The oscillation of harmonic and quasiregular mappings

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.