Does the generalized gamma distribution have a mgf?

[psie]

TL;DR Summary

I'm reading An Intermediate Course in Probability by Gut. I am confused about a statement made concerning the generalized gamma distribution and its existence of a moment generating function.

I quote from An Intermediate Course in Probability by Gut:

Another class of distributions that possesses moments of all orders but not a moment generating function is the class of generalized gamma distributions whose densities are $$f(x)=Cx^{\beta-1}e^{-x^{\alpha}},\quad x>0,\tag1$$where $\beta>-1$, $0<\alpha<1$, and $C$ is a normalization constant (that is chosen such that the total mass equals $1$).

It is clear that all moments exist, but, since $\alpha<1$, we have $$\int_{-\infty}^\infty e^{tx}x^{\beta-1}e^{-x^\alpha}\, dx=\infty\tag2$$ for all $t>0$, so that the moment generating function does not exist.

First, I don't think it is clear that all moments exist. Integrating $(1)$ and making the substitution $y=x^\alpha$, and rewriting the integral in terms of a gamma integral, I get that $C=\Gamma(\beta/\alpha)$. The only condition I get is $\beta/\alpha>0$, so I don't see that the restriction $\beta>-1$ makes sense. I also don't see why $\alpha<1$ would make sense. Does anyone know what the proper bounds should be on $\alpha$ and $\beta$? In view of my finding that $\beta/\alpha>0$, I don't see why they both could be negative too.

Second, with the observations made above, does the moment generating function exist?

 

[psie]

I think the correct $C$ should be $\alpha/\Gamma(\beta/\alpha)$. And $\beta>-1$ actually makes sense if we want all moments to exist. Moreover, $0<\alpha<1$ is only specified I think to show this is when the integral diverges.