An $L_p$ norm inequality related to extremal polynomials

Abstract

Let $E$ be a Jordan rectifiable curve in the complex plane and let $G$ be the bounded component of $\mathbb{C}\backslash E$. Now let $n\in \mathbb{N}$, and let $m_{n,E}$ denote the extremal constants defined by \begin{equation*}m_{n,E}=\inf \left\{ \left\Vert \dfrac{D_{E,\rho }\left( z\right) }{D_{E,\rho }\left( 0\right) }-P_{n}\left( z\right) \right\Vert_{L^{p}\left(G,\rho \right) }:P_{n}\left( \xi \right) =1\right\}\end{equation*}where $\xi $ is a fixed complex number.where $\rho $ is a weight function, $D_{E,\rho }\left( \cdot \right)$ is the so called {Szeg\"{o}} function, $z\in G$, $p\geq 2.$ The infimum is taken over all polynomials $P_{n}$ of degree $n$. The $L_{p}$ associated extremal polynomials $\left\{Q_{n}\right\}_{n=1,2....}$ satisfies \begin{equation*} m_{n,E}=\left\Vert \dfrac{D_{E,\rho }\left( z\right) }{D_{E,\rho }\left(0\right) }-Q_{n}\left( z\right) \right\Vert_{L^{p}\left( G,\rho \right) }.\end{equation*} We define the functions, if $p\in $ $\mathbb{N}$ \begin{equation*}J_{n}\left( z\right) =\int_{\xi_{G}}^{z}Q_{n}^{p}\left( t\right) dt;\;z\in G\end{equation*} which are of course well defined polynomials for any $n\in \mathbb{N}$. Following the same convention , we define the function \begin{equation*}\Phi \left( z\right) =\int\limits_{\xi_{G}}^{z}\left( \dfrac{D_{E,\rho}\left( t\right) }{D_{E,\rho }\left( 0\right) }\right) ^{p}dt,\end{equation*} Our main target in this paper is to show that when $m_{n,E}\longrightarrow0, $ then \begin{equation*}J_{n}\left( z\right) \text{ }\longrightarrow \Phi \left( z\right)\end{equation*} uniformly on compact subsets of $G.$