Constrained Moser-Trudinger-Onofri inequality and a uniqueness criterion for the mean field equation

We establish Moser-Trudinger-Onofri inequalities under constraint of a deviation of the second order moments from $0$, which serves as an intermediate one between Chang-Hang’s inequalities under first and second order moments constraints. A threshold for the deviation is a uniqueness criterion for the mean field equation $$-a{\mathrm{\Delta }}_{{\mathbb{S}}^2}u+1=e^{2u}\mathrm{o}\mathrm{n}{\mathbb{S}}^2$$ when the constant $a$ is close to $\frac{1}{2}$.