 Solution

Neither proper gamma prior yields the condition in part (c). The risk in part (a) is unbounded as $\theta \to \infty$ because $\beta >0$. The risk in part (b) is analytic, tends to zero at infinity, and is not identically zero, so its prior average is strictly below its supremum for a gamma prior with positive density everywhere. Thus this sufficient condition proves neither estimator minimax for any $\alpha ,\beta >0$.