Let ∆ be the unit disc of C and let f, g ∈ Hol(∆,∆) be such that f ◦ g = g ◦ f . For A > 1, let FixA(f) := {p ∈ ∂∆ | limr→1 f(rp) = p, limr→1 |f ′(rp)| ≤ A}. We study the behavior of g on FixA(f). In particular, we prove that g(FixA(f)) ⊆ FixA(f). As a consequence, besides conditions for FixA(f)∩FixA(g) 6= ∅, we prove a conjecture of C. Cowen in case f and g are univalent mappings.

We introduce and investigate classes of (p,q)-starlike harmonic univalent functions defined by subordination. first obtained a coefficient characterization these functions. give necessary sufficient convolution conditions, distortion bounds, compactness extreme points for the with negative coefficients.

Recently, many papers in the theory of univalent functions have been devoted to mapping and characterization properties of various linear integral or integro-differential operators in the class S (of normalized analytic and univalent functions in the open unit disk U), and in its subclasses (as the classes S∗ of the starlike functions and K of the convex functions in U). Among these operators, ...

Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi Author Tri Quach Name of the doctoral dissertation Computational Methods in Conformal and Harmonic Mappings Publisher School of Science Unit Department of Mathematics and Systems Analysis Series Aalto University publication series DOCTORAL DISSERTATIONS 226/2015 Field of research Mathematics Manuscript submitted 14 April 2015 Date of...