For the equation of constant mean curvature with prescribed constant contact angle boundary condition, using the unique continuation of analytic function, we get a minimum principle for a combination of the solution and its gradient. Thus we get the endpoint case for $P$-function (Sperb, 1981) and in fact answer an open question which appeared twenty years ago in Payne \& Philippin, 1977, 1979 and Sperb, 1981. As an application, sharp size and shape estimetes for capillary free surface without gravity are obtained.