Let $(X_{i})_{i\geq 1}$ be a sequence of centered real-valued martingale-differences with respect to some filtration $(\mathcal{F}_{i})_{i \geq 1}$. Define $S_{n} = \sum_{i=1}^{n}X_{i}$ and $\Sigma^{2}=\sum_{i=1}^{n}E[X^2_{i}]$. Furthermore, assume that the variables are a.s. bounded so that $|{X_{i}}| \leq M$ for some positive $M$. Let $h >0$ be some real number.

I am interested in the exponential inequalities of the form 

\begin{equation}
\label{eq:main_ineq}
E[\exp(hS_{n})] \leq \exp(\frac{\Sigma^2}{M^2} (\exp(hM)-hM-1))
\end{equation}

Such upper bound holds for example if $(X_{i})_{i\geq 1}$ are i.i.d. (see ex. Equation 1 in [here][1]in book Concentration Inequalities', by Boucheron, Lugosi and Massart page 25-26. ). Also, if we assume natural (for martingale differences) but stronger condition that a.s. holds that $\sum_{i=1}^{n}E_{i-1}[X^2_{i}] \leq \Sigma^{2}$ then the aforementioned inequality also holds (see for example [here][2] in the proof of the deviation bound of Theorem 3.2). 

But are there results under the condition on the variance (+ maybe some additional assumption which does not involve uniform bound on $E_{i-1}[{X_{i}^2}]$)? 


  [1]: https://epubs.siam.org/doi/abs/10.1137/1130013
  [2]: https://projecteuclid.org/download/pdf_1/euclid.aop/1176988477