  In this article we present various uniqueness and existence results for
Q-curvature type equations with a Paneitz operator on $\s^n$ in axially
symmetric function spaces. In particular, we show uniqueness results for $n=6,
8$ and improve the best constant of Beckner's inequality in these dimensions
for axially symmetric functions under the constraint that their centers of mass
are at the origin. As a consequence, the associated first Szeg\"o limit theorem
is also proven for axially symmetric functions.
