I will present  a variant of Aubin-Onofri inequality on the unit sphere,  which is  the optimal  Moser-Trudinger inequality on the sphere.  The inequality is a two dimensional version of the second Szego limit theorem regarding the Toeplitz determinants, which is useful in the study of  isospectral compactness for metrics defined on compact surfaces, among other applications. 

This is a joint work with Alice Chang from Princeton University.