Generalization of Some Well-Known Polynomial Inequalities for the Modified Smirnov Operator

Let $P(z)$ be a polynomial of degree $n$. In this paper, we consider the modified Smirnov operator, which carries a polynomial $P(z)$ into ${\tilde{\mathbb{S}}}_a[P](z)=(1+az)P^{\mathrm{\prime }}(z)-naP(z),$ where $a$ is an arbitrary number in $\bar{\mathbb{D}}$. We estimate minimum and maximum moduli of modified Smirnov operator of $P(z)$ on the unit circle with restricted zeros and thereby obtain a generalization of some results of Dewan and Hans \cite{dewan2013some}. This study includes compact generalization of some well-known polynomial inequalities.