On Some Correspondence between Holomorphic Functions in the Unit Disc and Holomorphic Functions in the Left Halfplane
Volume 57 / 2009
Bulletin Polish Acad. Sci. Math. 57 (2009), 223-229
MSC: Primary 30C45.
DOI: 10.4064/ba57-3-4
Abstract
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 $f$ and $w\in\mathbb H$, $\exp(L(f)(w))=f(\exp w)$. In particular, we prove that the famous class $S$ of univalent functions on the unit disc is homeomorphic via $L$ to the class $S({\mathbb H})$ of all univalent functions $g$ on $\mathbb H$ for which $g(w+2\pi i)=g(w)+2\pi i$ and $\lim_{\mathop{\rm Re} z\to-\infty}(g(w)-w)=0$.
