Premium
On the Importance of the Harmonic Majorization for a Systematization of the Theory of Phragmén‐Lindelöf
Author(s) -
Scharm Rolf
Publication year - 1987
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19871330122
Subject(s) - mathematics , majorization , closure (psychology) , bounded function , mathematical proof , pure mathematics , plane (geometry) , type (biology) , exponential function , function (biology) , variety (cybernetics) , harmonic , discrete mathematics , combinatorics , calculus (dental) , mathematical analysis , law , geometry , medicine , ecology , statistics , physics , dentistry , quantum mechanics , evolutionary biology , political science , biology
We consider the function f ( z ) analytic in the half plane ℌ + ={ z : Im z >0} and continuous in its closure. Theorems of the P HRAGMén ‐L INDELÖF type are characterized by assuming growth restrictions for | f ( z )| in the closure of ℌ + , which are sufficient for it to be bounded or at most of exponential type in the whole half plane. The variety of the methods and results of the P HRAGMÉN ‐L INDELÖF theory suggest an effort to systematize this theory. In the present paper the wellknown majorization formulas for log | f ( z )| given by F. and R. N EVANLINNA in [10] will be fundamental. We apply the principle of harmonic majorization in a generalized form and show that it is more suitable than other approaches to derive a considerable number of known P HRAGMEN ‐L INDELÖF theorems and of their generalizations. The proofs become more straightforward and superfluous assumptions needed before can be omitted.