Mean-curvature comparison at tangential contact

Suppose two graph surfaces $z=f(x,y)$ and $z=g(x,y)$ are tangent at the origin, are oriented upward, and $g\ge f$ nearby. Then $g-f$ has a local minimum, so its Hessian matrix is positive semidefinite. At the common horizontal tangent plane,

The same contact does not order Gaussian curvature, because determinant is not monotone under addition of a positive-semidefinite matrix when the original Hessian is indefinite or negative definite. For example,satisfy $g\ge f$ and are tangent at zero, but $K_g(0)=1<4=K_f(0)$.