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$(j,k)$-symmetrical functions; holomorphic function; integral formulas; uniqueness theorem; mean value of a function; a variant of Schwarz lemma; fixed point; spectrum of an operator
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.
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