Normal Families of Holomorphic Functions Concerning Zero Numbers

In this paper,we study the normal families related with a Hayman conjecture of higher derivative concerning zero numbers,and get one normal criteria.Our result improve some earlier related result.     

On Sets of Density Zero

In this note we show that the normed space of the simple functions associated to the ring of all subsets of ? of density zero is baireled, answering in the affirmative a question raised in [6]. This fact provides a new extension of Nikodým boundedness theorem.     

On Zero Sets in the Dirichlet Space

We study the zero sets of functions in the Dirichlet space. Using Carleson??s formula for the Dirichlet integral, we obtain some new families of zero sets. We also show that any closed subset of $E \subset \mathbb {T}$ with logarithmic capacity zero is the set of accumulation points of the zeros of a function in the Dirichlet space. The zeros satisfy a growth restriction which depends on E.     

Functions Holomorphic along Holomorphic Vector Fields

The main result of the paper is the following generalization of Forelli’s theorem (Math. Scand. 41:358–364, 1977): Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with eigenvalues whose ratios are positive reals. Then any function φ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary. K.T. Kim and G. Schmalz were supported by the Scientific visits to Korea program of the AAS and KOSEF. E. Poletsky was supported by NSF Grant DMS-0500880. G. Schmalz gratefully acknowledges support and hospitality of the Max-Planck-Institut für Mathematik Bonn.     

Compact Sets of Holomorphic Mappings

Compactness of locally bounded sets of holomorphic functions with infinite dimensional domains is connected, using Heinrich's density condition, to the Schwartz and semi-Montel properties on the domain. The metrizability of bounded subsets for various spaces of holomorphic functions is investigated.     

On Weighted Spaces of Harmonic and Holomorphic Functions

Weighed spaces of harmonic and holomorphic functions on theunit disc are studied. We show that for all radial weights whichare not decreasing too fast the space of harmonic functionsis isomorphic to c₀. For the weights that we consider we completelycharacterize those spaces of holomorphic functions which areisomorphic to c₀. Moreover, we determine when the Riesz projection,mapping the weighted space of harmonic functions onto the correspondingspace of holomorphic functions, is bounded.     

Holomorphic Almost-Periodic Functions

The present paper gives a survey of up-to-date results of holomorphic almost-periodic functions and mappings in one and several complex variables.     

On the Bohr Radius for Two Classes of Holomorphic Functions

Using some multidimensional analogs of the inequalities of E. Landau and F. Wiener for the Taylor coefficients of special classes of holomorphic functions on Reinhardt domains we obtain some estimates for the Bohr radius.     

On Chebyshev Systems of Functions Holomorphic in the Unit Disk

Using conditions on the divided differences of higher orders, we introduce special Chebyshev systems of functions holomorphic in the unit disk. We consider various properties of these functions.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 229–240, April–June, 2005.     

On Some Sets of Group Functions

Let G be a group, let A be an Abelian group, and let n be an integer such that n –1. In the paper, the sets _n (G,A) of functions from G into A of degree not greater than n are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from G into A is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if G is finite, then the study of the set _n (G,A) is reduced to that of the set n(G/O ^p (G),A _p ) for primes p dividing G/G. Here O ^p (G) stands for the p-coradical of the group G, A _p for the p-component of A, and G for the commutator subgroup of G.     

Holomorphic Semi-almost Periodic Functions

We study the Banach algebras of bounded holomorphic functions on the unit disk whose boundary values, having, in a sense, the weakest possible discontinuities, belong to the algebra of semi-almost periodic functions on the unit circle. The latter algebra contains as a special case an algebra introduced by Sarason in connection with some problems in the theory of Toeplitz operators.     

Zeros of Generalized Holomorphic Functions

Let f be a generalized holomorphic function on a connected open set . It is proved that f equals zero if and only if there exists a smooth curve and a set A of positive (one-dimensional) measure such that f takes zero value on A. Also, a holomorphic generalized function different from zero on the disc, which takes zero values on a dense G _δ-set of the disc, is constructed. The generalized zero set of a holomorphic function is introduced and studied in an analogous way.     

全纯函数的正规族

本文证明了两个正规族定理,分别改进了张国明、孙伟和庞学诚的一个结果和徐焱的一个结果.     

Kuratowski Convergence of Holomorphic Functions

The notion of Kuratowski convergence is applied to describe a kind of convergence in the context of holomorphic functions. We associate it to a convenient topology, explore its relation with the compact-open topology, thus providing a new set theoretic point of view of this classic topology, and present it in the framework of set-valued mappings.