After a short survey of the history of infinite-dimensional holomorphy, including the basic notions and facts, the author proves two fixed point theorems where the first theorem may be viewed as a holomorphic version of Banach's fixed point theorem [cf. \textit{C. J. Earle} and \textit{R. S. Hamilton}, Global Analysis, Proc. Symp. Pure Math. 16, 61--65 (1970; Zbl 0205.14702)], and the second theorem [cf. \textit{V. Khatskevich}, \textit{S. Reich} and \textit{D. Shoikhet}, Integral Equations Oper. Theory 22, No. 3, 305--316 (1995; Zbl 0837.46033)] turns out to be a consequence of the first. In [\textit{L. A. Harris}, Am. J. Math. 93, 1005--1019 (1971; Zbl 0237.58010)], the author defined the numerical range of a holomorphic mapping which maps a ball of the Banach space \(X\) centered at its origin into \(X\). This notion generalizes the numerical range for bounded linear operators. Using an extended notion to more general domains, another fixed point theorem of holomorphic mappings is proved. The author shows a theorem concerning Bloch radii, this is a quantitative version of the inverse function theorem and a distortion form of Cartan's uniqueness theorem. These applications can be also found in [\textit{L. A. Harris}, \textit{S. Reich} and \textit{D. Shoikhet}, J. Anal. Math. 82, 221--232 (2000; Zbl 0972.46029)].