Note on Bernstein Inequalities Concerning Complex Polynomials | Chapter 1 | Research Updates in Mathematics and Computer Science Vol. 7

 

 

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.

 

Author(s) Details

Sangeeta Garg

Department of Computer Science, Faculty of Mathematics, Mewar Institute of Management, Vasundhara-4C, Ghaziabad, CCS University, Meerut, India.

 

Please see the book here:- https://doi.org/10.9734/bpi/rumcs/v7/342