Harmonic mappings in the exterior of the unit disk

Abstract

Abstract. In this paper we consider a class of univalent orientation-preserving harmonic functions defined on the exterior of the unit disk which satisfy the condition ∞ n=1 n p (|an| + |bn|) ≤ 1. We are interested in finding radius of univalence and convexity for such class and we find extremal functions. Convolution, convex combination, and explicit quasiconformal extension for this class are also determined.

Quasiconformal Extensions of Harmonic Univalent Mappings of the Interior and Exterior Unit Disk

“…Some properties of the harmonic meromorphic starlike functions in Σ H have been investigated by Jahangiri [11]. In Section 4, we find a sufficient condition for f ∈ Σ H to be extended to a quasiconformal mapping of C. Theorem 4.1 is given as a generalization of the case f ∈ Σ H , proved in [18]. Let Σ k (0 < k < 1) be the class of sense-preserving homeomorphisms h of the extended plane C onto itself, with h(z) = z+ ∞ n=0 a n z −n analytic univalent in ∆ and k-quasiconformal in C. Krzyż [12] investigated the convolution problem of functions in Σ k by the area theorem.…”