The extremal case is given by the Koebe function or one of its rotations. Home Questions Tags Users Unanswered. It seems like a klebe odd condition, unless you are assuming your functions to be real on the real axis. The removed set is shown below in blue:.
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Hermann Koebe owned a factory and was able to give his son a good education. Koebe attended a realgymnasium in Berlin. He entered this school in and there he studied religion, Latin and modern languages, history and geography, and mathematics and science.
The course, which was based more on practical applications than that of the more academic gymnasium, still qualified Koebe to enter university.
He studied first at Kiel University which he entered in but after one semester he moved to Berlin University where he was to study for five years. At Berlin his thesis was directed by Hermann Schwarz and his additional examiner for the oral on his thesis was Friedrich Schottky who had been appointed to Berlin in while Koebe was in the middle of his studies. Koebe was appointed to Leipzig University in as an extraordinary professor of mathematics.
He became an ordinary professor in when he accepted a position at Jena university. He returned to Leipzig, this time as an ordinary professor, in The article [ 5 ] describes his contributions in some detail and gives a list of 68 publications by Koebe. These are not, however, a collection of great works on a par with his proof of the uniformisation theorem. He did make other important contributions, however, and his circle domain conjecture is still being attacked. Freudenthal writes in [ 1 ]:- He tended to deal broadly with special cases of a general theory by a variety of methods It is unclear from what Freudenthal writes whether he is implying that Koebe required a wife to help organise his life but certainly he had no wife, remaining a bachelor all his life.
Certain Conditions for Starlikeness of Analytic Functions of Koebe Type
Koebe Domains for the Classes of Functions with Ranges Included in Given Sets