Handelman's Conjecture on integrability of h from integrability of h(x)/h(2x) under ratio-limit condition

Establish that if h:(0,∞)→(0,∞) is log convex, satisfies lim_{n→∞} h(n+1)/h(n) = 1, and h(x)/h(2x) is integrable on (0,∞), then h is integrable on (0,∞).

Background

Following the discussion after Theorem 9 in Handelman (1996), the paper states a conjecture linking the integrability of the ratio h(x)/h(2x) to the integrability of h itself, under a discrete ratio-limit condition on h at integers.

This conjecture is recorded in the introduction as Conjecture 1, and later shown to be true in Theorem 2.