Suppose there is a density $f(x; \theta)$ over $X\subseteq\mathbb{R}_{+}$ with parameter $\theta\in \Theta \subseteq \mathbb{R}_{+}$. Both sets are bounded. Suppose as well that $f(x; \theta)$ is twice continously differentiable and log-convex on $\theta$, for each $x$.
Is it still possible that $\mathbb{E}\left[\bigg(\frac{\partial}{\partial \theta} \log f(X; \theta)\bigg)^2\bigg|\theta\right]=-\mathbb{E}\left[\frac{\partial^2}{\partial \theta^2} \log f(X; \theta)\bigg|\theta\right]$ ?
If so, does this mean that the Fisher information is always zero?
This may be a dumb question, but I couldn't find an answer anywhere; and couldn't convince myself why would be true that the Fisher information is zero.