Let $Y_1, \ldots ,Y_n$ be iid with density $f(y;\theta)$.
We assume that $\dfrac\partial{\partial\theta}\log f(y ; \theta)$ and $\dfrac{\partial^2}{\partial\theta^2}\log f(y ; \theta)$ exist for all $\theta$. Consider a class $G$ of real functions $g(Y;\theta)$ such that:
1) $E_\theta[g(Y;\theta)] = 0$.
2) $\dfrac{\partial{g(Y;\theta)}}{\partial\theta}$ exists, is negative and bounded for all $\theta$ and $Y$.
3) $E_\theta[g^2(Y;\theta)] < \infty,\,$ for all $\theta$.
The goal is to show that the score function (derivative of the log likelihood with respect to $\theta$) is a member of $G$ that minimizes:
$ \frac{E_{\theta}[g^2(Y;\theta)]}{(E_\theta[\partial_\theta g(Y;\theta)])^2}$
Thanks for your help.