Sum of exponential series of equal mean and variance

Question

Assuming $A$ and $B$ are two non-negative real-valued random variables such that

1. $\mathrm{E}(A)=\mathrm{E}(B)$ (equal means)
2. $\mathrm{Var}(A)=\mathrm{Var}(B)<\epsilon$ (equal small variances)

is there a way to prove that $\frac{1}{N}\sum_{j=1}^Ne^{-a_{j}}$ and $\frac{1}{N}\sum_{j=1}^Ne^{-b_{j}}$ are close to each other where $a_j$ and $b_j$ are independent realizations taken from $A$ and $B$, respectively. ($N$ can be assumed to be large as well)