On a Banach space of Laplace-Stieltjes integrals

Let $\Omega$ be class of positive unbounded functions $\Phi$ on $(-\infty,\,+\infty)$ such that the derivative $\Phi'$ is positive, continuously differentiable and increasing to $+\infty$ on $(-\infty,+\infty)$, $\varphi$ be the inverse function to $\Phi'$, and $\Psi(x)=x-\frac{\Phi(x)}{\Phi'(x)}$ be the function associated with $\Phi$ in the sense of Newton. Let $F$ be nonnegative nondecreasing unbounded continuous on the right function on $[0,+\infty)$ and $f$ be a real-value function on $[0,+\infty)$. By $LS_{\Phi}(F)$ we denote the class of integrals $I(\sigma)=\int\nolimits_{0}^{\infty}f(x)e^{x\sigma}dF(x)$, convergent for all $\sigma\in{\Bbb R}$ such that $|f(x)|\exp\{x\Psi(\varphi(x))\}\to 0$ as $x\to+\infty$. Put $\|I\|_{\Phi}:=\sup\{|f(x)|\exp\{x\Psi(\varphi(x))\}:\,x\ge 0\}$. It is proved that if $\ln\,F(x)=o(x)$ as $x\to+\infty$ then $(LS_{\Phi}(F),\,\|\cdot\|_{\Phi})$ is a Banach space and it is studied its properties.