TY - JOUR
T1 - The oscillation of harmonic and quasiregular mappings
AU - Anderson, J. M.
AU - Hinkkanen, A.
PY - 2002/4
Y1 - 2002/4
N2 - 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.
AB - 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.
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U2 - 10.1007/s002090100326
DO - 10.1007/s002090100326
M3 - Article
AN - SCOPUS:0036557491
SN - 0025-5874
VL - 239
SP - 703
EP - 713
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
IS - 4
ER -