On Some Correspondence between Holomorphic Functions in the Unit Disc and Holomorphic Functions in the Left Halfplane

Ewa Ligocka¹

¹ Pasteura 4/6 m. 31 02-093 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 57 (2009) no. 3, pp. 223-229.

Résumé

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 $\lim_{\mathop{\rm Re} z\to-\infty}(g(w)-w)=0$.

Détail

DOI : 10.4064/ba57-3-4

Keywords: study correspondence between classes functions holomorphic unit disc functions holomorphic halfplane correspondence every mathbb exp exp particular prove famous class univalent functions unit disc homeomorphic via class mathbb univalent functions mathbb which lim mathop to infty w