I am not a mathematics student and unfortunately have some confusion about a (well-known) theorem about strong unimodality of distributions. First of all let me clarify some terminologies and then ask the question. My definitions and the following theorem are based on https://www.google.com/books/edition/Unimodality_Convexity_and_Applications/V3rNCgAAQBAJ?hl=en&gbpv=0
Unimodality : A real random variable $X$ or its distribution function $F$ is called unimodal about a mode (or vertex) $\nu$ if $F$ is convex on $(—\infty,\nu)$ and concave on $(\nu,\infty)$. (or simply it's density has a single peak at $\nu$).
Unfortunately since convolution of two unimodal distributions are not unimodal, we lead to the concept of strong unimodality as follows :
Strong unimodality : A distribution function $G$ is a strongly unimodal if the convolution $G * F$ is unimodal for every unimodal $F$.
There is a powerful theorem in this context known as Ibragimov theorem :
Theorem 1.10. A nondegenerate distribution function G is strongly unimodal if, and only if, G is continuous and its density g is logconcave (i.e., log g is concave).
Question 1 : Is this theorem true for any interval $[a, b]$ or $(a,b)\subset\mathbb{R}$? It seems that it is not true as Gaussian normal density is concave over $[a,b], (a,b)$ and there no single peak for it. So what is the precise form of the theorem?
Question 2 I think that the distribution $G$ is defined as $G(x)=\int_{\infty}^xg(x')dx'$, where $g$ is the density function and by definition $\frac{dG(x)}{dx}=g(x)>0$, so $G(x)$ can not be unimodal (having a single peak). Does the strong unimodality about $g$?
