A Comparative Study of Some Tur\(\acute{a}\)n-type Inequalities in the Realm of Complex Polynomials

Abstract

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.