Some inequalities for polynomials with restricted zeros

Abstract

By using the boundary Schwarz lemma, it was shown by Dubinin (J Math Sci 143:3069–3076, 2007) that if P(z) is a polynomial of degree n having all its zeros in $|z| \le 1,$ then for all z on $|z|=1$ for which $P(z)\ne 0,$

$$\begin{aligned} Re\bigg (\frac{zP^{\prime }(z)}{P(z)}\bigg )\ge \frac{n}{2}+\frac{1}{2}\left( \frac{|a_n|-|a_0|}{|a_n|+|a_0|}\right) . \end{aligned}$$

In this paper, by using simple techniques we generalize the above inequality, thereby give a simple proof of the above inequality. As an application of our result, we also obtain sharp refinements of some known results due to Malik (J Lond Math Soc 1:57–60, 1969), Aziz and Rather (Math Ineq Appl 1:231–238, 1998). These results take into account the size of the constant term and the leading coefficient of the polynomial P(z).