  Our main result is an extension of the classical Cauchy inequality for the
case of bounded densities. In particular, this implies subharmonicity of the
function $M_n(E)$, where $V_n(x)$ is the critical Riesz potential in $R^n$
($\alpha=n$) of a density $0\leq \rho\leq 1$ and $M_n(t)$ is the profile
function: the solution of $y'(t)=1-y^{n/2}(t)$, $y(0)=0$. We show thath this
result is optimal (in the sense that $M_n(E)$ is harmnoic for characteristic
functions of a ball) and give thereby an affirmative answer to one question
posed by B. Gustafsson and M. Putinar (Ind. Univ. Math. J., 52(2003), 527-568).
