On the Inequalities Concerning Polynomials

Abstract

If $P(z)=a_n\prod \nolimits _{j=1}^n(z-z_j)$ is a complex polynomial of degree n having all its zeros in $|z|\le K,K\ge 1$ then Aziz (Proc Am Math Soc 89:259–266, 1983) proved that

$$\begin{aligned} \max _{|z|=1}{|P'(z)|}\ge \frac{2}{1+K^n}\sum _{j=1}^n\frac{K}{K+|z_j|}\max _{|z|=1}{|P(z)|}. \end{aligned}$$

(0.1)

In this paper we sharpen the inequality (0.1) and further extend the obtained result to the polar derivative of a polynomial. As a consequence we also derive two results on the generalization of Erdös–Lax type inequality for the class of polynomials having no zeros in the disc $|z|<K,\;K\le 1$.