L^q inequalities for the {s^{th}} derivative of a polynomial

Let f(z) be an analytic function on the unit disk \{z\in\mathbb{C},\ |z|\leq ۱\}, for each q>۰, the \|f\|_{q} is defined as follows\begin{align*}\begin{split}&\left\|f\right\|_q:=\left\{\frac{۱}{۲\pi}\int_۰^{۲\pi}\left|f(e^{i\theta})\right|^qd\theta\right\}^{۱/q},\\ \ ۰۰,\begin{align*}\left\|p'\right\|_{q}\leq \frac{n}{\|k+z\|_q}\|p\|_{q}.\end{align*}In this paper, we shall present an interesting generalization and refinement of this result which include some previous results.