Artin states that:
Theorem 1.10: If $f(x)$ is log convex on a certain interval, and if $c$ is any real number $\neq 0$, then both the functions $f(x+c)$ and $f(cx)$ are log convex on the corresponding intervals.
I do not really understand the part of "on the corresponding intervals". Does it mean:
(Reformulation): If $f$ is log convex on $(a,b)$, and if $c\not = 0$, then $x\mapsto f(x+c)$ is convex on $(a-c,b-c)$ and $x\mapsto f(cx)$ is log convex on $(a/c,b/c)$?
By "log convex", he mean "logarithmically convex". A function $f:A\to \mathbb{R}$ is said to be logarithmically convex if $$ f(\lambda x+(1-\lambda)y)\leq f(x)^{\lambda}f(y)^{1-\lambda} $$ for all $x,y\in A$ and all $\lambda\in (0,1)$.
