Maximum curves and isolated points of entire functions

Author:
T. F. Tyler

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2561-2568

MSC (2000):
Primary 30C80

Published electronically:
February 29, 2000

MathSciNet review:
1662226

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Abstract | References | Similar Articles | Additional Information

Given , curves belonging to the set of points were defined by Hardy to be maximum curves. Clunie asked the question as to whether the set could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions and , if the maximum curve of is the real axis, conditions are found so that the real axis is also a maximum curve for the product function . By means of these results an entire function of infinite order is constructed for which the set has an infinite number of isolated points. A polynomial is also constructed with an isolated point.

**1.**J. M. Anderson, K. F. Barth, and D. A. Brannan,*Research problems in complex analysis*, Bull. London Math. Soc.**9**(1977), no. 2, 129–162. MR**0440018****2.**G.H. Hardy. ``The Maximum Modulus of an Integral Function'' . Quart. J.Math. (41) (1909),**3.**W. K. Hayman,*A characterization of the maximum modulus of functions regular at the origin*, J. Analyse Math.**1**(1951), 135–154 (English, with Hebrew summary). MR**0045209****4.**E.C. Titchmarsh. The Theory of Functions, 2nd ed, Oxford University Press, London, 1939 (reprinted 1975).

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Additional Information

**T. F. Tyler**

Affiliation:
University of Greenwich, 30, Park Row, Greenwich, London SE10 9LS, United Kingdom

Email:
T.F.Tyler@gre.ac.uk

DOI:
http://dx.doi.org/10.1090/S0002-9939-00-05315-6

Keywords:
Maximum curves,
isolated points,
analytic functions

Received by editor(s):
June 15, 1998

Received by editor(s) in revised form:
October 5, 1998

Published electronically:
February 29, 2000

Communicated by:
Albert Baernstein II

Article copyright:
© Copyright 2000
American Mathematical Society