Relations of Borel Type for Generalizations of Exponential Series

  • O. B. Skaskiv
  • О. M. Trusevich

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, +∞).
Published
25.11.2001
How to Cite
SkaskivO. B., and TrusevichО. M. “Relations of Borel Type for Generalizations of Exponential Series”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 53, no. 11, Nov. 2001, pp. 1580-4, https://umj.imath.kiev.ua/index.php/umj/article/view/4379.
Section
Short communications