  A classical result of Aubin states that the constant in
Moser-Trudinger-Onofri inequality on $\mathbb{S}^{2}$ can be imporved for
furnctions with zero first order moments of the area element. We generalize it
to higher order moments case. These new inequalities bear similarity to a
sequence of Lebedev-Milin type inequalities on $\mathbb{S}^{1}$ coming from the
work of Grenander-Szego on Toeplitz determinants (as pointed out by Widom). We
also discuss the related sharp inequality by a perturbation method.
