Some sufficient conditions for univalence and starlikeness
Abstract. Let S denote the family of functions f, holomorphic and univalent in the open unit disk U, and normalized by f(0) = 0, f'(0) = 1.
We prove that an Ahlfors-regular set (with dimension one) E ⊂ Rn which verifies a βq-version of P. W. Jones' geometric lemma is included in an Ahlfors-regular curve Γ.This theorem is due to G. David and S. Semmes, we give a more direct proof.
In this paper we consider non-normalized univalent subordination chains and the connection with the Loewner differential equation on the unit ball in . To this end, we study the most general form of the initial value problem for the transition mapping, and prove the existence and uniqueness of solutions. Also we introduce the notion of generalized spirallikeness with respect to a measurable matrix-valued mapping, and investigate this notion from the point of view of non-normalized univalent subordination...
We give an example relating to the regularity properties of mappings with finite distortion. This example suggests conditions to be imposed on the distortion function in order to avoid "cavitation in measure".