So the question gave $D_1$ and $D_2$ as unbiased, efficient and consistent estimators of $\delta$ . $D_3$ is a new estimator which is obtained by taking a weighted average of $D_1$ and $D_2$ with one quarter of the weight placed on $D_1$. Now, the question is can I prove that $D_3$ is an efficient estimator of $\delta$?
I know for that for $D_3$ to be efficient, $\operatorname{Var} (D_3)$ has to be less than $\operatorname{Var} (D_1)$ and $\operatorname{Var} (D_3)$ has to be less than $\operatorname{Var} (D_2)$. I also know that
$$\operatorname{Var} (D_3) = \operatorname{Var} \left(\frac{1}{4D_1}+\frac{3}{4D_2}\right)$$
However, I'm not sure how to prove (or if it is even possible to prove) that $D_3$ is an efficient estimator.
You can use the properties of variance so that Var(D3)=(1/16)Var(D1)+(9/16)Var(D2). To prove the efficiency, you could perhaps use the Cramer-Rao bound and fisher information.