ON BOREL'S TYPE RELATION FOR THE LAPLACE-STIELTJES INTEGRALS - Скасків, Олег Богданович; Тарновецька, Ольга Юріївна; Трусевич, Оксана Мирославівна (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2019)
In article it is obtained sufficient conditions in order that relation $\ln F(x)\le (1+o(1))\lnμ(x,F)$ holds for each function of the form $F(x)=\sum_{n=0}^{+∞} f_n e^{xλ_n +τ(x)β_n}$ as $x→ +∞$ outside some set $E$ of zero linear density or of zero lower linear density, where $(f_n),$ $(λ_n),$ $(β_n)$ are positive sequences, and a differentiable function $τ: \mathbb{R}_+→ \mathbb{R}_+$ such that$τ′(x) \ge 1$ $(x\ge x_0),$ $μ(x,F)=\max\{ f_ne^{xλ_n +τ(x)β_n}: n \ge 0\} .$