  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 product 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.
