  The degree of mobility of a (pseudo-Riemannian) K\"ahler metric is the
dimension of the space of metrics h-projectively equivalent to it. We prove
that a metric on a closed connected manifold can not have the degree of
mobility $\ge 3$ unless it is essentially the Fubini-Study metric, or the
h-projective equivalence is actually the affine equivalence. As the main
application we prove an important special case of the classical conjecture
attributed to Obata and Yano, stating that a closed manifold admitting an
essential group of h-projective transformations is $(CP(n), g_{Fubini-Study})$
(up to a multiplication of the metric by a constant). An additional result is
the generalization of a certain result of Tanno 1978 for the pseudo-Riemannian
situation.
