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Suppose $f(x):R_+\rightarrow[0,1]$ such that $f'(x)>0$. There exists a function $H(x)=\frac{f(x)}{x}$. We know $H'(x)<0$ as long as $x\frac{f'(x)}{f(x)}<1$. Which general conditions function $f(x)$ need to satisfy for this to be true?

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  • $\begingroup$ what do you want exactly to be true ? $\endgroup$ – Fardad Pouran Apr 11 '15 at 19:54
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    $\begingroup$ @FardadPouran The OP is asking if $f: \Bbb R_{+} \to [0,1]$, what properties must $f$ satisfy for $x \frac{f'(x)}{f(x)} < 1$ to hold? I guess we are assuming $f$ is differentiable and nonzero. $\endgroup$ – layman Apr 11 '15 at 19:57
  • $\begingroup$ Yes $f$ is differentiable and nonzero. $\endgroup$ – etoposido Apr 12 '15 at 21:03
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$H'(x)<0 \rightarrow xf'(x)<f(x)$

Hence, $H'(x)<0 \iff \ln(f(x))<\ln(x)$ , which we arrive at by solving the differential equation yielded and, in turn, implies that $f(x)<x$

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