To Approximate the Polar Derivative of a Polynomial with Coefficients, one may use

Abstract

In this work, we look at the polynomial class  µ? no f degree n, having all zero son |z|=k,k?1,Novel inequalities are established between the uniform norm of a polynomial and its polar derivative under the condition that the number of zeros in the polynomial is constrained. Several previously demonstrated inequalities for restricted polynomials are improved upon by these findings, and new, sharper inequalities are generated, adding to the already extensive body of research on the topic. In particular situations, our theory also provides some interesting extensions of certain Zygmund type inequalities for polynomials. This research adds to the expanding body of knowledge on inequalities that link A team of researchers from the United Kingdom and the United States conducted this study. When a polynomial's zeros are limited, we create some additional requirements that link the polynomial's uniform-norm to its polar derivative. These restrictions are only true when the polynomial's zeros are limited. The findings are consistent with the newly discovered Erd osLax and Tur'an inequalities for restricted polynomials. Moreover, the revelations establish a number of disparities that are more pronounced than those previously recognized in the vast amount of study on this issue

Keywords: Zeros, polar derivatives, and complex domains