ON BOREL'S TYPE RELATION FOR THE LAPLACE-STIELTJES INTEGRALS

Abstract

In article it is obtained sufficient conditions in order that relation $\ln F(x)\le (1+o(1))\ln\mu(x,F)$ holds for each function of the form $F(x)=\sum_{n=0}^{+\infty} f_n e^{x\lambda_n +\tau(x)\beta_n}$ as $x\to +\infty$ outside some set $E$ of zero linear density or of zero lower linear density, where $(f_n),$ $(\lambda_n),$ $(\beta_n)$ are positive sequences, and a differentiable function $\tau: \mathbb{R}_+\to \mathbb{R}_+$ such that$\tau′(x) \ge 1$ $(x\ge x_0),$ $\mu(x,F)=\max\{ f_ne^{x\lambda_n +\tau(x)\beta_n}: n \ge 0\} .$