Legendre transform of a convex downward function

Question

I have the following function: $$\lim _{t \rightarrow \infty}-\frac{1}{t} \log \left\langle e^{-\lambda W(t)}\right\rangle=e(\lambda),$$ which is convex downward, where $\langle\exp [-\lambda W(t)]\rangle$ is the generating function of the random variable $W(t)$. We also know that the function has the following property: $$e(\lambda)=e(1-\lambda).$$ In the article the authors said that the Legendre transform $\hat{e}(w)$ is convex upward, $\hat{e}(w) \geqslant 0$ and $$\hat{e}(w)=\max _{\lambda}\{e(\lambda)-\lambda w\}.$$ Why these three statements are true?