Powers of an exponential product process that are submartingales

Let $X_1,X_2,\dots$ be i.i.d. exponential with rate $\lambda$, and set $M_0=1$, $M_n=M_{n-1}\cdot\tfrac12e^{\lambda X_n/2}$, using the filtration generated by the $X_i$. The process $M_n^p$ is a submartingale exactly for $a<p<b$ (with $0<a$). Find $a+b$.

$3$. By independence, $\mathbb{E}[M_{n+1}^p\mid\mathcal F_n]=M_n^p\,2^{-p}\tfrac{\lambda}{\lambda-\lambda p/2}=M_n^p\,\tfrac{2^{1-p}}{2-p}$ for $p<2$. The factor is at least $1$ when $2^{1-p}\ge2-p$; the curves meet at $p=0$ and $p=1$, and for $1<p<2$ the inequality holds strictly (it fails on $(0,1)$). So $a=1$, $b=2$.