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.

This is correct (if $f$ is real, of course). Use the Green formula to transform the LHS: $$\int_\Omega|\nabla u|^2dxdy=R\int_0^{2\pi}uu_rd\theta,$$ where $R$ is the radius of the disc. Now expand $$u(r,\theta)=\sum_{-\infty}^\infty r^{|n|}c_ne^{in\theta},$$ where $c_n=\overline{c_{-n}}$, because $f$ is real. Now compute, using orthogonality: $$R\int_0^{2\pi}uu_rd\theta=2\pi\sum_{-\infty}^\infty|n|R^{2n}|c_n|^2,$$ and $$\int_0^{2\pi}u_\theta^2d\theta=2\pi\sum_{-\infty}^\infty n^2R^{2n}|c_n|^2.$$ We conclude that LHS$\leq$RHS, with equality only if $c_n=0$ for $|n|\geq 2$.