Coefficient Inequalities for Concave and Meromorphically Starlike Univalent Functions

Let \ID denote the open unit disk and f:\,\ID\TO\BAR\IC be meromorphic and univalent in \ID with the simple pole at $p\in (0,1)$ and satisfying the standard normalization $f(0)=f'(0)-1=0$. Also, let $f$ have the expansion $$f(z)=\sum_{n=-1}^{\infty}a_n(z-p)^n,\quad |z-p|<1-p,$$ such that $f$ maps \ID onto a domain whose complement with respect to \BAR{\IC} is a convex set (starlike set with respect to a point w_0\in \IC, w_0\neq 0 resp.). We call these functions as concave (meromorphically starlike resp.) univalent functions and denote this class by $Co(p)$ $(\Sigma^s(p, w_0)$ resp.). We prove some coefficient estimates for functions in the classes where the sharpness of these estimates is also achieved

On the generalized Zalcman functional λan2−a2n−1\lambda a_n^2-a_{2n-1} in the close-to-convex family

Let ${\mathcal S}$ denote the class of all functions $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$ analytic and univalent in the unit disk \ID. For $f\in {\mathcal S}$, Zalcman conjectured that $|a_n^2-a_{2n-1}|\leq (n-1)^2$ for $n\geq 3$. This conjecture has been verified only certain values of $n$ for $f\in {\mathcal S}$ and for all $n\ge 4$ for the class $\mathcal C$ of close-to-convex functions (and also for a couple of other classes). In this paper we provide bounds of the generalized Zalcman coefficient functional $|\lambda a_n^2-a_{2n-1}|$ for functions in $\mathcal C$ and for all $n\ge 3$, where $\lambda$ is a positive constant. In particular, our special case settles the open problem on the Zalcman inequality for $f\in \mathcal C$ (i.e. for the case $\lambda =1$ and $n=3$).Comment: 14 pages. The article has been with a journa

Bohr radius for subordination and KK-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 $\varphi$, and the purpose is to look into two cases: when $\varphi$ 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 (\varphi(0),\partial\varphi(\ID)) ~\mbox{ for $r\leq r^*$} and give estimates for the largest possible $r^*$ depending only on the geometric property of \varphi (\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 $K\rightarrow \infty$.Comment: 15 pages; To appear in Bulletin of the Malaysian Mathematical Sciences Societ

Schwarz's Lemmas for mappings satisfying Poisson's equation

For $n\geq3$, $m\geq1$ and a given continuous function $g:~\Omega\rightarrow\mathbb{R}^{m}$, we establish some Schwarz type lemmas for mappings $f$ of $\Omega$ into $\mathbb{R}^{m}$ satisfying the PDE: $\Delta f=g$, where $\Omega$ is a subset of $\mathbb{R}^{n}$. Then we apply these results to obtain a Landau type theorem.Comment: 14 page

John disk and KK-quasiconformal harmonic mappings

The main aim of this article is to establish certain relationships between $K$-quasiconformal harmonic mappings and John disks. The results of this article are the generalizations of the corresponding results of Ch.~Pommerenke \cite{Po}.Comment: 18 pages, 1 figur

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.Comment: 16 pages; This paper was with Studia Scientiarum Mathematicarum Hungarica since May 2017; Finally returned by saying that they could not find a suitable referee

On Harmonic ν\nu-Bloch and ν\nu-Bloch-type mappings

The aim of this paper is twofold. One is to introduce the class of harmonic $\nu$-Bloch-type mappings as a generalization of harmonic $\nu$-Bloch mappings and thereby we generalize some recent results of harmonic $1$-Bloch-type mappings investigated recently by Efraimidis et al. \cite{EGHV}. The other is to investigate some subordination principles for harmonic Bloch mappings and then establish Bohr's theorem for these mappings and in a general setting, in some cases.Comment: 17 pages; Comments are welcom

Uniformly locally univalent harmonic mappings associated with the pre-Schwarzian norm

In this paper, we consider the class of uniformly locally univalent harmonic mappings in the unit disk and build a relationship between its pre-Schwarzian norm and uniformly hyperbolic radius. Also, we establish eight ways of characterizing uniformly locally univalent sense-preserving harmonic mappings. We also present some sharp distortions and growth estimates and investigate their connections with Hardy spaces. Finally, we study subordination principles of norm estimates.Comment: 28 pages; The article is to appear in Indagationes Mathematica

Representation formula and bi-Lipschitz continuity of solutions to inhomogeneous biharmonic Dirichlet problems in the unit disk

The aim of this paper is twofold. First, we establish the representation formula and the uniqueness of the solutions to a class of inhomogeneous biharmonic Dirichlet problems, and then prove the bi-Lipschitz continuity of the solutions.Comment: 24 pages; To appear in the Journal of Mathematical Analysis and Application

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 $D_f^{\epsilon}=zf_{z}-\epsilon\overline{z}f_{\overline{z}}~(|\epsilon|=1)$ and $F_{\lambda}(z)=(1-\lambda)f+\lambda D_f^{\epsilon}~(0\leq\lambda\leq 1)$ 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).Comment: 14 pages; To appear in the Bulletin of the Korean Mathematical Societ