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On the Factorization of Functions Bounded and Analytic in a Half Plane

✍ Scribed by Rolf Scharm

Publisher: John Wiley and Sons
Year: 1988
Tongue: English
Weight: 453 KB
Volume: 138
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.19881380114

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✦ Synopsis

We consider functions f l , fa analytic in the upper half plane and continuous in its closure and investigate the following problem:

Suppose that the product flfa is bounded in the half plane and both factors are bounded on the real axis; which assumptions on the growth of fl are sufficient for fz to be bounded or to have a special growth?

We derive a general representation formula for such a factor fl containing two important special cases. One of them is used to prove our main result: If the factor f l is a t most of finite order then both f l and fa are either of exponential type or of the same intermediate type of an integral order greater than one.

Furthermore, we modify a factorization theorem of I. V. OSTROVSKII [5] for factors which are assumed to be bounded in a strip contained in the half plane. As an essential tool we use a new lemma of PHRAOMEN-LINDELOF type which is of interest by itself; namely the growth restriction is imposed only on the real part of the function under consideration.

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