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Analyzing a Seneta's conjecture by using the Williamson transform
Edward Omey, Meitner Cadena · 2022-11-09 · via math.ST updates on arXiv.org

Considering slowly varying functions (SVF), Seneta in 2019 conjectured the following implication, for $α\geq1$, $$ \int_0^x y^{α-1}(1-F(y))dy\textrm{ is SVF}\ \Longrightarrow\ \int_{[0,x]}y^αdF(y)\textrm{ is SVF, as } x\to\infty,$$ where $F(x)$ is a cumulative distribution function on $[0,\infty)$. Complementary results related to this transform and particular cases of this extended conjecture are discussed.

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