  In this work we prove a strong maximum principle for fractional elliptic
problems with mixed Dirichlet-Neumann boundary data which extends the one
proved by J. D\'avila to the fractional setting. In particular, we present a
comparison result for two solutions of the fractional Laplace equation
involving the spectral fractional Laplacian endowed with homogeneous mixed
boundary condition. This result represents a non-local counterpart to a Hopf's
Lemma for fractional elliptic problems with mixed boundary data.
