Spirallike nonanalytic functions

Abstract:

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