A Revised Proof of Spirallikeness
In this paper, we present a proof for spirallikeness of injective holomorphic functions which is based on topological considerations. The proof exploits a continuous extending method applied to certain paths in the unit disk. Introduction Given d p/2;p/2) and z C {0} let (0.1) Ld(z) {exp(et)z t 0] i.e. Ld is a logarithmic spiral. Geometrically, it intersects each half line with endpoint at the origin at the constant angle d. We recall that a domain O in the complex plane C is said to be d spirallike (with respect to the origin) if 0 O and for every z O, Ld(z) O. The class of all d spirallike domains O will be denoted by Z* d Write D(a, r) for the euclidian disk with center at a C and radius r 0, i.e. D(a, r) {z C |z a| r} and let D(a, r) {z C |z a| r} We denote by S the class of all injective holomorphic (univalent) functions f in the unit disk D D(0, 1) keeping the origin fixed. In the paper [13] L. Spacek initiated research of the functions described as follows. Definition 0.1. Given d p/2;p/2) a function f D C is said to be a d spirallike holomorphic function if f S and f(D) Z* d The class of all d spirallike holomorphic functions will be denoted by S* d The fundamental result relevant to this class reads as follows. Theorem 0.2. For every d p/2;p/2) and every holomorphic function f in D with f (0) 0, f S* d iff the condition (0.2) Re e idzf (z) f(z) 0 z D f 1(0)