Subharmonic functions that are harmonic when they are large

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.