In mathematics, the Earle–Hamilton fixed point theorem is a result in geometric function theory giving sufficient conditions for a holomorphic mapping of an open domain in a complex Banach space into itself to have a fixed point.
The result was proved in 1968 by Clifford Earle and Richard S. Hamilton by showing that, with respect to the Carathéodory metric on the domain, the holomorphic mapping becomes a contraction mapping to which the Banach fixed-point theorem can be applied.
If the diameter of D is less than R then, by taking suitable holomorphic functions g of the form with a in X* and b in C, it follows that and hence that In particular d defines a metric on D. The chain rule implies that and hence f satisfies the following generalization of the Schwarz-Pick inequality: For δ sufficiently small and y fixed in D, the same inequality can be applied to the holomorphic mapping and yields the improved estimate: The Banach fixed-point theorem can be applied to the restriction of f to the closure of f(D) on which d defines a complete metric, defining the same topology as the norm.
In the case of bounded symmetric domains with the Bergman metric, Neretin (1996) and Clerc (1998) showed that the same scheme of proof as that used in the Earle-Hamilton theorem applies.
The open semigroup of the complexification Gc taking the closure of D into D acts by contraction mappings, so again the Banach fixed-point theorem can be applied.