATO Advanced Study Institute) 86, Les Presses de l'Université de Montréal (1983; Zbl 0525.30001)]).
scientific article; zbMATH DE number 1910303