  The curvature $\mathcal K_T(w)$ of a contraction $T$ in the Cowen-Douglas
class $B_1(\mathbb D)$ is bounded above by the curvature $\mathcal K_{S^*}(w)$
of the backward shift operator. However, in general, an operator satisfying the
curvature inequality need not be contractive. In this note, we characterize a
slightly smaller class of contractions using a stronger form of the curvature
inequality. Along the way, we find conditions on the metric of the holomorphic
Hermitian vector bundle $E_T$ corresponding to the operator $T$ in the
Cowen-Douglas class $B_1(\mathbb D)$ which ensures negative definiteness of the
curvature function. We obtain a generalization for commuting tuples of
operators in the class $B_1(\Omega)$, for a bounded domain $\Omega$ in $\mathbb
C^m$.
