Bounding the MGF of a non-homogeneous Rademacher chaos of order two I am trying to bound a quantity of the form
$$
\mathbb{E}\left[ F\!\left( \sum_{i,j} a_{ij}\varepsilon_i\varepsilon_j'
+ \sum_{i,j,k} b_{ijk}\varepsilon_i\varepsilon_j\varepsilon_k'+ \sum_{i,j,k} b_{ijk}\varepsilon_i'\varepsilon_j'\varepsilon_k
+ \sum_{i,j,k,\ell} c_{ijk\ell}\varepsilon_i\varepsilon_j\varepsilon_k'\varepsilon_\ell' \right)\right] \tag{1}
$$
where $F$ is convex (in my case, $F=\exp$, so I'll focus on this below), $\varepsilon,\varepsilon'$ are two i.i.d. vectors, each of $n$ independent Rademacher variables, and the coefficients have pretty muchall the symmetry one could hope for.
Trying to proceed by first conditioning on $\varepsilon'$, I need to get a bound on something of the form
$$
\mathbb{E}\left[ \exp\left( \sum_{i} a_{i}\varepsilon_i
+ \sum_{i\neq j} b_{ij}\varepsilon_i\varepsilon_j \right)\right] \tag{2}
$$
If the exponent were homogeneous (no linear term), then I would know how to continue (or at least where to look for results to apply); but the factor that I have a non-homogeneous Rademacher chaos of order 2 here makes me hit a roadblock. Is there a standard way to handle those to get a bound on (2)?
 A: Well, so far I have experimented with some horrendous expressions when using decoupling on (2):
$$
\mathbb{E}_{\varepsilon}\left[ \exp\left( \sum_{i} a_{i}\varepsilon_i
+ \sum_{i\neq j} b_{ij}\varepsilon_i\varepsilon_j \right)\right] \leq
\mathbb{E}_{\varepsilon\varepsilon'}\left[ \exp\left( 2\sum_{i} a_{i}\varepsilon_i
+ 4\sum_{i, j} b_{ij}\varepsilon_i\varepsilon'_j \right)\right] 
$$
and then first conditioning on $\varepsilon$ (or $\varepsilon'$) to bound part of the expression, etc. I am not sure it's the way to go, as it gets very ugly very soon.
The other way I found is to use the elementary inequality
$$
xy \leq \frac{x^2+y^2}{2} \qquad x,y\in\mathbb{R}
$$
to get
$$
\mathbb{E}\exp\left( \sum_{i} a_{i}\varepsilon_i+ \sum_{i\neq j} b_{ij}\varepsilon_i\varepsilon_j \right)
\leq
\frac{1}{2}\mathbb{E}\left[ \exp\left( 2\sum_{i} a_{i}\varepsilon_i \right)\right]
+\frac{1}{2}\mathbb{E}\left[ \exp\left(2\sum_{i\neq j} b_{ij}\varepsilon_i\varepsilon_j \right)\right] 
$$
and then use separate estimates for homogeneous monomials on both summands.
