Related papers: Improved Bohr inequalities for certain class of ha…

We give coefficient estimates for a class of close-to-convex harmonic mappings, and discuss the Fekete-Szeg\H{o} problem of it. We also introduce two classes of polyharmonic mappings $\mathcal{HS}_{p}$ and $\mathcal{HC}_{p}$, consider the…

Let $f(\bfz,\bar\bfz)$ be a mixed strongly polar homogeneous polynomial of $3$ variables $\bfz=(z_1,z_2, z_3)$. It defines a Riemann surface $V:=\{[\bfz]\in \BP^{2}\,|\,f(\bfz,\bar\bfz)=0 \}$ in the complex projective space $\BP^{2}$. We…

We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of…

It is proved an inequality - integrated analogue of the Hardy inequality and as application simplified proof of the theorem of S. A. Vinogradov for the bounded Toeplitz operators on the space of functions analytic and bounded in the unit…

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…

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…

We present a unified strategy to derive Hardy-Poincar\'e inequalities on bounded and unbounded domains. The approach allows proving a general Hardy-Poincar\'e inequality from which the classical Poincar\'e and Hardy inequalities immediately…

The third real de Rham cohomology of compact homogeneous spaces is studied. Given $M=G/K$ with $G$ compact semisimple, we first show that each bi-invariant symmetric bilinear form $Q$ on $\mathfrak{g}$ such that…

A monotonicity property of Harnack inequality is proved for positive invariant harmonic functions in the unit ball.

The paper establishes a new family of sharp analytic inequalities. Together with the fractional Sobolev inequalities of Almgren and Lieb, they form a complete class of analytic inequalities, referred to as the chord Sobolev inequalities. A…

Let $L_H$ denote the set of all normalized locally one-to-one and sense-preserving harmonic functions in the unit disc $\Delta$. It is well-known that every complex-valued harmonic function in the unit disc $\Delta$ can be uniquely…

Let ${\mathcal S}$ be the class of analytic functions $f$ in the unit disk ${\mathbb D}$ with $f({\mathbb D}) \subset \overline{\mathbb D}$. Fix pairwise distinct points $z_1,\ldots,z_{n+1}\in \mathbb{D}$ and corresponding interpolation…

We make a careful analysis of Bohr's inequality, in the line started by Kayumov and Ponnusamy, where some extra summand (depending on the function) is added in the right-hand side of the inequality. We analyse the inequality when smaller…

We find best constants in several dilation invariant integral inequalities involving derivatives of functions. Some of these inequalities are new and some were known without best constants. The contents: 1. Estimate for a quadratic form of…

Let $n$ be a positive integer. Let $\mathbf U$ be the unit disk, $p\ge 1$ and let $h^p(\mathbf U)$ be the Hardy space of harmonic functions. Kresin and Maz'ya in a recent paper found the representation for the function $H_{n,p}(z)$ in the…

We introduce and study properties of certain new harmonic function spaces on products of upper half-spaces.Norm estimates for the so-called expanded Bergman projections are obtained.Sharp theorems on multipliers acting on certain Sobolev…

Recently, the Wang et al. \cite{wwrq} proposed a coefficient conjecture for the family ${\mathcal S}_H^0(K)$ of $K$-quasiconformal harmonic mappings $f = h + \overline{g}$ that are sense-preserving and univalent, where…

We introduce the volume-preserving equivalence among symmetric matrix-valued map-germs which is the unimodular version of Bruce's $\mathcal{G}$-equivalence. The key concept to deduce unimodular classification out of classification relative…

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…

In this work we present a definition for coherence and compatibility of multilinear mappings and homogenous polynomial classes. These definitions are more restricted than the ones proposed before. We began analyzing this new definition in a…