The univalent Bloch-Landau constant, harmonic symmetry and conformal glueing

Bohr radius for locally univalent harmonic mappings

We consider the class of all sense-preserving harmonic mappings $f= h+\overline{g}$ of the unit disk $\ID$, where $h$ and $g$ are analytic with $g(0)=0$, and determine the Bohr radius if any one of the following conditions holds: \bee $h$ is bounded in $\ID$. $h$ satisfies the condition ${\rm Re}\, h(z)\leq 1$ in $\mathbb{D}$ with $h(0)>0$. both $h$ and $g$ are bounded in $\ID$. $h$ is bounded and $g'(0)=0$. \eee We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of $f$ in $\ID$ is strictly less than $1$. In addition, we determine the…

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.

A subclass of harmonic univalent mappings with a restricted analytic part

In this article, a subclass of univalent harmonic mapping is introduced by restricting its analytic part to lie in the class $\mathcal{S}^δ[α]$, $0\leq α< 1$, $-\infty < δ< \infty$ which has been introduced and studied by Kumar \cite{Kumar87} (see also \cite{Mishra95}, \cite{MishraChoudhury95}, \cite{MishraDas96}, \cite{MishraGochhayat06}). Coefficient estimations, growth and distortion properties, area theorem and covering estimates of functions in the newly defined class have been established. Furthermore, we also found bounds for the Bloch's constant for all functions in that family.

On harmonic Bloch-type mappings

Let $f$ be a complex-valued harmonic mapping defined in the unit disk $\mathbb D$. We introduce the following notion: we say that $f$ is a Bloch-type function if its Jacobian satisfies $$ \sup_{z\in\mathbb D}(1-|z|^2)\sqrt{|J_f(z)|}<\infty. $$ This gives rise to a new class of functions which generalizes and contains the well-known analytic Bloch space. We give estimates for the schlicht radius, the growth and the coefficients of functions in this class. We establish an analogue of the theorem which states that an analytic $φ$ is Bloch if and only if there exists $c>0$ and a univalent $ψ$ such…

Landau 常数的计算机辅助下界

A computer-assisted lower bound for Landau's constant

本文通过计算机辅助方法，利用正 Toeplitz 矩矩阵和线性规划对偶性，将 Landau 常数和局部单叶 Bloch 常数的下界改进至 0.51 以上，并给出严格验证。

关于伪双曲度量下的尖锐Lipschitz连续性问题

The sharp Lipschitz continuity problem with respect to the pseudo hyperbolic metric

本文解决了Huang等人提出的开放问题，建立了局部单叶调和Bloch映射在伪双曲度量下的尖锐Lipschitz连续性，并推广到调和$(K,K_0)$-拟正则Bloch型映射，当$K_0=0$时在渐近尖锐意义下给出解答。