While reading a research paper on log convexity, I encountered a preposition (which is my question). I tried to prove it. I'm not getting any idea how to proceed. The statement is as follows:
Suppose $f: [a, b] \rightarrow \mathbb{R}^{+}$ is continuous and has a constant sign. Then $f$ is log-convex (strictly log-convex) $ \iff x \mapsto \frac{f(x+h)}{f(x)}$ is non-decreasing (increasing) on $[a, b-h]$ for each fixed $h > 0$.
The answer was in front of eyes only. This answer is completely due to Martin's comments. I'm writing this answer as it requires the concept of Wright convex.
A function $f:[a,b] \mapsto \mathbb{R}$ is said to be Wright convex if, for every $\delta >0$ and $x \leq y$ with $x$ and $y+\delta \in [a,b]$ we have $f(x+h) - f(x) \leq f(y+h) - f(y)$
$log f(x)$ is convex $\iff$ $log f(x)$ is Wright convex as $f$ is continuous (Due to Jensen). i.e., for $ x \leq y,$ $\log f(x+h) - \log f(x) \leq \log f(y+h) - \log f(y) \iff \frac{f(x+h)}{f(x)} \leq \frac{f(y+h)}{f(y)}.$ The hypothesis constant sign says $\frac{f(x+h)}{f(x)}$ is positive which allowed us writing $\log f(x+h) - \log f(x) = \log \frac{f(x+h)}{f(x)}.$
For the result of Jensen, see CONVEX FUNCTIONS AND THEIR APPLICATIONS A contemporary approach ( by Constantin P. Niculescu, Lars-Erik Persson) Theorem 1.1.4. or J. L. W. V. Jensen, Sur les fonctions convexes et les inegalités entreles valeurs moyennes, Acta Math., 30 (1906), 175-193.