An application $h$ from a non-trivial interval $J$ of $\mathbf{R}$ to $\mathbf{R}$ is said to be log-convex if, and only if, it takes values in $\mathbf{R}_+^*$ and $\ln \circ h$ is convex on $J$.
More generally, determine, if $T \in \mathbf{R}_+^*$, all applications $g$ from $]-T, +\infty[$ to $\mathbf{R}$, log-convex and satisfying
$$\forall t \in ]-T, +\infty[, (t+T)g(t) = (t+2T)g(t+2T).$$