A Sharp Bound for the Growth of Minimal Graphs

We consider minimal graphs $$u = u(x,y) > 0$$ over unbounded domains $$D \subset R^2$$ bounded by a Jordan arc $$\gamma $$ on which . prove sort of reverse Phragmén-Lindelöf theorem showing that if D contains sector $$\begin{aligned} S_{\lambda }=\{(r,\theta )=\{-\lambda /2<\theta<\lambda /2\},\quad \pi <\lambda \le 2\pi , \end{aligned}$$ then the rate growth is at most $$r^{\pi /\lambda }$$