Find all harmonic functions $u$ in a half-plane $H$ so that $u=0$ on $\partial H$ and $\vert u(x)\vert \le \vert x\vert$ in $H$.
This domain is not bounded. If it's bounded, then after using the maximum principe and the minimum principle, we conclude $u\equiv 0$ and the condition that $\vert u(x)\vert \le \vert x\vert$ will hold automatically.
I guess we should use the maximum principle for the unbounded case, but I don't remember the description of it and don't know how to process.
Any help will be appreciated.