Various derivative estimates for functions of exponential type in a half-plane are proved in this paper. The reader will also find a related result about functions analytic in a quadrant. In addition, the paper contains a result about functions analytic in a strip. Our main tool in this study is the Schwarz-Pick theorem from the geometric theory of functions. We also use the Phragmén-Lindelöf principle, which is of course standard in such situations.

Let $C$ be the extended complex plane; $G\subset C$ a finite Jordan with $0\in G$; $w=\varphi \left(z\right)$ the conformal mapping of $G$ onto the disk $B\left(0;{\rho }_0\right):=\left\}w\left|w\right|<{\rho }_0\right\{$ normalized by $\varphi \left(0\right)=0$ and ${\varphi }^{\text{'}}\left(0\right)=1$. Let us set ${\varphi }_p\left(z\right):={\int }_0^z{\left[{\varphi }^{\text{'}}\left(\zeta \right)\right]}^{2/p}\mathrm{d}\zeta$, and let ${\pi }_{n,p}\left(z\right)$ be the generalized Bieberbach polynomial of degree $n$ for the pair $(G,0)$, which minimizes the integral $\underset{G}{\int \int }{\left|{\varphi }_p^{\text{'}}\left(z\right)-P_n^{\text{'}}\left(z\right)\right|}^p\mathrm{d}{\sigma }_z$ in the class of all polynomials of degree not exceeding $\le n$ with $P_n\left(0\right)=0$, $P_n^{\text{'}}\left(0\right)=1$. In this paper we study the uniform convergence of the generalized Bieberbach polynomials ${\pi }_{n,p}\left(z\right)$ to ${\varphi }_p\left(z\right)$ on $\overline{G}$ with interior and exterior zero angles and determine its dependence on the...

On montre que tout point de Misiurewicz dans l’ensemble de Mandelbrot $M$ possède un système fondamental de voisinages connexes dans $M$.