A Note On Generalized $L_p$ Inequalities for the polar derivative of a polynomial

Abstract: Let ( P(z) ) be a polynomial of degree ( n ) and $\alpha \in \mathbb{C}$. The polar derivative of ( P(z) ) is denoted by ( D_\alpha P(z) ) and is defined as $D_\alpha P(z) = nP(z) + \alpha z P'(z).$ The polar derivative ( D_\alpha P(z) ) is a polynomial of degree at most ( n - 1 ) and it generalizes the ordinary derivative ( P'(z) ). In this paper, we establish some ( L_p ) inequalities for the polar derivative of a polynomial with all its zeros located within a prescribed disk. Our results refine and generalize previously known findings.