Bieberbach conjecture, Bohr radius, Bloch constant and Alexander's theorem in infinite dimensions

In this paper, we investigate holomorphic mappings $F$ on the unit ball $\mathbb{B}$ of a complex Banach space of the form $F(x)=f(x)x$, where $f$ is a holomorphic function on $\mathbb{B}$. First, we investigate criteria for univalence, starlikeness and quasi-convexity of type $B$ on $\mathbb{B}$. Next, we investigate a generalized Bieberbach conjecture, a covering theorem and a distortion theorem, the Fekete-Szeg\"{o} inequality, lower bound for the Bloch constant, and Alexander's type theorem for such mappings.