Regularly varying sequences and entire functions of finite order

Slavko Simi\'c

Abstract: We present a method for estimating the asymptotic behavior of: $$ f^\alpha(x):=\sum_{n=1}^\infty n^\alpha l_n a_n x^n,\ \ x\to \infty,\ \ \alpha \in R, $$ related to a given entire function $f(x):=\sum_{n=1}^\infty a_n x^n$ of finite order $\rho$, $0<\rho<+\infty$, $a_n\ge 0$, $n\in N$; where $(l_n)$, $n\in N$, are slowly varying sequences in Karamata's sense.

Classification (MSC2000): 30D20; 26A12

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