About Borel type relation for some positive integrals
DOI:
https://doi.org/10.64700/altay.20Keywords:
maximal term, functional series, Borel relation, stability of maximal termAbstract
The manuscript contains new results describing asymptotic behavior of functions which are represented by integrals of the form \(F(x)=\int_{0}^{+\infty}{a(t)f(x+t)\nu(dt)},\) where \(\nu\) is locally finite measure on \(\mathbb{R}_+\), \(a\) is positive \(\nu\)-measurable function, \(f\) is positive and increasing to \(+\infty\) in \([0,+\infty)\) function such that \(f(0)=1\) and \(\ln{f(x)}\) is convex on the interval \([0,+\infty)\) function. The obtained main result was applied to the study of the stability of the maximum term of the series of the form \(F(x)=\sum\limits_{n=0}^{\infty}a_nf(\lambda_n + x),\quad a_n\geq 0\ (n\geq 0).\)
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Copyright (c) 2025 Oleh Skaskiv, Andriy Bandura, Andriy Bodnarchuk
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