ulan Li

Solution to an open problem on convolutions of harmonic mappings

The convolution of two univalent harmonic mappings of the unit disc onto convex domains is not necessarily univalent. Dorff, Nowak and Wołoszkiewicz formulated an open problem concerning the convolution of two right half-plane mappings when one of them is fixed. In this article we solve this open problem.

Convolutions of slanted half-plane harmonic mappings

Abstract Let S0(Hγ) denote the class of all univalent, harmonic, sense-preserving and normalized mappings f of the unit disk D onto the slanted half-plane Hγ: = {w:Re(eiγw) > −1/2} with an additional condition fz(0) = 0. Functions in this class can be constructed by the shear construction due to Clunie and Sheil-Small which allows by examining their conformal counterpart. Unlike the conformal case, convolution of two univalent harmonic convex mappings in D is not necessarily even univalent in D. In this paper, we fix f0 ∈ S0(H0) and show that the convolutions of f0 and some slanted half-plane harmonic mapping are still convex in a particular direction. The results of the paper enhance the interest among harmonic mappings and, in particular, solve a recent open problem of Dorff et al. in a more general setting. Finally, we present some basic examples of functions and their corresponding convolution functions with specified dilatations, and illustrate them graphically with the help of MATHEMATICA software. These examples explain the behaviour of the image domains.

Discreteness of Möbius groups in infinite dimension

In this article, we prove that three statements of discreteness of Möbius groups in infinite dimension are equivalent, which are generalizations of the corresponding results in finite dimension. We also give an example to show that one discrete statement of finite dimension does not hold in infinite dimension.

Injectivity of sections of convex harmonic mappings and convolution theorems

We consider the class H0 of sense-preserving harmonic functions

Injectivity of sections of univalent harmonic mappings

f = h + \bar g

Sections of stable harmonic convex functions

defined in the unit disk |z| < 1 and normalized so that h(0) = 0 = h′(0) − 1 and g(0) = 0 = g′(0), where h and g are analytic in the unit disk. In the first part of the article we present two classes PH0(α) and GH0(β) of functions from H0 and show that if f ∈ PH0(α) and F ∈ GH0(β), then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections (partial sums)

Generalized Zalcman conjecture for convex functions of order α

{s_{n,n}}\left( f \right)\left( z \right) = {s_n}\left( h \right)\left( z \right) + \overline {{s_n}\left( g \right)\left( z \right)} ,