Extensions of the maximum principle for vector-valued analytic and harmonic functions
The maximum principle in the complex plane is well known: if \(f(z)\) is a complex valued analytic function on a domain \(G\) in the complex plane and if \(f(z)\) attains its maximum value on \(G\), then \(f(z)\) is a constant function. If \(f(z)\) has values in a complex Banach space, the maximum principle holds iff the Banach space is complex strictly convex. The author extends this theorem to show that a complex Banach space \(X\) is strictly convex if all non-constant analytic functions, defined on the open unit disc in the complex plane, with values in \(X\), have a certain mean growth. This result cannot be extended to the quasi-Banach space and it cannot be uniformized.
scientific article; zbMATH DE number 765143