Building New Log-Concave Sequences

Question

A sequence of positive numbers $a_1,a_2, \ldots$ is called log-concave if $a_k^2\geq a_{k-1}a_{k+1}$. I vaguely remember coming across results involving products of log-concave sequences being log-concave. I can't remember the sources or even if this is even imagined! For example, suppose $a_k$ and $b_k$ are log-concave. The term-wise product $c_k=a_kb_k$ is log-concave since \begin{align*} c_k^2&=a_k^2b_k^2\\ &\geq a_{k-1}a_{k+1}b_{k-1}b_{k+1}\\ &=c_{k-1}c_{k+1} \end{align*} Anyhow, I vaguely remember a theorem stating the Cauchy product $c_k=\sum_{j=0}^ka_jb_{k-j}$ of two log-concave sequences is also log-concave. Is this true?