Calc. Var. 6, 95–104 (1998) c  Springer-Verlag 1998 An improved Moser-Aubin-Onofri inequality for axially symmetric functions on S 2 J. Feldman, R. Froese, N. Ghoussoub, C. Gui Department of Mathematics, The University of British Columbia, Vancouver, Canada Received October 17, 1996 / Accepted November 11, 1996 Summary. We show that for some α, 1 2 ≤ α< 16 25 , the following inequality holds: α 2  1 −1 (1 − x 2 )|g ′ (x )| 2 dx +  1 −1 g(x ) dx − ln 1 2  1 −1 e 2g(x ) dx ≥ 0, for every function g on (−1, 1) satisfying ‖g‖ 2 =  1 −1 (1 − x 2 )|g ′ (x )| 2 dx < ∞ and  1 −1 e 2g(x ) xdx = 0. This improves a result of Chang and Yang [C-Y2] in the axially symmetric case. 1. Introduction Let S 2 be the 2-dimensional sphere and let J α denote the functional on the Sobolev space H 1,2 (S 2 ) defined by J α (g)= α  S 2 |∇g| 2 d ω +2  S 2 g d ω − ln  S 2 e 2g d ω Here d ω denotes Lebesgue measure on the unit sphere, normalized to make  S 2 d ω = 1. The finiteness of the last term follows from the exceptional case of the Sobolev imbedding theorem ([A] Theorem 2.46). In [M], Moser showed that J 1 is bounded below by a —necessarily non- positive— constant C 1 . Later, Onofri [O] used the conformal structure on S 2 to show that C 1 can be taken to be 0. Other proofs were also given by Osgood- Phillips-Sarnak [O-P-S] and by Hong [H]. Prior to that, Aubin [A] had shown that by restricting J α to the class G of functions g in H 1,2 (S 2 ) for which e 2g has centre of mass equal to zero, (that is  S 2 e 2g xd ω = 0), then for any α> 1 2 , the functional J α is again bounded below