Subharmonic functions that are harmonic when they are large

Abstract

Suppose that $u$ is subharmonic in the plane and such that, for some $c>1$ and sufficiently large $K_0=K_0(c)$, $u$ is harmonic in the disc $\Delta (z,\tau (z)^{-c})$ whenever $u(z)>B(|z|,u)-K_0\log \tau (z)$, where $\tau (z)=\max \{|z|,B(|z|,u)\}$ and $B(r,u)=\max _{|z|=r}u(z)$. It is shown that if in addition $u$ satisfies a certain lower growth condition, then there are ‘Wiman–Valiron discs’ in each of which $u$ is the logarithm of the modulus of an analytic function, and that the derivatives of the analytic functions have regular asymptotic growth.