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!
Some hypothesis about the specific functions
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real-analysis
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0Always glad to help! :) – 2012-01-12