  We present some comparison results for solutions to certain non local
elliptic and parabolic problems that involve the fractional Laplacian operator
and mixed boundary conditions, given by a zero Dirichlet datum on part of the
complementary of the domain and zero Neumann data on the rest. These results
represent a non local generalization of a Hopf's lemma for elliptic and
parabolic problems with mixed conditions. In particular we prove the non local
version of the results obtained by J. D\'avila and J. D\'avila-L. Dupaigne for
the classical case respectively.
