For axisymmetric
f
∈
C
∞
(
S
2
)
f \in {C^\infty }({S^2})
we find conditions to make
f
f
the scalar curvature of a metric pointwise conformal to the standard metric of
S
2
{S^2}
. Closely related to these results, we prove that in the inequality (Moser [8])
\[
∫
S
2
e
u
≤
C
e
‖
∇
u
‖
2
2
/
16
π
∀
u
∈
H
1
2
(
S
2
)
with
∫
S
2
u
=
0
,
\int _{{S^2}} {{e^u} \leq C{e^{\left \| {\nabla u} \right \|_2^2/16\pi \quad }}\forall u \in H_1^2({S^2})} {\text { with }}\int _{{S^2}} {u = 0} ,
\]
, the best constant
C
=
Vol(
S
2
)
C = {\text {Vol(}}{{\text {S}}^2}{\text {)}}
.