A hypergeometic E-function is
$${}_pF_q\left(\left.\begin{array}{ll}a_{1}, \ldots, & a_{p} \\b_{1}, \ldots, & b_{q}\end{array} \right\rvert\, \lambda z^{q-p}\right)=\sum_{n=0}^{\infty} \frac{\left(a_{1}\right)_{n} \cdots\left(a_{p}\right)_{n}}{\left(b_{1}\right)_{n} \cdots\left(b_{q}\right)_{n}} \lambda^{n} z^{n(q-p)}$$
for integers $0 \leqslant p<q$, rational parameters $a_{1}, \ldots, a_{p} \in \mathbb{Q}$ and $b_{1}, \ldots, b_{q} \in \mathbb{Q} \backslash \mathbb{Z}_{\leqslant 0}$, and an algebraic scalar $\lambda$, where $(x)_{n}=x(x+1) \cdots(x+n-1)$ denotes the rising Pochhammer symbol of a rational number $x$.
In this notation, the confluent hypergeometric function is $_1F_2$ with $b_2=1$. I am interested primarily in bounding the confluent hypergeometric function above by some function for all real or all rational parameters and arguments. Later, I will consider $_pF_q$ in general. I suspect the bound will be of the form $Ce^{|x|}$ where $C$ is a constant dependent on the parameters and argument. Ideally, this constant will decrease rapidly as one of the parameters, or both parameters tend to infinity - perhaps a denominator growing factorially as one or both of the parameters grow.
I am thinking that I will have to use some combination of a known asymptotic expansion for large $z$, and an analysis of the series expansion for small $z$ but I would really appreciate some guidance about how this is to be done correctly. Perhaps we will have to limit the parameters somehow further, but before I do that, I want to know if any nice bound is known.