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](https://epubs.siam.org/doi/abs/10.1137/1130013)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](https://projecteuclid.org/download/pdf_1/euclid.aop/1176988477) 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 $E\_{i-1}[{X\_{i}^2}]$)?



Exponential upper bounds for sums of martingale differences
Let $J$ take the value $M$ with probability $1/M^2$ and the value 1 with probability $1-1/M^2$. Let $R\_i$ be i.i.d. $\pm 1$ valued random variables of of mean zero, and define $X\_i:=JR\_i$. Then for $h=1/M$ it is easy to see your inequality is not satisfied, as the LHS is at least $M^{-2}(e/2)^n$, while the RHS is at most $\exp(2n/M^2)$.


