In this talk, I will present a recent work confirming the best constant of a Moser-Trudinger type inequality conjectured by A. Chang and P. Yang. The proof is based on a new and powerful lower bound of total mass for mean field equations. Other applications of the lower bound include the classification of certain Onsager vortices on the sphere, the radially symmetry of solutions to Gaussian curvature equation on the plane, classification of solutions for mean field equations on tori and the sphere, etc. The resolution of several interesting problems in these areas will be presented. The work is jointly done with Amir Moradifam from UC Riverside.