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 $D_{\alpha }P(z):=nP(z)+(\alpha -z)P^{\mathrm{\prime }}(z)$. The polar derivative $D_{\alpha }P(z)$ is a polynomial of degree at most $n-1$ and it generalizes the ordinary derivative $P^{\mathrm{\prime }}(z)$. 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.