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

Abstract

We consider functions f1, f2 analytic in the upper half plane and continuous in its closure and investigate the following problem: Suppose that the product f1f2 is bounded in the half plane and both factors are bounded on the real axis; which assumptions on the growth of f1 are sufficient for f2 to be bounded or to have a special growth? We derive a general representation formula for such a factor f1 containing two important special cases. One of them is used to prove our main result: If the factor f1 is at most of finite order then both f1 and f2 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 PHRAGMEN-LINDELOF type which is of interest by itself; namely the growth restriction is imposed only on the real part of the function under consideration.