We will discuss a sequence of Moser-Trudinger inequalities on the 2-sphere generalizing Aubin's inequality for functions with vanishing first order moments of the area elements to higher order cases. An interesting phenomenon is the appearance of minimal number of nodes for numerical integration on the 2-sphere (which is still not known). Our approach also gives a new explanation and proof of the sequence of Lebedev-Milin type inequalities on the unit circle, which previously were derived from Szego limit theorem (as observed by Widom). These inequalities are partly motivated from the study of isospectral problem on surfaces by Osgood-Phillips-Sarnak in late 80’s.