Some geometric relations for equipotential curves

Let $U(\boldsymbol r),\boldsymbol r\in\Omega\subset \mathbb R^2$ be a harmonic function that solves an exterior Dirichlet problem. If all the level sets of $U(\boldsymbol r),\boldsymbol r\in\Omega$ are smooth Jordan curves, then there are several geometric inequalities that correlate the curvature $\kappa(\boldsymbol r)$ with the magnitude of gradient $|\nabla U(\boldsymbol r)|$ on each level set ("equipotential curve"). One of such inequalities is $\langle [\kappa(\boldsymbol r)-\langle\kappa(\boldsymbol r)\rangle][|\nabla U(\boldsymbol r)|-\langle |\nabla U(\boldsymbol r)|\rangle]\rangle\geq0$, where $\langle \cdot\rangle$ denotes average over a level set, weighted by the arc length of the Jordan curve. We prove such a geometric inequality by constructing an entropy for each level set $U(\boldsymbol r)=\varphi$, and showing that such an entropy is convex in $\varphi$. The geometric inequality for $\kappa(\boldsymbol r)$ and $|\nabla U(\boldsymbol r)|$ then follows from the convexity and monotonicity of our entropy formula. A few other geometric relations for equipotential curves are also built on a convexity argument.