  In this paper in the cylindrical domain we consider a fractional elliptic
operator with Dirichlet conditions. We prove, that the first eigenvalue of the
fractional elliptic operator is minimised in a circular cylinder among all
cylindrical domains of the same Lebesgue measure. This inequality is called the
Rayleigh-Faber-Krahn inequality. Also, we give Lyapunov and Hartmann-Wintner
inequalities for the fractional elliptic boundary value problem.
