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Yesterday I found the question https://math.stackexchange.com/questions/95784/a-question-about-the-relation-between-two-classes-of-functions which is strongly related to my former question Example of a special function. It's funny because Mark's question (unfortunately there weren't any answers for his question) is quite similar to the one I would like to ask now.
I strongly believe that for any continuous function $f$ which is:
1. positive and strictly increasing in $(0,a)$ for some $a>0$,
2. $f(0)=0$,
3. $\lim\limits_{x\rightarrow0^{+}}\frac{x}{f(x)}=0$
4. there doesn't exist an interval $(0,b)$, $b>0$, such that the map $x\rightarrow\frac{x}{f(x)}$ is strictly increasing for all $x\in(0,b)$,
one can always construct the continuous function $g$ satisfying $f>g$ in the interval $(0,a)$ for which $\lim\limits_{x\rightarrow0^{+}}\frac{f(x)}{g(x)}=1$, $g$ satisfies $1.$-$3.$ and there exists some interval $(0,\alpha)$, $\alpha>0$, such that the map $x\rightarrow\frac{x}{g(x)}$ is strictly increasing for all $x\in(0,\alpha)$.
I found the form of such functions $g$ for the functions $f$ given by Mr. Piau and Mr. Nicolas as the answers to my question Example of a special function but I couldn't find a general method to proof my hypothesis. Could someone give me some hints or maybe even counterexamples if my hypothesis is wrong?
Thank you in advance for your help!

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    Always glad to help! :)2012-01-12

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