Improvement of some Turan-type inequalities for polynomials

Mir, Mohd Yousf; Shah, W. M.

doi:10.22075/ijnaa.2025.35981.5335

Improvement of some Turan-type inequalities for polynomials

Articles in Press

Document Type : Research Paper

Authors

Mohd Yousf Mir ¹

W. M. Shah ²

¹ Jammu and Kashmir Institute of Mathematical Sciences, University of Kashmir, 190008, India

² Department of Mathematics, Central University of Kashmir, Ganderbal-191201, India

10.22075/ijnaa.2025.35981.5335

Abstract

For a polynomial $P(z)=a_n\prod\limits_{j=1}^n(z-z_j)$ of degree $n$ having all zeros in $|z|\leq k, k\geq 1$, it is known that $$|P'(z)|\geq \dfrac{2}{1+k^n}\sum\limits_{j=1}^n\dfrac{k}{k+|z_j|} \max_{|z|=1}|P(z)|.$$
In this paper we prove an improvement of this inequality and also extend the same to the polar derivative of a polynomial, thereby generalizing some Tur'an-type inequalities for polynomials.

Keywords

Polynomial

Inequalities

Polar derivative

Zeros

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