A generalization of the isoperimetric inequality in the hyperbolic plane (Q1204071)

Let a closed curve in the hyperbolic plane \(H\) be given by a smooth immersion \(f\) of a circle \(C\) into \(H\). Let \(L\) equal the length of \(f(C)\), then \[ L^ 2 \geq 4\pi \int_ H w^ 2_ f(p) dH_ p + \int_{H\times H} m_ f(p,q) dH_ p \wedge dH_ q \] where \(w_ f(p)\) denotes the winding number w.r.t. \(p \in H\setminus f(C)\) and \(m_ f(p,q) = w_ f(p)w_ f(q)\), \(p,q \in H\setminus f(C)\). Equality holds if and only if \(f(c)\) is a geodesic circle traversed in the same direction a number of times and \(dH_ p\) denotes the area element of \(H\) in \(p\).