Note on Bernstein Inequalities Concerning Complex Polynomials

Abstract

Let \(p(z)\) be a polynomial of degree \(n\) having no zero in \(|z|<1\), then Erdös conjectured and later Lax [Bull. Amer. Math. Soc., \(50(1944), 509-513\) ] prove that
\[
\max _{|z|=1}\left|p^{\prime}(z)\right| \leq \frac{n}{2} \max _{|z|=1}|p(z)|
\]

This Erdös-Lax's inequality was generalized for the first time by Malik [J. London Math. Soc., 1(1969), 57-60] that if \(p(z)\) is a polynomial of degree \(n\) having no zero in \(|z|<k, k \geq 1\), then
\[
\max _{|z|=1}\left|p^{\prime}(z)\right| \leq \frac{n}{1+k} \max _{|z|=1}|p(z)|
\]

For the class of polynomials not vanishing in \(|z|<k, k \leq 1\), the precise estimate for maximum of \(\left|p^{\prime}(z)\right|\) on \(|z|=1\), in general, does not seem to be easily obtainable. But for the particular class of polynomials having all its zeros on \(|z|=k, k \leq 1\), Govil [J. Math. and Phy. Sci., 14(1980), 183-187] was able to prove that
\[
\max _{|z|=1}\left|p^{\prime}(z)\right| \leq \frac{n}{k^{n-1}+k^n} \max _{|z|=1}|p(z)| .
\]

In this article, we compare some inequalities of later type concerning the ordinary and polar derivatives of the polynomial.