Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros

Abstract

In this article, we extend inequalities concerning the polar derivative of a polynomial to integral mean for the class of polynomials with s-fold zero at the origin and the remaining zeros inside some closed disk of radius kk for k≥1k\ge 1 and k≤1k\le 1, and we thereby obtain bounds that depend on some coefficients of the underlying polynomial. Our results not only extend some known polynomial inequalities but also reduce some interesting results in particular cases. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to previously established results.

Keywords

• turán-type inequalities
• integral inequalities
• polar derivative
• s-fold zero
• 30a10
• 30c10
• 30d15