Locally univalent functions with locally univalent derivatives

Abstract:

S. M. Shah and S. Y. Trimble have discovered that the behavior of an analytic function $f(z)$ is strongly influenced by the radii of univalence of its derivatives ${f^{(n)}}(z)\;(n = 0,1,2, \ldots )$. In this paper many of Shah and Trimble’s results are extended to large classes of locally univalent functions with locally univalent derivatives. The work depends on the concept of the ${\mathcal {U}_\beta }$-radius of a locally univalent function that is introduced and developed in this paper. Ch. Pommerenke’s definition of a linear invariant family of locally univalent functions and the techniques of that theory are employed in this paper. It is proved that the universal linear invariant families ${\mathcal {U}_\alpha }$ are rotationally invariant. For fixed $f(z)$ in ${\mathcal {U}_\alpha }$, it is shown that the function $r \to {\text {order}}\;[f(rz)/r]\;(0 < r \leqq 1)\;$ is a continuous increasing function of r.