Is there a modification of Laplace's method for obtaining the asymptotic behaviour of Riemann-Stieltjes integrals?
In particular, I am interested in asymptotic behaviour of the Riemann-Stieltjes integral $$L(x)=\int_{0}^{T} te^{-xt} \text{d} g(t)$$ as $x\rightarrow +\infty$, where $g$ is a nondecreasing function. Or at least an upper bound for $L(x)$, which is sharper than stemed from the mean value theorem, $L(x)=O(1/x)$.