A reverse cos\(\pi \rho \) theorem and a question of Fryntov (Q428949)

The authors prove the following theorem. Suppose that \(u\) is subharmonic in the plane and that \(G\) is an open set in which \(m \{ \theta: r e^{i \theta} \in G \} \geq 2 (\pi - \alpha)\) for all large \(r\) and for some number \(\alpha\) satisfying \(0 < \alpha < \pi\). Given any \(\lambda \) satisfying \(0 < \alpha \lambda \leq \pi/2\), either \(A(r, u) > \cos (\alpha \lambda) B(r, u)\) for certain arbitrary large values of \(r\) or \(\lim_{r \to \infty} \frac{B(r, u)}{r^\lambda}\) exists and is either positive or \(+ \infty\). Here \(A(r, u) =\inf_\theta u(r e^{i \theta}), B(r, u) = \sup_\theta u(r e^{i \theta})\). The authors further discuss a connection between this theorem and conjectures of A. Eremenko, A. Fryntov and T. Kövari.