Asymptotics of moment generating function

Question

Consider r.v. $\xi$ with known c.d.f. $F$ and p.d.f. $f$. Let the corresponding moment generating function $M(z)$ be finite for all $z \in \mathbb{R}$. I am interested in deriving the asymptotics of $M(z)$ as $z \rightarrow \infty$ from asymptotics of $1-F(x)$ or $f(x)$ as $x \rightarrow \infty$.

Please advise where to read more about this relationship?

I am almost sure that this relationship exists, I will write arguments below.

Firstly, there are tauberian theorems that study the relationship of $F(x)$ and left tail of $M(z)$. Secondly, the right tail of $M(z)$ gets heavier as right tail of $f(x)$ gets heavier:

1. For Gumbel-like distribution with $F(\xi < x) = 1 - \exp \left\{-\exp \left\{x\right\} \right\}$ follows: $M(z) = \Gamma (z+1) \sim e^{z\ln{z}}$ as $z \rightarrow \infty$.
2. For $N(0,1)$ follows: $M(z) = e^{z^2/2} $.
3. For $Exp(1)$ m.g.f. $M(z)$ is finite only for $z<1$.

Edit 1. I think I have found the approach to this problem. Let us denote $1 - F(x)$ for $S(x)$ and transform m.g.f. using integration by parts as follows:

$$\mathbb{E}(e^{i\xi}) = \int\limits_{\mathbb{R}} e^{tx}dF(x) = t\int\limits_{\mathbb{R}} S(x) e^{tx}dx.$$

Now let us rewrite $S(x)$ as $e^{\ln(S(x))}$. This is reasonable, since the function $S(x)$ usually has an exponential form. Therefore:

$$\mathbb{E}(e^{i\xi}) = t\int\limits_{\mathbb{R}} e^{tx+\ln S(x)}dx.$$

We to find such $a$ and $b$ that $tx+\ln S(x) = t^b\left (xt^a + g(xt^a) \right),$ at least, asymptotically. With the change of variables $xt^a \rightarrow z$ the integral takes the following form:

$$\mathbb{E}(e^{i\xi}) = t^{1-a}\int\limits_{\mathbb{R}} e^{ t^b\left (z + g(z) \right)}dx.$$

Finally, using Laplace method we can obtain:

$$\mathbb{E}(e^{i\xi}) \sim Ct^{1-a} \cdot t^{-b/2} e^{ t^b\left (z^* + g(z^*) \right)}, t\rightarrow \infty,$$ where $z^*$ if point of maximum of $z + g(z)$.

I am still wondering if there are other approaches.

Edit 2. There is even a simpler method. See general Laplace approximation formula (2.5) in "R. Butler. Saddlepoint approximations with applications."

Edit 3. Edit 2 is wrong. There is no simpler method. Moreover, approach in the Edit 1 only works for Weibull-like distributions. Therefore, question is still open.