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A Strong Version of Liouville's Theorem

descriptionPublicationkeyboard_double_arrow_right Article 01 Aug 2008 English Publisher:Informa UK LimitedJournal:The American Mathematical Monthly, volume 115, pages 583-595 (issn: 0002-9890, eissn: 1930-0972, Copyright policy )
Authors: Hansen, Wolfhard;

doi: 10.1080/00029890.2008.11920570

A Strong Version of Liouville's Theorem

Abstract

1. THE MAIN RESULT. Liouville's theorem states that every bounded holomor phic function on C is constant. Let us recall that holomorphic functions / on open subsets U of the complex plane have the mean value property, that is, for every closed disk B(z,r) in U, the value of / at its center z is equal to the average of the values of f on the circle S(z,r) = dB(z,r). In this paper, we shall present an elementary proof for the following stronger result (where, as usual, we do not distinguish between C and R2):

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This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
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popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
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