On a conjecture of Seneta

In this short note we prove that $h_\beta(x) = \beta \int_0^x y^{\beta-1} \overline F(y) \mathrm{d} y$ is regularly varying with index $\rho \in [0,\beta)$ if and only if $V_\beta (x) = \int_{[0,x]} y^\beta \mathrm{d} F(y)$ is regularly varying with the same index. This implies an extended version of a recent conjecture by Seneta.