Upper bound for m.g.f $X$ is a discrete random variable from power series family (e.g., binomial, poisson etc.). is it possible to find an upper bound for the m.g.f of $X$?
N.B: from stack exchange I obtained the following statement:
Suppose $X$ is a bounded zero-mean random variable taking on values in $[−σ,σ]$ and let $G(t)=E[e^{tX}]$ be its MGF. From a bound used in a proof of Hoeffding's Inequality, we have that $G(t)=E[e^{tX}] \le \exp(σ^2t^2/2)$
But this is for bounded r.v. Can we found some similar m.g.f bound for discrete r.v from psd family?

This is not possible.
The most general discrete variable $X$ is supported on a countable set of real numbers $x_1, x_2, \ldots, x_n, \ldots$ with associated positive probabilities $p_i = e^{q_i}.$  By definition,
$$\phi_X(t) = E\left[e^{tX}\right] = \sum_{i} p_i e^{t x_i} = \sum_{i} \exp(q_i + t x_i),$$
which is a sum of strictly positive values.
To say that this variable is unbounded means for every natural number $N$ there exists an index $i(N)$ for which $|x_{i(N)}| \gt N.$  Pick any real number $t$ for which
$$t \operatorname{sgn}(x_{i(N)}) \gt \frac{\log(N) - q_{i(N)}}{N}$$
and by neglecting all the terms in the sum but the one with index $i(N),$ estimate
$$\phi_X(t) \ge  \exp\left({q_{i(N)} + tx_{i(N)}}\right) \gt e^{\log(N)}= N.$$
This proves $\phi_X$ is unbounded.  (Indeed, it shows $\phi_X$ is unbounded on at least one of the intervals $[0,\infty)$ and $(-\infty, 0]$.)
