  We consider two disjoint and homotopic non-contractible embedded loops on a
Riemann surface and prove the existence of a non-contractible orbit for a
Hamiltonian function on the surface whenever it is sufficiently large on one of
the loops and sufficiently small on the other one. This gives the first example
of an estimate from above for a generalized form of the
Biran-Polterovich-Salamon capacity for a closed symplectic manifold.
