We show that axially symmetric solutions on S to a constant Q-curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter α in front of the Paneitz operator belongs to the interval This is in contrast to the case α D 1, where there exists a family of solutions, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on S. As a consequence, we prove an improved Beckner’s inequality on S for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when α D 1=5 by exploiting Pohozaev-type identities, and prove the existence of a non-constant axially symmetric solution for α 2 .1=5; 1=2/ via a bifurcation method. (Fomula Presented)