## Abstract

If u(z) is harmonic in ℝ^{2} 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 < r_{1} < r_{2} < 1 then a bound for osc(u, r_{2}) can be obtained in terms of [osc(u, r_{1})]^{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 ℝ^{n}. 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 language | English (US) |
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Pages (from-to) | 703-713 |

Number of pages | 11 |

Journal | Mathematische Zeitschrift |

Volume | 239 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2002 |

## ASJC Scopus subject areas

- General Mathematics