  The standard Bonnet-Myers theorem says that if the Ricci scalar of a
Riemannian manifold is bounded below by a positive number, then the manifold is
compact. Moreover, a bound of its diameter is pointed out. The theorem was
extended to Finsler manifolds. In this paper we prove that if a certain
condition on the average of the Ricci scalar holds, then the Finsler manifold M
is compact if the Ricci scalar is bounded above by the same positive number. An
upper bound of the diameter is also found. With no condition on Ricci scalar
itself but with a different one on its average, we find that the Finsler
manifold M is again compact. This time no bound of the diameter is found.The
proofs are given in the Finslerian setting and are based on the index form
along geodesics.
