On Erdos – Lax and Turan Type Inequalities of a Polynomial
DOI:
https://doi.org/10.9734/bpi/nramcs/v3/6092FKeywords:
Bernstein’s inequality, Erdos – Lax inequality, Turan’s inequality, polynomial, zerosAbstract
Let p(Z) be a polynomial of degree n if p(Z) has no zero in the open unit disk then
\(\max _{|z|=1}\left|p^{\prime}(z)\right| \leq \frac{n}{2} \max _{|z|=1}|p(z)\)
But if p(Z) has all its zeros in the closed unit disk then
\(\max _{|=|=1}\left|p^{\prime}(z)\right| \geq \frac{n}{2} \max _{|z|=1}|p(z)\) .
The above inequalities are respectively the well-known Erdos – Lax inequality and the Turan’s inequality. A natural question that follows is to investigate the extension of these inequalities for open or closed disk of radius K, K > 0 In literature, we find extensions of Erdos – Lax inequality for a polynomial p(Z) of degree n having no zero in the open disk of radius K, K \(\geq\) 1. For K < 1, a similar extension does not seem to exist in general. In this paper, we discuss in brief why such an extension seems unattainable in general for K < 1 . Further, we also give a brief account of the existence of extensions of Turan’s inequality for a polynomial p(Z) of degree n having all its zeros in the closed disk of radius K for every value of K > 0 in completion.
