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On the Importance of the Harmonic Majorization for a Systematization of the Theory of Phragmén-Lindelöf

✍ Scribed by Rolf Scharm

Publisher
John Wiley and Sons
Year
1987
Tongue
English
Weight
697 KB
Volume
133
Category
Article
ISSN
0025-584X

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✦ Synopsis

We consider the function f ( z ) analytic in the half plane Q+ = { z : Im z -0) and continuous in its closure. Theorems of the PHEAGMJ~N-LINDELOF type are characterized by assuming growth restrictions for If(z)l in the closure of @+, which are sufficient for i t to be bounded or at most of exponential type in the whole half plane. The variety of the methods and results of the h a a -MEX-LINDELOF theory suggest a n effort to systematize this theory. I n the present paper the wellknown mnjorizntion formulas for log If(z)l given by F. and R. XEVANLINNA in [lo] will be fundamentiil. We apply the principle of harmonic majorizntion in sgeneralized form and shorn that it is more suitable than other approaches to derive a considerable number of knOWnPHaAGMEN-LINDE-LOF theorems and of their generalizations. The proofs become more straightforward nnd superfluous assumptions needed before cim be omibted.

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