by using generalized salagean differential operator a newclass of univalent holomorphic functions with fixed finitely manycoefficients is defined. coefficient estimates, extreme points,arithmetic mean, and weighted mean properties are investigated.

The paper studies semi-almost periodic holomorphic functions on a polydisk which have, in a sense, the weakest possible discontinuities on the boundary torus. The basic result used in the proofs is an extension of the classical Bohr approximation theorem for almost periodic holomorphic functions on a strip to the case of Banach-valued almost periodic holomorphic functions.

It is a well-known and elementary fact that a holomorphic function on a compact complex manifold is necessarily constant. The purpose of the present article is to investigate whether, or to what extent, a similar property holds in the setting of holomorphically foliated spaces.

exists. The function f is continuously differentiable when it is differentiable and f ′ is continuous. A k-times continuously differentiable function is C, and a continuous function is C. A V -valued function f is weakly C when for every λ ∈ V ∗ the scalar-valued function λ◦ f is C. This sense of weak differentiability of a function f does not refer to distributional derivatives, but to differe...

We show that the duals of Banach algebras scalar-valued bounded holomorphic functions on open unit ball BE a space E lack weak⁎-strongly exposed points. Consequently, we obtain some an arbitrary have Daugavet property which extends observation P. Wojtaszczyk [56]. Moreover, present new denseness result by proving set norm-attaining vector-valued dual is dense provided its predual has metric π-p...

The Liouville theorem states that bounded holomorphic complex functions are necessarily constant. Holomorphic functions fulfill the socalled Cauchy-Riemann (CR) conditions. The CR conditions mean that a complex z-derivative is independent of the direction. Holomorphic functions are ideal for activation functions of complex neural networks, but the Liouville theorem makes them useless. Yet recen...

We establish basic results of complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds (such as algebras of Bohr’s holomorphic almost periodic functions on tube domains or algebras of all fibrewise bounded holomorphic functions arising, e.g., in the corona problem for H). In particular, in this context we obtain results on holomorphic extension fr...

The aim of the present article is to solve in the negative a well-known open problem raised by Bierstedt, Meise and Summers in [BMS1] (see also [BM1]). We construct a countable inductive limit of weighted Banach spaces of holomorphic functions, which is not a topological subspace of the corresponding weighted inductive limit of spaces of continuous functions. As a consequence the topology of th...

Almost periodic divisors, holomorphic functions, and holomorphic mappings * Favorov S.Ju. Abstract We prove that to each almost periodic, in the sense of distributions, divisor d in a tube T Ω ⊂ C m one can assign a cohomology class from H 2 (K, Z) (actually, the first Chern class of a special line bundle over K generated by d) such that the trivial cohomology class represents the divisors of a...

There are surprisingly rich properties of these holomorphic functions. The possibility of holomorphic continuation of holomorphic functions forces us to consider multi-valued holomorphic functions. The concept of Riemann Surfaces was introduced to understand such phenomena. The ideas of branch cuts and branch points immediately relate topology of these surfaces to complex variables. The possibi...