Let \(p(z)\) be a polynomial of degree \(n\) having all its
zeros in \(|z| \leq 1\), then famous inequality due to Turán [Compos. Math. 7
(1939), 89-95] is \[\max_{ |z|=1 }\left|p^{\prime}(z)\right| \geq \frac{n}{2}
\max _{|z|=1}|p(z)| \text {. }\]
This Turán'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 all its zeros in \(|z| \leq k, k \leq 1\), then \[\max_{
|z|=1 }\left|p^{\prime}(z)\right| \geq \frac{n}{1+k} \max _{|z|=1}|p(z)| \text
{. }\] While for the case \(k \geq 1\), Govil [Proc. Amer. Math. Soc. 41(1973),
543-546] prove that \[\max_{ |z|=1 }\left|p^{\prime}(z)\right| \geq
\frac{n}{1+k^n} \max _{|z|=1}|p(z)| \text {. }\]
The above inequalities play a vital role in approximation
theory. Frequently, further progress in this theory has depended on first
obtaining a corresponding generalization or analogue of these inequalities. In
this article, we discuss in brief some of the recent improvements of the above
inequalities particularly the later type and make a comparative study of them
using an example with detail graphical illustrations.
Author(s) Details
Robinson Soraisam
Department of Mathematics, National Institute of Technology Manipur, Langol
795004, India.
Barchand Chanam
Department of Mathematics, National Institute of Technology Manipur, Langol
795004, India.
Please see the book here:- https://doi.org/10.9734/bpi/mcscd/v1/8954A
