Bernstein type $L_{p}$ inequalities for composition of polynomials - Malik, Shabir Ahmad (Sarajevo Journal of Mathematics, 2024)
Let $P(z)=a_{0}+\sum\limits_{v=μ}^{n} a_v z^v\in\mathscr P_{n,~μ}$ and $P(z) eq 0$ for $|z|<k$, where $k≥ 1$, then for $0≤ p≤∞$, Gardner and Weems (J. Math. Anal. Appl. 219 (1998), 472-478) proved that \begin{align*} \|P'\|_{p}≤\frac{n}{\|k^{μ}+z\|_{p}}\|P\|_{p}. \end{align*} In this note, we consider a more general class of polynomials $P° Q\in\mathscr P_{mn,~μ}$ defined by $(P° Q)(z)=P(Q(z))$, where $Q\in \mathscr P_m$ and provide an extension of the above inequality and related results.