Ishfaq Nazir, Irfan Ahmad Wani, Firdose Ahmad

On Some Turan-type Inequalities for Derivative of a Polynomial

Introduction

On some turan-type inequalities for derivative of a polynomial. Explore Turan-type inequalities for polynomial derivatives. This paper refines Aziz's inequality, extends it to polar derivatives, and offers significantly enhanced precision with numerical examples.

Abstract

If $P(z) = a_{n}\prod_{\nu=1}^{n} (z - z _{\nu} )$ is a complex polynomial of degree $n$ having all its zeros in $|z| \leq K,$ $K \geq 1$ then Aziz (Proc Am Math Soc 89:259-266, 1983) proved that \begin{align*} \max_{|z|=1} |P'(z)| \geq \frac{2}{1+K^{n}} \sum_{\nu=1}^{n}\frac{K}{K+|z_{\nu}|} \max_{|z|=1} |P(z)|. \tag{0.1} \end{align*} This paper presents a comprehensive analysis that encompasses the refinement of inequality (0.1) while also extending the well-established Turan's inequality. Furthermore, we broaden the scope of our findings by applying them to the polar derivative of a polynomial. Our investigation reveals that the bounds derived from our results exhibit a significantly enhanced level of precision compared to inequality (0.1). To illustrate this, we provide a numerical example to underscore the superior performance of our findings.

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