Landau’s theorem for polyharmonic mappings

In this paper, we first investigate coefficient estimates for bounded polyharmonic mappings in the unit disk D$\mathbb{D}$. Then, we obtain two versions of Landau’s theorem for polyharmonic mappings F$F$, and for the mappings of the type L(F)$L\left(F\right)$, where L$L$ is the differential operator of Abdulhadi, Abu Muhanna and Khuri. Examples and numerical estimates are given.     

拟共形映射面积偏差条件下的Schwarz型定理

关于拟共形映射在一些面积偏差条件下,得到了Schwarz型定理．     

Distortion theorems for subclasses of starlike mappings along a unit direction in ℂ

In this paper, we obtain a distortion theorem of Jacobian matrix for biholomorphic subclasses of starlike mappings along a unit direction on the unit polydisc. These results extend the classical distortion theorem of starlike mappings to higher dimensions. We then give an upper bound estimate for biholomorphic subclasses of starlike mappings along a unit direction in a complex Banach space.     

第一类典型域上的Bloch常数

引入第一类典型域R_I(m,n)上的全纯映照子族H_k(R_I(m,n)),当k→+∞时,该映照族就是R_I(m,n)上的局部双全纯映照族.建立了H_k(R_I(m,n))上的Bonk偏差定理.当k=1和k→+∞时,其结果分别都回到了FitzGerad和龚升关于典型域R_I(m,n)上的Bonk偏差定理.当m=n=1时,其结果又回到了Liu和Minda在单位圆盘上的偏差定理.应用偏差定理,给出了映照族H_k(R_I(m,n))上的Bloch常数估计,其结果补全了从k=1和k→+∞之间的R_I(m,n)上Bloch常数估计的所有结果,而且把单位球上的Bloch常数估计推广到R_I(m,n)上.     

The Distortion Theorems for Harmonic Mappings with Negative Coefficient Analytic Parts

Some sharp estimates for coefficients, distortion and the growth order are obtained for harmonic mappings $f ∈ TL^α_H$which are locally univalent harmonic mappings in the unit disk $\mathbb{D}:=\{z:|z| < 1\}$. Moreover, denoting the subclass $TS^α_H$ of the normalized univalent harmonic mappings, we also estimate the growth of $|f|,$ $f ∈ TS^α_H,$ and their covering theorems.     

Compatible mappings of type (B) and common fixed point theorems in Saks spaces

In this paper we first introduce the concept of compatible mappings of type (B) and compare these mappings with compatible mappings and compatible mappings of type (A) in Saks spaces. In the sequel, we derive some relations between these mappings. Secondly, we prove a coincidence point theorem and common fixed point theorem for compatible mappings of type (B) in Saks spaces.     

The Distortion Theorems for Harmonic Mappings with Negative Coefficient Analytic Parts

Some sharp estimates for coefficients, distortion and the growth order are obtained for harmonic mappings $f \in TL^{\alpha}_H,$ which are locally univalent harmonic mappings in the unit disk $\mathbb{D}:=\{z:|z|<1\}.$ Moreover, denoting the subclass $TS^{\alpha}_H$ of the normalized univalent harmonic mappings, we also estimate the growth of $|f|,$ $f \in TS^α_H,$ and their covering theorems.     

多重调和映射的等周型和Fejer-Riesz型不等式

本文讨论多重调和映射的等周型和Fejer-Riesz型不等式.首先,本文改进Kalaj和Meˇstrovi′c的相应结果,并将其结果推广到多重调和映射.其次,本文证明Pavlovi′c和Dostani′c的相应结果对于多重调和映射也是成立的.最后,本文建立关于多重调和映射的Fejer-Riesz型不等式.     

有限偏差函数与调和函数

分别借助解析函数与调和函数两类函数的Dirichlet积分,利用相关文献给定边界值的拟共形映射极值伸缩商的估计方法,通过有限偏差函数和拟共形映射的关系估计了具有给定边界值的有限偏差函数的极值伸缩商.得到了解析函数的Dirichlet积分在有限偏差函数下具有拟不变性,同时给出有限偏差函数极值伸缩商的下界估计.     

On polyharmonic helices in space forms

Archiv der Mathematik - In this article, we study polyharmonic curves of constant curvature where we mostly focus on the case of curves on the sphere. We classify polyharmonic curves of constant...     

