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

Let $P(z)$ be a polynomial of degree $n$ and $\alpha \in \mathbb{C}$. The polar derivative of $P(z)$, denoted by $D_\alpha P(z)$ and is defined by $D_\alpha P(z): = nP(z) + (\alpha -z)P'(z)$. 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.