On Strong Asymptotic Tracts of Functions Holomorphic in a Disk

Abstract

For entire functions of finite order the Ahlfors classical theorem about finiteness of the set of asymptotic values is well known. In 1999, the first author introduced the concept of the strong asymptotic value of entire functions and obtained an analogue of the Ahlfors theorem for distinct strong asymptotic spots of entire functions of infinite order [14].In 2001 we introduced the concept of a strong asymptotic spot in a point z0 for functions holomorphic in the disk. For such asymptotic spots we obtained an analogue of the Ahlfors theorem for functions holomorphic in the disk.In the case of holomorphic functions of order ρ<1 MacLane proved that the set of distinct asymptotic tracts corresponding to the point z0 is finite [12]. He also obtained an estimate for their quantity. We introduce the concept of a strong asymptotic tracts for functions holomorphic in a disk and obtain an analogue of the Ahlfors theorem for their quantity for functions holomorphic in a disk.Mathematics Subject Classification 200030D3030D35Key words and phrasesasymptotic valuesstrong asymptotic tracts