Some properties of functionals on level sets

In the paper we consider special functionals on a planar domain $G$ constructed by means of the distance to the boundary $\partial G$ and a classical warping function. The functionals depending on the distance function are considered for simply-connected domains. We also study the functionals depending on the warping function for a finite-connected domain. We prove that the property of isoperimetric monotonicity with respect to a free parameter gives rise to another monotonicity, namely, the monotonicity of the functionals considered as the functions of the sets defined on subsets of the domain. Some partial cases of the inequality were earlier obtained by Payne. We note that the inequalities were successfully applied for justifying new estimates for the torsional rigidity of simply-connected and multiply-connected domains. In particular, we construct new functionals of domains monotone in both its variables. Moreover, we find sharp estimates of variation rate of the functions, that is, we obtain sharp estimates of their derivatives.