10.57647/cna.2025.fyph-17u2.8

Integral Bounds for Generalized Polar Derivatives via Bernstein and Tur´an Inequalities

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  • Shahadat Ali*1,
  • Arshad Usmani2,
  • Rajbahadur Singh2,
  1. Department of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, India
  2. Department of Mathematics, Monad University, N.H. 9, Delhi Hapur Road, Village & Post Kastla, Kasmabad, P.O Pilkhuwa - 245304, Dist. Hapur (U.P), India

Received: 2024-09-30

Accepted: 2025-02-10

Published in Issue 2025-06-30

Copyright (c) 2025 Shahadat Ali, Arshad Usmani, Rajbahadur Singh (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Integral Bounds for Generalized Polar Derivatives via Bernstein and Tur´an Inequalities. (2025). Communications in Nonlinear Analysis, 13(1). https://doi.org/10.57647/cna.2025.fyph-17u2.8

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Abstract

Let $P(z) = \sum_{\mu=0}^{n}a_v z^{v}$ be a polynomial degree $n$, and $\beta$ be any complex number. Then $nP(z) + ({\beta} - z)P^{\prime}(z)$ is the polar derivative of $P(z)$, usually denoted by $D_{\beta} P(z)$. The derivative $D_{\beta} P(z)$ is the polynomial of degree at most $n-1$. The ordinary derivative follows from polar derivative. In this work, we establish integral mean inequalities of Tur\'an and Erd\H os--Lax type for the Generalized polar derivative of a polynomial, incorporating specific coefficients of the polynomial. Our findings refine several earlier results in the literature, and notably, one of our theorems extends and sharpens the well-known inequality by Khangembam Babina Devi and Barchand Chanam as a particular case. These improvements take into account both the location of the zeros and the structure of the coefficients of the polynomial. To illustrate the effectiveness and enhanced sharpness of our results, we also include numerical simulations and graphical comparisons with existing inequalities.

Keywords

  • Bernstein-type inequalities,
  • Tur´an-type inequalities,
  • Polar derivative,
  • Polynomials,
  • Generalized Polar Derivative,
  • MSC 2020: 30A10, 30C10, 30C15, 60E05, 65D15
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