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On some correspondence between holomorphic functions in the unit disc and holomorphic functions in the left halfplane

Ligocka E.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

Vol. 57, no 3-4

223-229

We study a correspondence L between some classes of functions holomorphic in the unit disc and functions holomorphic in the left halfplane. This correspondence is such that for every ƒ and w ∈ H, exp(L(ƒ)(w)) = ƒ(exp w). In particular, we prove that the famous class S of univalent functions on the unit disc is homeomorphic via L to the class S(H) of all univalent functions g on H for which = g(w) + 2 πi) = g(w) + 2 πi and limRe z →-- ∞(g(w) - w)=0.