The numerical range of holomorphic mappings arises in many aspects of nonlinear analysis, finite and infinite dimensional holomorphy, and complex dynamical systems. In particular, this notion plays a crucial role in establishing exponential and produ
ct formulas for semigroups of holomorphic mappings, the study of flow invariance and range conditions, geometric function theory in finite and infinite dimensional Banach spaces, and in the study of complete and semi-complete vector fields and their applications to starlike and spirallike mappings, and to Bloch (univalence) radii for locally biholomorphic mappings. In the present paper we establish lower and upper bounds for the numerical range of holomorphic mappings in Banach spaces. In addition, we study and discuss some geometric and quantitative analytic aspects of fixed point theory, nonlinear resolvents of holomorphic mappings, Bloch radii, as well as radii of starlikeness and spirallikeness.
We present a rigidity property of holomorphic generators on the open unit ball $mathbb{B}$ of a Hilbert space $H$. Namely, if $finHol (mathbb{B},H)$ is the generator of a one-parameter continuous semigroup ${F_t}_{tgeq 0}$ on $mathbb{B}$ such that fo
r some boundary point $tauin partialmathbb{B}$, the admissible limit $K$-$limlimits_{ztotau}frac{f(x)}{|x-tau|^{3}}=0$, then $f$ vanishes identically on $mathbb{B}$.
