arXiv subjects

ZhiHong Liu

Publications and source records attributed to ZhiHong Liu.

5 recordsLinked to original sources

arXiv2019

On univalent log-harmonic mappings

We consider the class univalent log-harmonic mappings on the unit disk. Firstly, we obtain necessary and sufficient conditions for a complex-valued continuous function to be starlike or convex in the unit disk. Then we present a general idea, for example, to construct log-harmonic Koebe mapping, log-harmonic right half-plane mapping and log-harmonic two-slits mapping and then we show precise ranges of these mappings. Moreover, coefficient estimates for univalent log-harmonic starlike mappings are obtained. Growth and distortion theorems for certain special subclass of log-harmonic mappings are studied. Finally, we propose two conjectures, namely, log-harmonic coefficient and log-harmonic covering conjectures.

math.CV↗

arXiv2019

Bohr radius for subordination and $K$-quasiconformal harmonic mappings

The present article concerns the Bohr radius for $K$-quasiconformal sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ for which the analytic part $h$ is subordinated to some analytic function $φ$, and the purpose is to look into two cases: when $φ$ is convex, or a general univalent function in $\ID$. The results state that if $h(z) =\sum_{n=0}^{\infty}a_n z^n$ and $g(z)=\sum_{n=1}^{\infty}b_n z^n$, then $$\sum_{n=1}^{\infty}(|a_n|+|b_n|)r^n\leq \dist (φ(0),\partialφ(\ID)) ~\mbox{ for $r\leq r^*$} $$ and give estimates for the largest possible $r^*$ depending only on the geometric property of $φ(\ID)$ and the parameter $K$. Improved versions of the theorems are given for the case when $b_1 = 0$ and corollaries are drawn for the case when $K\rightarrow \infty$.

math.CV↗

arXiv2018

Some properties of univalent log-harmonic mappings

We determine the representation theorem, distortion theorem, coefficients estimate and Bohr's radius for log-harmonic starlike mappings of order $α$, which are generalization of some earlier results. In addition, the inner mapping radius of log-harmonic mappings is also established by constructing a family of $1$-slit log-harmonic mappings. Finally, we introduce pre-Schwarzian, Schwarzian derivatives and Bloch's norm for non-vanishing log-harmonic mappings, several properties related to these are also obtained.

math.CV↗

arXiv2017

Radius of fully starlikeness and fully convexity of harmonic linear differential operator

Let $f=h+\overline{g}$ be a normalized harmonic mapping in the unit disk $\ID$. In this paper, we obtain the sharp radius of univalence, fully starlikeness and fully convexity of the harmonic linear differential operators $D_f^ε=zf_{z}-ε\overline{z}f_{\overline{z}}~(|ε|=1)$ and $F_λ(z)=(1-λ)f+λD_f^ε~(0\leqλ\leq 1)$ when the coefficients of $h$ and $g$ satisfy harmonic Bieberbach coefficients conjecture conditions. Similar problems are also solved when the coefficients of $h$ and $g$ satisfy the corresponding necessary conditions of the harmonic convex function $f=h+\overline{g}$. All results are sharp. Some of the results are motivated by the work of Kalaj et al. \cite{Kalaj2014} (Complex Var. Elliptic Equ. 59(4) (2014), 539--552).

math.CV↗

arXiv2015

Univalent harmonic mappings and lift to the minimal surfaces

We construct sense-preserving univalent harmonic mappings which map the unit disk onto a domain which is convex in the horizontal direction, but with varying dilatation. Also, we obtain minimal surfaces associated with such harmonic mappings. This solves also a recent problem of Dorff and Muir (Abstr. Appl. Anal. (2014)). In several of the cases, we illustrate mappings together with their minimal surfaces pictorially with the help of \texttt{Mathematica} software.

math.CV↗