  When the backward shift operator on a weighted space $H^2_w=\{f=\sum_{j=0}
^{\infty} a_jz^j : \sum_{j=0}^{\infty} |a_j|^2w_j < \infty\}$ is an
$n$-hypercontraction, we prove that the weights must satisfy the inequality
$$\frac{w_{j+1}}{w_j} \leq {\frac{1+j}{n+j}}.$$ As an application of this
result, it is shown that such an operator cannot be subnormal. We also give an
example to illustrate the important role that the $n$-hypercontractivity
assumption plays in determining the similarity of Cowen-Douglas operators in
terms of the curvatures of their eigenvector bundles.
