On (j,k)(j,k)-symmetrical functions

summary:n the present paper the authors study some families of functions from a complex linear space $X$ into a complex linear space $Y$. They introduce the notion of $(j,k)$-symmetrical function ($k=2,3,\dots$; $j=0,1,\dots,k-1$) which is a generalization of the notions of even, odd and $k$-symmetrical functions. They generalize the well know result that each function defined on a symmetrical subset $U$ of $X$ can be uniquely represented as the sum of an even function and an odd function

On strongly starlikeness of order α in several complex variables

In this paper we introduce the concept of strongly starlikeness of order α > 0, for holomorphic mappings defined on the unit ball of Cn. We obtain the distorsion and the covering theorems for strongly starlike mappings of order α ∈ (0,1] and we give a connection between strongly starlikeness and spirallikeness in Cn

Starlike mappings of order alpha on the unit ball in complex Banach spaces

In this paper, we will give the growth theorem of starlike mappings of order α on the unit ball B in complex Banach spaces. We also give an analytic sufficient condition for a locally biholomorphic mapping on B to be a starlike mapping of order α

Applications of the Hadamard product in geometric function theory

summary:Let \Cal A denote the set of functions $F$ holomorphic in the unit disc, normalized clasically: $F(0)=0, F'(0)=1$, whereas A\subset \Cal A is an arbitrarily fixed subset. In this paper various properties of the classes $A_\alpha, \alpha \in C \{-1,-\frac{1}{2},\ldots\}$, of functions of the form $f=F*k_\alpha$ are studied, where $F\in .A$, $k_\alpha(z)=k(z,\alpha)=z+\frac{1}{1+\alpha}z^2+\ldots + \frac{1}{1+(n-1)\alpha}z^n+\ldots$, and $F*k_\alpha$ denotes the Hadamard product of the functions $F$ and $k_\alpha$. Some special cases of the set $A$ were considered by other authors (see, for example, [15],[6],[3])

Some Results of Fekete-Szegö Type. Results for Some Holomorphic Functions of Several Complex Variables

This paper is devoted to a generalization of the well-known Fekete-Szegö type coefficients problem for holomorphic functions of a complex variable onto holomorphic functions of several variables. The considerations concern three families of such functions f, which are bounded, having positive real part and which Temljakov transform Lf has positive real part, respectively. The main result arise some sharp estimates of the Minkowski balance of a combination of 2-homogeneous and the square of 1-homogeneous polynomials occurred in power series expansion of functions from aforementioned families

Distortion theorems for biholomorphic convex mappings in Cn

AbstractIn the paper the problem of sharp lower estimation for ‖Df(z)‖ in the class of normalized biholomorphic mappings f between the open unit ball Bn and convex domains in Cn has been considered

Starlikeness with respect to a boundary point and Julia's theorem in Cn

AbstractIn the paper necessary and sufficient conditions for the boundary starlikeness of holomorphic mappings in Cn are given. In the proof an n-dimensional version of Julia's theorem is used