Belonging of Laplace-Stieltjes integrals to convergence classes

For positive continuous functions $\alpha$ and $\beta$ increasing to $+\infty$ on $[x_0,+\infty)$ and the Laplace--Stieltjes integral $I(\sigma)=\int\limits_{0}^{\infty}f(x)e^{x\sigma}dF(x),\,\sigma\in{\Bbb R}$, a generalized convergence $\alpha\beta$-class is defined by the condition $$\int\limits_{\sigma_0}^{\infty}\dfrac{\alpha(\ln\,I(\sigma))}{\beta(\sigma)}d\sigma<+\infty.$$ Under certain conditions on the functions $\alpha$, $\beta$, $f$, and $F$, it is proved that the integral $I$ belongs to the generalized convergence $\alpha\beta$-class if and only if $\int\limits_{x_0}^{\infty}\alpha'(x)\beta_1 \left(\dfrac1{x}\ln\dfrac1{f(x)}\right)<+\infty,\,\beta_1(x)= \int\limits_{x}^{+\infty}\dfrac{d\sigma}{\beta(\sigma)}$. For a positive, convex on $(-\infty,\,+\infty)$ function $\Phi$ and the integral $I$, a convergence $\Phi$-class is defined by the condition $\int\limits_{\sigma_0}^{\infty}\dfrac{\Phi'(\sigma)\ln\,I(\sigma)}{\Phi^2(\sigma)}d\sigma<+\infty$, and it is proved that under certain conditions on $\Phi$, $f$ and $F$, the integral $I$ belongs to the convergence $\Phi$-class if and only if $\int\limits_{x_0}^{\infty}\dfrac{dx}{\Phi'\left(({1/x)\ln\,(1/f(x))}\right)}<+\infty$. Conditions are also found for the integral of the Laplace--Stieltjes type $\int\limits_{0}^{\infty} f(x)g(x\sigma)dF(x)$ to belong to the generalized convergence $\alpha\beta$-class if and only if the function $g$ belongs to this class.