We know that $m(T), p(T), M(T)$ are functions of variable $T$. Find these functions, as well as a constant $\delta>0$ such that $m(T)p(T)$ is maximized and $$mp+\delta m \leq M<e^{\frac{T}{1+\alpha/\ln\left(1+\frac{\epsilon^2}{2 m p^2}\right)}}$$ also, $0<p(T)<1$, and $p(T) \to 0$ and $m(T) \to \infty$ as $T\to \infty$.
I think we should say $M=e^{\frac{T}{1+\alpha/\ln\left(1+\frac{\epsilon^2}{2 m p^2}\right)}}-\epsilon$; $$mp+\delta m + \epsilon=e^{\frac{T}{1+\alpha/\ln\left(1+\frac{\epsilon^2}{2 m p^2}\right)}}$$
Then, we have to maximize $m p$ here and find $\epsilon$ and $\delta$. Please give me hints and answers even your answer is sub-optimal.