A Comparative Study of Some Tur\(\acute{a}\)n-type Inequalities in the Realm of Complex Polynomials | Chapter 5 | Mathematics and Computer Science: Contemporary Developments Vol. 1

 

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.

 

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