Bernstein type $L_{p}$ inequalities for composition of polynomials

Abstract

Let $P(z)=a_{0}+\sum\limits_{v=\mu}^{n} a_v z^v\in\mathscr P_{n,~\mu}$ and $P(z)\neq 0$ for $|z|<k$, where $k\geq 1$, then for $0\leq p\leq\infty$, Gardner and Weems (J. Math. Anal. Appl. 219 (1998), 472-478) proved that
\begin{align*}
\|P'\|_{p}\leq\frac{n}{\|k^{\mu}+z\|_{p}}\|P\|_{p}.
\end{align*}
In this note, we consider a more general class of polynomials $P\circ Q\in\mathscr P_{mn,~\mu}$ defined by $(P\circ Q)(z)=P(Q(z))$, where $Q\in \mathscr P_m$ and provide an extension of the above inequality and related results.