Nusrat Ahmad Dar, Idrees Qasim, Abdul Liman

Some lower bound estimates for the generalized derivative of a polynomial

Introduction

Some lower bound estimates for the generalized derivative of a polynomial. This paper extends an inequality for polynomials with restricted zeros to the generalized derivative, providing new lower bound estimates for generalized and polar derivatives.

Abstract

If $P(z)$ is a polynomial of degree $n$ having all its zeros in $\left|z\right|\leq k, k\leq 1$, then Rather et al. ( Some inequalities for polynomials with restricted zeros, Ann. Univ. Ferrara, 67 (2021), 183-189.) proved that for all $z$ on $\left|z\right|=1$ for which $P(z)\neq 0,$ \begin{align*} Re\left(z\frac{P^\prime(z)}{P(z)}\right) \geq \frac{n}{1+k}\left\lbrace 1 + \frac{k}{n}\left(\frac{k^{n}\left|a_n\right| - \left|a_0\right|}{k^{n}\left|a_n\right| + \left|a_0\right|}\right)\right\rbrace. \end{align*} In this paper, we extend this inequality to the generalised derivative by taking $s$-folded zeros at origin. As an application, we obtain some lower bound estimates for the generalized derivative and generalized polar derivative of a polynomial with restricted zeros, which include various results due to Tur\'{a}n, Malik, Dubinin, Aziz, Rather and Govil as special cases.

Review

This paper presents a significant extension of an inequality concerning polynomials with restricted zeros, initially established by Rather et al. The core contribution lies in generalizing the original lower bound estimate for $Re\left(zP^\prime(z)/P(z)\right)$ to the context of generalized derivatives, specifically by considering $s$-folded zeros at the origin. This approach allows the authors to derive new lower bound estimates for both the generalized derivative and the generalized polar derivative of polynomials, maintaining the critical condition that all zeros lie within $|z|\leq k, k\leq 1$. A major strength of this work, as highlighted in the abstract, is its unifying nature. The resulting inequalities are shown to encompass a wide array of existing results by prominent mathematicians such as Turán, Malik, Dubinin, Aziz, Rather, and Govil, as special cases. This indicates that the developed framework is broad and powerful, providing a more comprehensive perspective on lower bound estimates for polynomial derivatives. The systematic generalization of an important inequality, coupled with its ability to subsume numerous prior findings, underscores the paper's potential impact on the field of complex analysis and polynomial theory. Overall, the abstract suggests a valuable contribution to the literature. The extension to generalized derivatives and generalized polar derivatives, particularly with the $s$-folded zero consideration, addresses a relevant direction for research in polynomial inequalities. Assuming the detailed proofs and derivations are rigorous and well-presented, this paper promises to be a robust and insightful piece of work. The comprehensive nature, by unifying several special cases, makes it a noteworthy addition to the body of knowledge on polynomial inequalities with restricted zeros, and it is therefore recommended for publication.

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