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We continue our previous work on a problem of Janiec connected with a uniqueness theorem, of Cartan-Gutzmer type, for holomorphic mappings in ℂⁿ. To solve this problem we apply properties of (j;k)-symmetrical functions.

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 ($k=2,3,\cdots$; $j=0,1,\cdots ,k-1$) 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.

We continue E. Ligocka's investigations concerning the existence of m-valent locally biholomorphic mappings from multi-connected onto simply connected domains. We decrease the constant m, and also give the minimum of m in the case of mappings from a wide class of domains onto the complex plane ℂ.

Let $𝒜$ denote the set of functions $F$ holomorphic in the unit disc, normalized clasically: $F\left(0\right)=0,F^{\text{'}}\left(0\right)=1$, whereas $A\subset 𝒜$ is an arbitrarily fixed subset. In this paper various properties of the classes $A_{\alpha },\alpha \in C\{-1,-\frac{1}{2},...\}$, of functions of the form $f=F*k_{\alpha }$ are studied, where $F\in .A$, $k_{\alpha }\left(z\right)=k(z,\alpha )=z+\frac{1}{1+\alpha }{z}^2+...+\frac{1}{1+(n-1)\alpha }{z}^n+...$, 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]).