If \(B\) is the unit ball in \(\mathbb{C}^n\) equipped with some norm, let us define the following sets: \(\mathbb{S}= \{f:B\to \mathbb{C}^n:f\) is a biholomorphic mapping of \(B\) onto \(f(B)\), \(f(0)=0\), \(Df(0)= \text{Id}\}\), \(\mathbb{S}^* =\{f\in \mathbb{S} :f(B)\) is starlike with respect to \(0\}\), \(K=\{f\in \mathbb{S}:f(B)\) is convex}. When \(n=1\), \(f\in\mathbb{S}^*\) and \(f\in K\) are known to be equivalent to the positivity of the real part of \({zf'(z)\over f(z)}\) and \({zf''(z)\over f'(z)} +1\) respectively. The aim of this paper is to generalize these characterizations to higher dimensions.
After studying many examples showing that there is no obvious extension of these equivalences the authors define two classes of functions \(\mathbb{G}\) and \(\mathbb{F}\) characterized by similar positivity conditions and prove the inclusions: \(K\subset\mathbb{G}\subset\mathbb{S}^*\) and \(\mathbb{G} \subseteq \mathbb{F}\). In fact the authors suspect that these two classes are the same.