Related papers: Landau-type theorems for certain bounded bi-analyt…
The aim of this paper is to obtain the Schwarz-Pick type inequality for $\alpha$-harmonic functions $f$ in the unit disk and get estimates on the coefficients of $f$. As an application, a Landau type theorem of $\alpha$-harmonic functions…
A $2p$-times continuously differentiable complex-valued function $f=u+iv$ in a simply connected domain $\Omega\subseteq\mathbb{C}$ is \textit{p-harmonic} if $f$ satisfies the $p$-harmonic equation $\Delta ^pf=0.$ In this paper, we…
In this paper, we first establish two versions of Landau-Bloch type theorem for $(K,K')$-elliptic harmonic mappings with a bounded minimum distortion. Next, we provide several coefficient estimates and a conjecture for $(K,K')$-elliptic…
In this paper, we continue our investigation of polyharmonic mappings in the complex plane. First, we establish two Landau type theorems. We also show a three circles type theorem and an area version of the Schwarz lemma. Finally, we study…
With the uniform positions we prove theorems of Landau and Hardy-Littlwood type for Goldbach, Chen, Lemoine-Levy and other binary partitions of positive integers. We also pose some new conjectures.
In this paper, firstly we prove two refined Bohr-type inequalities associated with area for bounded analytic functions $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ in the unit disk. Later, we establish the Bohr-type operator on analytic functions…
In this paper, we investigate some properties of pluriharmonic mappings defined in the unit ball. First, we discuss some geometric univalence criteria on pluriharmonic mappings, and then establish a Landau-Bloch theorem for a class of…
For analytic functions f(z) in the closed unit disk \bar{U}, two boundary points z_1 and z_2 such that \alpha = (f'(z_1)+f'(z_2))/2 in f'(U) are considered. The object of the present paper is to discuss some interesting conditions for f(z)…
In one complex variable it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant $\rho$, independent…
We prove sharp Landau- and Schottky-type theorems for analytic functions which omit the $n$-th roots of unity. The proofs are based on a sharp lower bound for the Poincar\'e metric of the complex plane punctured at the roots of unity.
We discuss some properties of linear functionals on topological hyperbolic and topological bicomplex modules. The hyperbolic and bicomplex analogues of the uniform boundedness principle, the open mapping theorem, the closed graph theorem…
The main purpose of this paper is to develop some methods to investigate equivalent norms and Hardy-Littlewood type Theorems on Lipschitz type spaces of analytic functions and complex-valued harmonic functions. Initially, some…
Polyharmonic functions f of infinite order and type {\tau} on annular regions are systematically studied. The first main result states that the Fourier-Laplace coefficients f_{k,l}(r) of a polyharmonic function f of infinite order and type…
In this paper, we investigate the properties of locally univalent and multivalent planar harmonic mappings. First, we discuss the coefficient estimates and Landau's Theorem for some classes of locally univalent harmonic mappings, and then…
We obtain uniqueness theorems for harmonic and subharmonic functions of a new type. They lead to new analytic extension criteria and new conditions for stability of operator semigroups in Banach spaces with Fourier type.
We establish two-point distortion theorems for sense-preserving planar harmonic mappings $f=h+\overline{g}$ which satisfies the univalence criteria in the unit disc such that, Becker's and Nehari`s harmonic version. In addition, we find the…
Let $U$ be a bounded open subset of the complex plane and let $A_{\alpha}(U)$ denote the set of functions analytic on $U$ that also belong to the little Lipschitz class with Lipschitz exponent $\alpha$. It is shown that if $A_{\alpha}(U)$…
We prove several Liouville theorems for F-harmonic maps from some complete Riemannian manifolds by assuming some conditions on the Hessian of the distance function, the degrees of F(t) and the asymptotic behavior of the map at infinity. In…
Bieberbach's conjecture was very important in the development of Geometric Function Theory, not only because of the result itself, but also due to the large amount of methods that have been developed in search of its proof, it is in this…
Let $\mathcal{H}$ be the class of harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, where $h$ and $g$ are analytic in $\mathbb{D}$ with the normalization $h(0)=g(0)=h'(0)-1=0$. Let…