Let $X_1, \dots, X_n$ be dependent Bernoulli r.v. such that $\mathbb{E}[X_i]=p_i, i=1,\dots,n$. I want to find a tight bound to the mgf of the sum of such random variables, i.e. a bound $B$ function of some random variables $Y_1, \dots, Y_n$ (note that $Y_i$ may also be equal to $X_i, i=1,\dots,n$) s.t., \begin{equation} (1) \text{ } \mathbb{E}[e^{ \lambda \sum_i X_i} ] \leq B.\end{equation}
For negative associated random variables, i.e. variables $X,Y$ for which $Cov(X,Y) \le 0$, it holds that $ \mathbb{E}[e^{ \lambda (X+Y)} ] \le \mathbb{E}[e^{ \lambda X} ] \mathbb{E}[e^{ \lambda Y} ] (\bigstar)$; unfortunately the variables $X_1, \dots, X_n$ are not negative associated. In general, appling $(\bigstar)$ to the lhs in $(1)$ leads to a incorrect result thus it is not true that $\mathbb{E}[e^{ \lambda \sum_i X_i} ] \leq \prod_i \mathbb{E}[e^{ \lambda X_i}]$ for the considered random variables $X_1, \dots, X_n$. Is there a way to obtain the desired bound $B$ with other techniques? Any refrence will be very useful!