A Theorem of Galambos-Bojanić-Seneta Type

Abstract

In the theorems of Galambos-Bojanić-Seneta's type, the asymptotic behavior of the functions $c_{\left[x\right]},\hspace{0.17em}\hspace{0.17em}x\ge 1$, for $x\to +\infty$, is investigated by the asymptotic behavior of the given sequence of positive numbers $\left(c_n\right)$, as $n\to +\infty$ and vice versa. The main result of this paper is one theorem of such a type for sequences of positive numbers $\left(c_n\right)$ which satisfy an asymptotic condition of the Karamata type ${\underset{¯}{\lim }}_{n\to \infty }$ $c_{\left[\lambda n\right]}/c_n>1$, for .