On a conjecture of Seneta on regular variation of truncated moments

We prove that h?(x) = ??x0 y??1F?(y)dy is regularly varying with index ? [0, ?) if and only V?(x) ?[0,x] y?dF(y) the same index, where > 0, F(x) a distribution function of nonnegative random variable, F?(x) 1?F(x). This contains at ?= 1 result Rogozin [8] on relative stability, 2 new, equivalent characterization domain attraction normal law. For 0 our implies recent conjecture by Seneta [9].