Regularization inequalites for one-dimensional Cauchy-type measures

In the paper we investigate various inequalities for the one-dimensional Cauchy measure. We also consider analogous properties for one-dimensional sections of multidimensional isotropic Cauchy measure. The paper is a continuation of our previous investigations \cite{BZ}, where we found, among intervals with fixed measure, the ones with the extremal measure of the boundary. Here for the above mentioned measures we investigate inequalities that are analogous to those found for Gaussian measures by Borell in \cite{B} and by Landau and Shepp in \cite{LS}.