Analitične funkcije s konstantno normo - Globevnik, Josip (visokošolska dela, 2008)
A complex-valued analytic function ?$f$?, defined on a domain ?$D$? in the complex plane, satisfying ?$\|f(ζ)\|=1$? ?$(ζ \in D)$? is necessarily a constant. In general this is no longer true if the function has values in a complex Banach space. We study analytic functions from ?$D$? into a complex Banach space whose norm is constant on ?$D$?. Various necessary and sufficient conditions are obtained to assure that a nonconstant analytic function from ?$D$? to a complex Banach space has constant norm on ?$D$?. We study also the analytic functions ?$f$? from ?$D$? to a complex Banach space having the following property: there exist a complex-valued analytic function ?$φ$? on ?$D$? such that ?$\|f(z)\|=|φ(z)|$? ?$(z\inD)$?.
Keywords: analitične funkcije s konstantno normo; complex analysisu; kompleksna analiza; vektorske analitične funkcije