Extensions of the Maximum Principle for Vector-Valued Analytic and Harmonic Functions

A complex Banach space X is complex strictly convex if and only if X-valued analytic functions on the open unit disc in the complex plane satisfy a certain mean growth condition. This extends the Thorp-Whitley maximum modulus theorem. A corresponding characterization of strict convexity in real Banach spaces in terms of a mean growth condition of vector-valued harmonic functions is also obtained