The oscillation of harmonic and quasiregular mappings

J. M. Anderson, Aimo H J Hinkkanen

Research output: Contribution to journalArticlepeer-review

Abstract

If u(z) is harmonic in ℝ2 with u(0) = 0 and r > 0 we set M(u,r) = sup {u(z) : |z| < r}, osc(u,r) = sup {u(z): |z| < r} - inf {u(z): |z| < r}. A result is obtained which shows, in particular that if M(u, 1) < ∞ and 0 < r1 < r2 < 1 then a bound for osc(u, r2) can be obtained in terms of [osc(u, r1)]α M(u, 1) 1-α for a suitable constant α < 1, so that the logarithm of the oscillation has an approximate convexity property. The proof uses classical inequalities of Hadamard and Borel-Carathéodory and this suggests a generalization to quasiregular mappings in ℝn. Such results are obtained, though necessarily in a less precise form because of the lack of good explicit estimates for A-harmonic measures in spherical ring domains.

Original languageEnglish (US)
Pages (from-to)703-713
Number of pages11
JournalMathematische Zeitschrift
Volume239
Issue number4
DOIs
StatePublished - Apr 1 2002

ASJC Scopus subject areas

  • Mathematics(all)

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