Optimal Monotonicity of $L^p$ Integral of Conformal Invariant Green Function

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.