Related papers: A note on the Schwarz lemma for harmonic functions
In this note we establish a Schwarz type inequality for holomorphic mappings between unit balls $B_n$ and $B_m$ in corresponding complex spaces.
This paper introduces a unified framework for Bohr-type inequalities by incorporating multiple Schwarz functions into the majorant series for $K$-quasiconformal harmonic mappings in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| <…
The main purpose of this paper is to develop some methods to investigate the Schwarz type lemmas of holomorphic mappings and pluriharmonic mappings in Hilbert and Banach spaces. Initially, we extend the classical Schwarz lemmas of…
In these notes we consider power series representations of functions on the unit disk in the complex plane which define harmonic and holomorphic functions and related matters concerning boundary values, Poisson kernels, and so on.
In this paper, we establish a boundary Schwarz lemma for solutions to non-homogeneous biharmonic equations.
In this paper, we consider transversally harmonic maps between Riemannian manifolds with Riemannian foliations. In terms of the Bochner techniques and sub-Laplacian comparison theorem, we are able to establish a generalization of the…
For $-1\leq B<A\leq 1$, let $\mathcal{C}(A,B)$ denote the class of normalized Janowski convex functions defined in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ that satisfy the subordination relation $1+zf''(z)/f'(z)\prec…
The main purpose of this paper is to establish a Schwarz lemma for the solutions to the Dirichlet problems for the invariant Laplacians. The obtained result of this paper is a generalization of the corresponding known results [11, Theorem…
The well-known Schwarz-Pick lemma states that any analytic mapping $\phi$ of the unit disk $U$ into itself satisfies the inequality $$|\phi'(z)|\leq \frac{1-|\phi(z)|^2}{1-|z|^2}, \quad z\in U.$$ This estimate remains the same if we…
We produce a Schwarz lemma for the symmetrized tridisc \[ \mathbb G_3 =\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|< 1, i=1,2,3 \}. \] We show that an interpolating function related to the Schward lemma for $\mathbb G_3$ is not…
We prove the stronger version of Harnack's inequality for positive harmonic functions defined on the unit disc.
The purpose of this note is to discuss the real analogue of the Schwarz lemma from complex analysis.
In this paper, Jack lemma is used for obtaining several differential inequalities over analytic functions that later on, lead to new criteria for univalency in the unit disk.
Let $\mathcal{A}$ be the class of analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalized conditions $f(0)=0$, $f'(0)=1$. For $-\pi/2<\alpha<\pi/2$ and $0\le \beta<1$, let…
In this article, Bohr type inequalities for some complex valued harmonic functions defined on the unit disk are given. All the results are sharp.
Let $\mathcal{A}$ denote the class of analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ normalized by $f(0)=0$, $f'(0)=1$. For $-\pi/2<\alpha<\pi/2$, let $\mathcal{S}_{\alpha}$ be the subclass of $\mathcal{A}$…
We prove some sharp inequalities for complex harmonic functions on the unit disk. The results extend a M. Riesz conjugate function theorem and some well-known estimates for holomorphic functions. We apply some of results to the…
The aim of this paper is to establish some properties of solutions to the Dirichlet-Neumann problem: $(\partial_z\partial_{\overline{z}})^2 w=g$ in the unit disc $\ID$, $w=\gamma_0$ and…
Here we shall introduce the concept of harmonic balls/spheres in sub-domains of $\R^n$, through a mean value property for a sub-class of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept…
Let $f(z)=h(z)+\overline{g(z)}$ be a harmonic mapping of the unit disk $U$. In this paper, the sharp coefficient estimates for bounded planar harmonic mappings are established, the sharp coefficient estimates for normalized planar harmonic…