I am interested in the following Poincaré-type inequality, $$ \int_{S(r)} \lvert u-\bar{u}\rvert^2 d\sigma \leq C(N) \int_{S(r)} |u_{\theta}|^2 d\sigma$$ where $\bar{u} = \frac{1}{\lvert S(r)\rvert}\int_{S(r)} u d\sigma$ and $u_{\theta}$ denotes the tangential derivative of $u$. The domain $S(r)$ is just the $N$-dimensional sphere with radius $r$. The function $u$ can be assumed to be smooth for the purpose of this question.

In the case $S(r)$ is $1$ dimensional (i.e. a circle), then the constant $C(1)=1$ and this is the Wirtinger's inequality. Are there any references where I can find the best constant in higher dimensions?