New extremal polynomials and their approximation properties

M. Küçükaslan, F. G. Abdullayev

Let $G\subset\mathbb C$ be a simply connected region whose boundary $L:=\partial G$ is a Jordan curve and $z_0\in G$ be an arbitrary fixed point. Let $w=\varphi(z)$ be the conformal mapping of $G$ onto the disk $D(0,r_0):=\{w:|w|<r_0\}$, satisfying $\varphi(z_0)=0$, $\varphi'(z_0)=1$. Let us consider the following extremal problem: \begin{equation}abel{eq1} \|ǎrphi-P_n\|_{L'_p(G)}:=\|ǎrphi'-P'_n\|_{L-p(G)}o\min,\quad p>0, \end{equation} in the class of all polynomials satisfying $P_n(z_0)=0$ and $P'_n(z_0)=1$. There exists a polynomial $\Pi_{n,p}(z)$ furnishing to the \eqref{eq1} and $\Pi_{n,p}(z)$ is determined uniquely when $p>1$. This kind of polynomials will be called $p$-Bieberbach polynomials. In this work, we investigate the approximation properties of the polynomials $\{\Pi_{n,p}(z)\}$ to the $\varphi$ in the $L^1_p$- and $C$-norms for some regions of the complex plane.