Subharmonic functions that are harmonic when they are large (Q461369)

Let \(\Delta(a,r)\) be the open disc with radius \(r\) and centre \(a\). For a subharmonic function \(u\), let \(B(r,u)=\max_{|z|=r}u(z)\) and \(\tau=\tau(z)=\max\{|z|,B(|z|,u)\}\). 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 \(u(z)>B(|z|,u)-K_0\log\tau(z)\). It is shown that if in addition \(u\) satisfies the following lower growth condition: there are sequences \(0<\mathcal{R}_n<\mathcal{R}'_n\) satisfying \(\mathcal{R}_n\to\infty\) and \(\liminf_{n\to\infty}\mathcal{R}'_n/\mathcal{R}_n>1\) such that, with \(\mathcal{I}=\cup_{n=1}^\infty(\mathcal{R}_n,\mathcal{R}'_n)\), we have \[ \liminf\limits_{r\to\infty,r\in\mathcal{I}}\frac{B(r,u)}{(\log r)^3(\log\log r)^2(\log\log\log r)^{1+\epsilon_0}}>0\, \] for some \(0<\epsilon_0<1\), 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.