二维Lipschitz区域上一类带L^p边值的非齐次多调和Neumann问题

该文运用层位势方法研究了二维Lipschitz区域上一类带L^p边值的非齐次多调和Neumann问题.利用多层S位势,给出了该类问题的惟一积分表示解,其中,多层S位势是经典单层位势的高阶类似物,通过多调和基本解加以定义.     

Divergence symmetries of semilinear polyharmonic equations involving critical nonlinearities

It is shown that a Lie point symmetry of the semilinear polyharmonic equations involving nonlinearities of power or exponential type is a variational/divergence symmetry if and only if the equation parameters assume critical values. The corresponding conservation laws for critical polyharmonic semilinear equations are established.     

Refined Hyers-Ulam approximation of approximately Jensen type mappings

In 1940 S.M. Ulam proposed the famous Ulam stability problem. In 1941 D.H. Hyers solved this problem for additive mappings subject to the Hyers condition on approximately additive mappings. In this paper we generalize the Hyers result for the Ulam stability problem for Jensen type mappings, by considering approximately Jensen type mappings satisfying conditions weaker than the Hyers condition, in terms of products of powers of norms. This process leads to a refinement of the well-known Hyers-Ulam approximation for the Ulam stability problem. Besides we introduce additive mappings of the first and second form and investigate pertinent stability results for these mappings. Also we introduce approximately Jensen type mappings and prove that these mappings can be exactly Jensen type, respectively. These stability results can be applied in stochastic analysis, financial and actuarial mathematics, as well as in psychology and sociology.     

Conformal Killing Vector Fields and Rellich Type Identities on Riemannian Manifolds, II

We propose a general Noetherian approach to Rellich integral identities. Using this method we obtain a higher order Rellich type identity involving the polyharmonic operator on Riemannian manifolds admitting homothetic transformations. Then we prove a biharmonic Rellich identity in a more general context. We also establish a nonexistence result for semilinear systems involving biharmonic operators.     

Solvability of some boundary-value problems for polyharmonic equation with Hadamard-Marchaud boundary operator

We investigate solvability conditions for certain non-classical boundary-value problems for the polyharmonic equation. As the boundary operators we consider fractional differential operators in the sense of Hadamard-Marchaud. The considered problems generalize well-known Dirichlet and von Neumann boundary-value problems for boundary operators of fractional type.     

多复变数的抛物星形映射

本文给出了抛物星形映射,ρ次的抛物星形映射,抛物形β型螺形映射和ρ次的抛物形β型螺形映射的定义.证明了推广的Rope-Suffridge算子保持这些映射的几何性质不变.     

Higher order Poisson kernels and Lp polyharmonic boundary value problems in Lipschitz domains

In this article, we introduce higher order conjugate Poisson and Poisson kernels, which are higher order analogues of the classical conjugate Poisson and Poisson kernels, as well as the polyharmonic fundamental solutions, and define multi-layer potentials in terms of the Poisson field and the polyharmonic fundamental solutions, in which the former is formed by the higher order conjugate Poisson and the Poisson kernels. Then by the multi-layer potentials, we solve three classes of boundary value problems(i.e., Dirichlet, Neumann and regularity problems) with L~p boundary data for polyharmonic equations in Lipschitz domains and give integral representation(or potential) solutions of these problems.     

拟共形映照的双曲面积偏差估计

The distortion property of hyperbolic area of planar quasiconformal mappings is studied in this paper.In the case of radial quasiconformal mappings and angular deformed quasiconformal mappings their hyperbolic area distortions are estimated quite sharply.The result can be applied to judge whether the hyperbolic area of a planar subset is explodable.     

Pizzetti Series and Polyharmonicity Associated with the Dunkl Laplacian

In this paper we are concerned with the Pizzetti series associated with the Dunkl Laplacian denoted Δ _k . We study the convergence of this series and we give some applications. Next we establish some properties of polyharmonic functions associated with Δ _k , especially, we establish Liouville type results.     

多复变数星形映射在某方向上精确的偏差定理

研究了Cn中单位多圆柱上星形映射在某方向上精确的偏差定理．给出了复Banach空间单位球的某方向上精确的偏差的上界,同时给出了下界的猜测．