# Article

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Keywords:
\$(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
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.
References:
[1] J. Dieudonne: Grundzüge der modernen Analysis. II Auflage, VEB Deutscher Verlag der Wissenschaften, Berlin, 1972. MR 0474358 | Zbl 0264.26001
[2] W. Fulton J. Harris: Representation theory. Graduate Text Math., Spгinger, 1991. MR 1153249
[3] E. Janiec: Some uniqueness theorems concerning holomorphic mappings. Demonstratio Math. 23, 4 (1990), 879-892. MR 1124740 | Zbl 0755.32002
[4] R. Mortini: Lösung der Aufgabe 901. El. Math. 39 (1984), 130-131.
[5] J. Mujica: Complex analysis in Banach spaces. Noгth-Holland, Amsterdam, New York, Oxfoгd. Zbl 0586.46040
[6] A. Pfluger: Varianten des Schwarzschen Lemma. El. Math. 40 (1985), 46-47. MR 0803075 | Zbl 0566.30021
[7] W. Rudin: The fixed-point sets of some holomorphic maps. Bull. Malaysian Math. Soc. (2) 1 (1978), 25-28. MR 0506535 | Zbl 0413.32012
[8] W. Rudin: Real and complex analysis. (second edition). McGraw-Hill Inc, 1974. MR 0344043 | Zbl 0278.26001

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