Hülder Conditions and the Topology of Simply Connected Domains*

Let ƒ be regular univalent and normalized in the unit disc U (i.e. ƒ ∊ S) and continuous on U ∈ T, where T denotes the boundary of U.Recently Essén proved [5] a conjecture of Piranian [7] stating that if the derivative of ƒ ∊ S is bounded in U and ƒ(z1) = ƒ(z2) = … = ƒ(zn) for Zj ∊ T, 1 ≤ j ≤ n, then n ≤ 2. In fact, Essén proved a more general result, using a deep result on harmonic functions. The aim of the following article is to replace Essén's proof by a completely different proof which is based only on Goluzin's inequalities and is much more elementary.