Extremal problems of Turán-type involving the location of all zeros of a polynomial

Abstract

If $P(z)=a_n\prod _{v=1}^n(z-z_v)$ is a polynomial of degree n having all its zeros in $|z|\le k,k\ge 1$ then Aziz [Inequalities for the derivative of a polynomial. Proc Am Math Soc. 1983;89(2):259–266] proved that $\underset{|z|=1}{max}|P^{\prime }(z)|\ge \frac{2}{1+k^n}\sum _{v=1}^n\frac{k}{k+|z_v|}\underset{|z|=1}{max}|P(z)|.$ Recently, Kumar [On the inequalities concerning polynomials. Complex Anal Oper Theory. 2020;14(6):1–11 (Article ID 65)] established a generalization of this inequality and proved under the same hypothesis for a polynomial $P(z)=a_0+a_{1}z+a_2{z}^2+\cdots +a_n{z}^n=a_n\prod _{v=1}^n(z-z_v)$, that $\begin{array}{rl}& \underset{|z|=1}{max}|P^{\prime }(z)|\\ & \ge (\frac{2}{1+k^n}+\frac{(|a_n|k^n-|a_0|)(k-1)}{(1+k^n)(|a_n|k^n+k|a_0|)})\sum _{v=1}^n\frac{k}{k+|z_v|}\underset{|z|=1}{max}|P(z)|.\end{array}$ In this paper, we sharpen the above inequalities and further extend the obtained results to the polar derivative of a polynomial. As a consequence, our results also sharpens considerably some results of Dewan and Upadhye [Inequalities for the polar derivative of a polynomial. J Ineq Pure Appl Math. 2008;9:1–9 (Article ID 119)].

Keywords:

• Polar derivative of a polynomial
• Rouché's theorem
• Turán-type inequalities

AMS Subject Classifications:

• Primary 30A10
• 30C10
• Secondary 30D15

Acknowledgments

The authors would like to thank the anonymous referee for many useful comments. The first author was supported by the National Board for Higher Mathematics (R.P), Department of Atomic Energy, Government of India (No. 02011/19/2022/R&D-II/10212).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.