On Some Convexity Questions of Handelman

We resolve some questions posed by Handelman in 1996 concerning log convex integrable functions. In particular, we give a negative answer to a question he posed concerning the integrability of $h^2(x)\mathrm{/}h(2x)$ when $h$ is integrable and log convex and $h(n{)}^{1\mathrm{/}n}$ converges to 1.