Handelman's Question on integrability of h(x)/h(2x) for log-convex h with lim h(n)^{1/n} = 1

Determine whether, for a log-convex function h:(0,∞)→(0,∞) that is integrable on (0,∞) and satisfies lim_{n→∞} h(n)^{1/n} = 1, the ratio h(x)/h(2x) is integrable on (0,∞).

Background

Handelman introduced a maximal function H_h(a) = sup_{b≥a} h(b)/h(a+b) and observed that for log-convex h this simplifies to H_h(x) = h(x)/h(2x). Motivated by this, he posed a question relating integrability of h to integrability of h(x)/h(2x).

This paper revisits that question directly in the introduction, presenting it as Question 1 before constructing a counterexample in Section 2.1.