Relations of Borel Type for Generalizations of Exponential Series

Abstract

We prove that the condition $\sum\nolimits_{n = 1}^{ + \infty } {\left( {n{\lambda }_n } \right)^{ - 1} < + \infty }$ is necessary and sufficient for the validity of the relation ln F(σ) ∼ ln μ(σ, F), σ → +∞, outside a certain set for every function from the class $H_ + \left( {\lambda } \right)\mathop = \limits^{{df}} \cup _f H\left( {{\lambda,}f} \right)$. Here, H(λ, f) is the class of series that converge for all σ ≥ 0 and have a form

$$F\left( {\sigma} \right) = \sum\limits_{n = 0}^{ + \infty } {a_n f\left( {{\sigma \lambda}_n } \right),\quad a_n \geqslant 0,\;n \geqslant 0,}$$

and f(σ) is a positive differentiable function increasing on [0, +∞) and such that f(0) = 1 and ln f(σ) is convex on [0, +∞).