  It is shown that if the Kato constant of the negative part of the Ricci
curvature below a positive level is small, then the volume of the corresponding
manifold can be bounded above in terms of the Kato constant and the total Ricci
curvature. Together with the results from [5] and [6], this yields a
generalization of the famous Bonnet-Myers theorem. Connections to some earlier
generalizations are discussed.
