Une construction de fonctions plurisousharmoniques nous permet, en utilisant les techniques de Hörmander, d’obtenir un résultat de $d^{\text{'}\text{'}}$-cohomologie à croissance. Les méthodes de B. Malgrange nous fournissent alors deux applications aux systèmes différentiels à coefficients constants.

Si $B$ est une boule ouverte contenue dans le domaine euclidien $\Omega$, tout filtre sur $B$, tendant non tangentiellement vers un point de $\partial \Omega \cap \partial B$, converge vers un point minimal dans le compactifié de Martin de $\Omega$. On donne une application, et une variante dans le cas plan, et on termine par un contre-exemple apportant une solution négative à un problème de R.S. Martin. L’idée générale de l’article est d’établir des variantes des inégalités de Harnack pour déterminer la frontière de Martin du domaine.

Dati due elementi $f$ e $g$ in un'algebra uniforme $A$, sia $G=f(M_A/f({\partial }_A)$. Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione $\lambda \to \log \max g(f^{-1}(\lambda ))$ è subarmonica su $G$ e che l’applicazione $\lambda \to g(f^{-1}(\lambda ))$ è analitica nel senso di Oka.

We give a characterization of functions that are uniformly approximable on a compact subset $K$ of ${ℝ}^n$ by biharmonic functions in neighborhoods of $K$.

Let a < 0, Ω = ℂ -(-∞, a] and U = z: |z| < 1. We consider the class $S_H(U,\Omega )$ of functions f which are univalent, harmonic and sense preserving with f(U) = Ω and satisfy f(0) = 0, $f_z\left(0\right)>0$ and $f_{z̅}\left(0\right)=0$. We describe the closure $\overline{S_H(U,\Omega )}$ of $S_H(U,\Omega )$ and determine the extreme points of $\overline{S_H(U,\Omega )}$.

Let a < 0 < b and Ω(a,b) = ℂ - ((-∞, a] ∪ [b,+∞)) and U= z: |z| < 1. We consider the class $S_H(U,\Omega (a,b))$ of functions f which are univalent, harmonic and sense-preserving with f(U) = Ω and satisfying f(0) = 0, $f_z\left(0\right)>0$ and $f_z̅\left(0\right)=0$.

A holomorphic function $f$ on a simply connected domain $\Omega$ is said to possess a universal Taylor series about a point in $\Omega$ if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta $K$ outside $\Omega$ (provided only that $K$ has connected complement). This paper shows that this property is not conformally invariant, and, in the case where $\Omega$ is the unit disc, that such functions have extreme angular boundary behaviour.