We give a complete and transparent proof of Koebe's General Uniformisation Theorem that every planar Riemann surface is biholomorphic to a domain in the Riemann sphere ℂ̂, by showing that a domain with analytic boundary and at least two boundary components on a planar Riemann surface is biholomorphic to a circular-slit annulus in ℂ.

Let $P\left(z\right)={\sum }_{\nu =0}^n{a}_{\nu }{z}^{\nu }$ be a polynomial of degree at most $n$ which does not vanish in the disk $\left|z\right|<1$, then for $1\le p<\infty$ and $R>1$, Boas and Rahman proved $${∥P\left(Rz\right)∥}_p\le ({∥R^n+z∥}_p/{∥1+z∥}_p){∥P∥}_p.$$ In this paper, we improve the above inequality for $0\le p<\infty$ by involving some of the coefficients of the polynomial $P\left(z\right)$. Analogous result for the class of polynomials $P\left(z\right)$ having no zero in $\left|z\right|>1$ is also given.

Étant donné une fonction $w\left(x\right)\ge 0$ paire et continue, on se demande si une fonction entière $\phi \left(z\right)\not\equiv 0$ de type exponentiel $a$ existe telle que $\phi \left(x\right)\exp w\left(x\right)$ soit borné pour $-\infty \&lt;x\&lt;\infty$. L’existence d’une telle $\phi$ est équivalente à celle d’une fonction croissante $\rho \left(t\right)$ sur $[0,\infty )$ telle que $\rho \left(t\right)=\theta \left(t\right)$, que $\frac{\rho \left(t\right)}{t}\to \frac{a}{\pi }$ pour $t\to \infty$, et que $w\left(x\right)+{\infty }_0^{\infty }log|1-\frac{x^2}{t^2}|d\rho \left(t\right)\le C^{\mathrm{te}}$, $x\in \mathbf{R}$, pourvu que $w\left(x\right)$ satisfasse à une condition de régularité assez peu restrictive, décrite au début de l’article. On démontre que l’existence d’une telle $\rho$ est à son tour équivalente à ce que la fonction $\frac{1}{\pi }{\int }_{-\infty }^{\infty }\frac{\left|\Im z\right|}{|z-t{|}^2}w\left(t\right)dt-a\left|\Im z\right|$ admette une majorante surharmonique...

Under mild conditions on the weight function K we characterize lacunary series in the so-called ${}_K$ spaces.

Starting from Lagrange interpolation of the exponential function ${\mathrm{e}}^z$ in the complex plane, and using an integral representation formula for holomorphic functions on Banach spaces, we obtain Lagrange interpolating polynomials for representable functions defined on a Banach space $E$. Given such a representable entire funtion $f:E\to ℂ$, in order to study the approximation problem and the uniform convergence of these polynomials to $f$ on bounded sets of $E$, we present a sufficient growth condition on the interpolating...