Characterization of holomorphic functions in terms of their moduli

Abstract

It was shown in (Boche, H. and Pohl, V., 2005, Spectral factorization in the disk algebra. Complex Variables. Theory and Applications, 50, 383–387.) that if the modulus |f| of a function is continuous in the closure of the unit disk, the function f itself needs not to be continuous there, in general. This article shows that if the modulus of continuity of a function is a weak regular majorant, the continuity of the modulus always implies the continuity of the function itself.

Keywords:

• Disk algebra
• Lipschitz continuity
• Majorants
• AMS Subject Classifications: 30H05
• 46E35

Notes

H. Boche is also with the Fraunhofer Institute for Telecommunications, Heinrich-Hertz-Institute Berlin,Germany and with the Sino-German Mobile Communications Lab, Berlin, Germany.

Additional information

Notes on contributors

Volker Pohl

H. Boche is also with the Fraunhofer Institute for Telecommunications, Heinrich-Hertz-Institute Berlin,Germany and with the Sino-German Mobile Communications Lab, Berlin, Germany.