The oscillation of harmonic and quasiregular mappings

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) < infinity 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))](alpha) M(u, 1)(1-alpha) for a suitable constant alpha < 1, so that the logarithin of the oscillation has an approximate convexity property. The proof uses classical inequalities of Hadamard and Borel-Caratheodory 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.