Maximum curves and isolated points of entire functions
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- by T. F. Tyler PDF
- Proc. Amer. Math. Soc. 128 (2000), 2561-2568 Request permission
Abstract:
Given $M(r,f)=\max _{|{z}|=r}\left ( |{f(z)}|\right )$ , curves belonging to the set of points $\mathcal {M}=\left \{ z:|{f(z)}|=M(|{z}|,f)\right \}$ were defined by Hardy to be maximum curves. Clunie asked the question as to whether the set $\mathcal { M}$ 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 $f_1(z)$ and $f_2(z)$, if the maximum curve of $f_1(z)$ is the real axis, conditions are found so that the real axis is also a maximum curve for the product function $f_1(z)f_2(z)$ . By means of these results an entire function of infinite order is constructed for which the set $\mathcal {M}$ has an infinite number of isolated points. A polynomial is also constructed with an isolated point.References
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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
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2561-2568
- MSC (2000): Primary 30C80
- DOI: https://doi.org/10.1090/S0002-9939-00-05315-6
- MathSciNet review: 1662226