A Seneta’s Conjecture and the Williamson Transform

Considering slowly varying functions (SVF), Seneta in 2019 conjectured the following implication, for α ≥ 1, ∫ x 0 yα−1(1 − F(y))dy is SVF ⇒ ∫ x 0 yαdF(y) is SVF, as x → ∞, where F(x) is a cumulative distribution function on [0,∞). By applying the Williamson transform, an extension of this conjecture is proved. Complementary results related to this transform and particular cases of this extended conjecture are discussed.