Spirallike nonanalytic functions

Abstract

Let f ( z ) = u ( x , y ) + i υ ( x , y ) f(z) = u(x,y) + i\upsilon (x,y) be a complex function defined in the unit disc U . f U.f is said to belong to the class C 1 ( U ) {C^1}(U) if the functions u ( x , y ) u(x,y) and υ ( x , y ) \upsilon (x,y) have continuous first order partial derivatives in U U . We determine sufficient conditions for functions in the class C 1 ( U ) {C^1}(U) to be univalent and to map U U onto spirallike domains. These conditions are similar to those in the analytic case as given by Spaček and Rakhmanov.