Published online by Cambridge University Press: 24 October 2008
1. A theorem of Montel (14), states that if f(z) is analytic and bounded in the half-strip
and if there exists an x0 in (a, b) such that
as y → ∞, then
as y → ∞ l. u. on (a, b)(locally uniformly on (a, b), i.e. uniformly on every compact subset of (a, b)). Bohr (3), has proved a version of this result applicable to functions analytic, but not necessarily bounded, in S(a, b) and Hardy, Ingham and Pólya (10), have considered whether or not f(z) can be replaced by |f(z)| in (1·1) and (1·2). They show that this is not so but prove that if f(z) is analytic and bounded in S(a, b) and if
as y → ∞ for three distinct values x1, x2 and z3 in (a, b) then there exist constants A and B such that
as y → ∞, l.u. on (a, b). Cartwright (Theorem 5, (6)) has proved that if f(z) is analytic and satisfies |f(z)| < 1 in, S(a, b) and if for some x0 in (a, b)
asy → ∞, then
as y → ∞ co, l.u. on (a, b) and Bowen (5), has shown that if (1·4) is replaced by the weaker condition
as n → ∞ for some suitable sequence of points Zn in S(a, b) then (1·5) is still valid. In sections 2–5 of this paper we shall consider whether or not these results are valid |f(z)| is replaced by a subharmonic function.
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