  Both analytic and geometric forms of an optimal monotone principle for
$L^p$-integral of the Green function of a simply-connected planar domain
$\Omega$ with rectifiable simple curve as boundary are established through a
sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the
Huber analytic and geometric isoperimetric inequalities under finiteness of the
positive part of total Gauss curvature of a conformal metric on $\Omega$.
Consequently, new analytic and geometric isoperimetric-type inequalities are
discovered. Furthermore, when applying the geometric principle to
two-dimensional Riemannian manifolds, we find fortunately that $\{0,1\}$-form
of the induced principle is midway between Moser-Trudinger's inequality and
Nash-Sobolev's inequality on complete noncompact boundary-free surfaces, and
yet equivalent to Nash-Sobolev's/Faber-Krahn's
eigenvalue/Heat-kernel-upper-bound/Log-Sobolev's inequality on the surfaces
with finite total Gauss curvature and quadratic area growth.
