  Let $\Phi$ be a real valued function of one real variable, let $L$ denote an
elliptic second order formally self-adjoint differential operator with bounded
measurable coefficients, and let $P$ stand for the Poisson operator for $L$. A
necessary and sufficient condition on $\Phi ensuring the equivalence of the
Dirichlet integrals of $\Phi\circ Ph$ and $P(\Phi\circ h)$ is obtained. We
illustrate this result by some sharp inequalities for harmonic functions.
