The growth theorem for holomorphic convex mappings

Abstract

In the case of one complex variable, the following growth theorem is well known. If f (z) = z + … is holomorphic and univalent in the unit disk $\Delta = \left\{ {z \in \mathbb{C}:\left| z \right| < 1} \right\}$, then

$$\frac{r}{{{{(1 + r)}^2}}} \leqslant \left| {f(z)} \right| \leqslant \frac{r}{{{{(1 - r)}^2}}},\left| z \right| = r < 1.$$