For a nonnegative nondecreasing unbounded right-continuous function F on [0, +∞), an entire transcendental function \(f\left(z\right)={\sum }_{k=0}^{\infty }{f}_{k}{z}^{k}\) with fk ≥ 0 for all t ≥ 0, and a function a(x) nonnegative on [0, +∞), the integral I(r) = \({\int }_{0}^{\infty }a\left(x\right)f\left(xr\right)dF\left(x\right)\) is called a Laplace–Stieltjes-type integral. Suppose that for a positive function Φ unbounded on (–∞, +∞), the derivative Φ′ is a positive continuously differentiable function increasing to +∞. We establish the conditions under which the following relation is true: \({\int }_{{r}_{0}}^{\infty }\frac{{\Phi }^{\prime}\left(r\right)\mathrm{ln}I\left(r\right)}{{\Phi }^{2}\left(r\right)}dr<+\infty \) .
