I suppose $f$ is a real function (otherwise this is wrong). For real functions we actually
have equality in the UNIT disc. Indeed, by Green's formula,
$$\int_\Omega|\nabla u|^2dxdy=\int_0^{2\pi}uu_rd\theta.$$
Expand $f$ into Fourier series, then
$$u(r,\theta)=\sum_{-\infty}^\infty r^{|n|}c_ne^{in\theta}.$$
Now using the orthogonality of the exponentials, and the fact that $f$ is real, so $c_n=\overline{c_{-n}}$, we compute
$$\int_0^{2\pi} u_\theta^2d\theta=2\pi\sum_{-\infty}^\infty n^2|c_n|^2,$$
and 
$$\int_0^{2\pi} uu_rd\theta=2\pi\sum_{-\infty}^\infty n^2|c_n|^2.$$
So the things are equal.