Solution
= Solution
Put $k=m+1$ and $L=\lambda+X$. The posterior risk is
$$
\tfrac12\{e^{-ra}M(r)+e^{ra}M(-r)\},
$$
where $M(s)=(L/(L-s))^k$. Differentiating in $a$ gives
$$
e^{2ra}=\frac{M(r)}{M(-r)}=\left(\frac{L+r}{L-r}\right)^k.
$$
Thus
$$
\boxed{\hat\theta=\frac{m+1}{2r}\log\frac{\lambda+X+r}{\lambda+X-r}.}
$$