  There are two parts of this paper. First, we discovered an explicit formula
for the complex Hessian of the weighted log-Bergman kernel on a parallelogram
domain, and utilised this formula to give a new proof about the strict
convexity of the Mabuchi functional along a smooth geodesic. Second, when a
C^{1,1}-geodesic connects two non-degenerate energy minimizers, we also proved
this strict convexity, by showing that such a geodesic must be non-degenerate
and smooth.
