A best constant and the Gaussian curvature

For axisymmetric <!-- MATH $f \in {C^\infty }({S^2})$ --> we find conditions to make the scalar curvature of a metric pointwise conformal to the standard metric of . Closely related to these results, we prove that in the inequality (Moser [8]) <!-- MATH \begin{displaymath} \int_{{S^2}} {{e^u} \leq C{e^{\left\| {\nabla u} \right\|_2^2/16\pi \quad }}\forall u \in H_1^2({S^2})} {\text{ with }}\int_{{S^2}} {u = 0} , \end{displaymath} -->