A lower bound for the logarithmic energy on $\mathbb{S}^2$ and for the Green energy on $\mathbb{S}^n$

We show an alternative proof of the sharpest known lower bound for the logarithmic energy on the unit sphere $\mathbb{S}^2$. We then generalize this proof to get new lower bounds for the Green energy on the unit $n$-sphere $\mathbb{S}^n$.