# Maximum curves and isolated points of entire functions

Tyler, T.F.
(2000)
*Maximum curves and isolated points of entire functions.*
Proceedings of the American Mathematical Society, 128 (9).
pp. 2561-2568.
ISSN 0002-9939 (Print), 1088-6826 (Online)
(doi:10.1090/S0002-9939-00-05315-6)

## Abstract

Given M(r; f) =maxjzj=r (jf(z)j) , curves belonging to the set of points M = fz : jf(z)j = M(jzj; f)g were de�ned by Hardy to be maximum curves. Clunie asked the question as to whether the set M 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 f1(z) and f2(z), if the maximum curve of f1(z) is the real axis, conditions are found so that the real axis is also a maximum curve for the product function f1(z)f2(z). By means of these results an entire function of in�nite order is constructed for which the set M has an in�nite number of isolated points. A polynomial is also constructed with an isolated point.

Item Type: | Article |
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Uncontrolled Keywords: | maximum curves, isolated points, analytic functions |

Subjects: | Q Science > QA Mathematics |

Pre-2014 Departments: | School of Computing & Mathematical Sciences |

Related URLs: | |

Last Modified: | 14 Oct 2016 09:00 |

URI: | http://gala.gre.ac.uk/id/eprint/425 |

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