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
