The relationship of the Gaussian curvature with the curvature of a Cowen-Douglas operator

It has been recently shown that if K is a sesqui-analytic scalar valued non-negative definite kernel on a domain ? in Cm, then the function (K2?i?�jlogK)mi,j=1, is also a non-negative definite kernel on ?. In this paper, we discuss two consequences of this result. The first one strengthens the curvature inequality for operators in the Cowen-Douglas class B1(?) while the second one gives a relationship of the reproducing kernel of a submodule of certain Hilbert modules with the curvature of the associated quotient module.