Suppose that $X$ is a nonnegative random variable.
We say that $X$ has moments of all orders if $$ E(X^p) <\infty $$ for all $p$. We say that $X$ has exponential moments if $$ E(e^{aX}) < \infty $$ for all (nonnegative) $a$.
My question is about something in between moments and exponential moment. Suppose we know, for example, that $X$ satisfies $$ E\big(e^{a(ln(X))^2}\big)<\infty. $$ This appears to be strictly between existence of moments and existence of exponential moments in the sense that is weaker than having exponential moments and stronger than having moments. Is there a name for this property?
There are many things "between" polynomial and exponential moments I guess. For instance: using Taylor's formula we see: for any $a>0$ $$E[e^{aX}] = \sum_{n=0}^\infty \frac{a^n}{n!}E[X^n].$$
Here we see that $E[X^n]<\infty$ for every $n\geq 0$ is not enough to guarantee exponential moments as you know.
Define the sequence $b_n := E[X^n]$, $n\geq 0$ and assume $b_n<\infty$ for all $n\geq 0$. Then $X$ has polynomial moments of all orders by assumption. A sufficient condition for $X$ to have exponential moments is thus $$\sum_{n=0}^\infty \frac{a^n}{n!}b_n<\infty,$$ for instance: $b_n \sim C^n (n!)^{1-\varepsilon}$ for any $C>0$ and $\varepsilon>0$ (that is, the polynomial moments can still grow really fast to infinity but at maximum as a factorial). So anything that does not make the sum converge will be weaker than exponential moments. I'm not sure if I am answering your question, in any case I hope it helped a bit more! :)