prove $ ||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$

Question

$$ ||u||_{L_2(\Gamma)} \leq (2||v||^2_{L_2(\Omega)} + 2||v||_{L_2(\Omega)} ||\nabla v||_{L^2(\Omega)})^{\frac{1}{2}}$$ for all $u \in H^1(\Omega)$ for the open unit circle $\Omega$ in $\mathbb{R^2}$.

My work: So I already prove the following identity and I want to use it for this problem.

$$ v(1,\phi)^2 = \int_0^1 \frac{\delta}{\delta r}(r^2v(r,\phi)^2 dr $$

for functions $ v \in C^1(\bar{\Omega})$.