Approximation properties of a class of extremal polynomials over smooth Jordan curves (Q2739141)

Let \(D\subset \mathbb C\) be a simple connected domain with \(0\in D\) bounded by a Jordan curve \(\Gamma \in \mathcal C^{1+\alpha}\), \(0<\alpha< 1.\) Let \(\varphi: D\to \{|w|<1\}\) be the conformal mapping such that \(\varphi (0)=0\), \(\varphi'(0)>0\). Put \(f_2(z):=\int_0^z[\frac {\varphi'(\xi)}{\varphi'(0)}]^{\frac 12} d\xi\), \(z\in D\). Let \(\Pi_n\) be the class fo all polynomials \(p\) of degree \(\leq n\) with \(p(0)=0\), \(p'(0)=1\). Denote by \(p_{n,2}\) the unique polynomial of the class \(\Pi_n\) such that NEWLINE\[NEWLINE\int_{\Gamma}|p'_{n,2}(z)|^2|dz|= \inf_{p\in\Pi_n}\int_{\Gamma}|p'(z)|^2|dz|.NEWLINE\]NEWLINE Main result: There exists a constant \(C>0\) such that NEWLINE\[NEWLINE\|f_2-p_{n,2}\|_{\Gamma}\leq Cn^{-\alpha}\log (n+1),\quad n\geq 1.NEWLINE\]