Relations of Borel Type for Generalizations of Exponential Series - Skaskiv, O. B.; Trusevich, О. M. (Ukrains’kyi Matematychnyi Zhurnal, 2001)
We prove that the condition \(\sum olimits_{n = 1}^{ + ∞ } {\left( {n{λ }_n } \right)^{ - 1} < + ∞ }\) 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 \(H_ + \left( {λ } \right)\mathop = \limits^{{df}} \cup _f H\left( {{λ,}f} \right)\) . Here, H(λ, f) is the class of series that converge for all σ ≥ 0 and have a form $$F\left( σ \right) = \sum\limits_{n = 0}^{ + ∞ } {a_n f\left( {{σ λ}_n } \right),\quad a_n \geqslant 0,\;n \geqslant 0,}$$ 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, +∞).