Analyzing a Seneta's conjecture by using the Williamson transform

Abstract

Considering slowly varying functions (SVF), Seneta in 2019 conjectured the following implication, for $\alpha \ge 1$, $${\int }_0^x{y}^{\alpha -1}(1-F(y))dy\text{ is SVF}⟹{\int }_{[0,x]}{y}^{\alpha }dF(y)\text{ is SVF, as }x\to \infty ,$$ 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.