Need help about harmonic functions!

Question

I have trouble on solving the following problem:

Show that there doesn't exist a non-constant function $u$ such that $u$ is harmonic on C and for $z=x+iy$ in C that $u(z)>4x^2+9y^2+1$.

Answer 1

If $u$ is harmonic, then $u(0) = \frac1{2\pi}\int_0^{2\pi} u(r e^{i\theta}) \, d\theta$ for any $r > 0$. Now think about what happens as $r \to \infty$: $$ u(0) \ge \frac1{2\pi}\int_0^{2\pi} 4r^2\cos^2\theta + 9r^2\sin^2\theta \, d\theta \ge \frac1{2\pi}\int_0^{2\pi} 4r^2\cos^2\theta + 4r^2\sin^2\theta \, d\theta = 4r^2 .$$ This is true for all $r>0$, hence $u(0)$ must be infinite.