When solving a problem, I came to a need of having a function $f$ satisfying the following properties:
(a) $f:\left(1,\infty\right)\rightarrow\left(0,\infty\right)$ is a $C^{2}$ function,
(b) $f'>0$ on $\left(1,\infty\right)$ and $\lim_{x\rightarrow\infty}f(x)=\infty$,
(c) $\dfrac{\left(f'(x)\right)^{2}-f''(x)}{f'(x)}\leq\dfrac{1}{x}$ on $\left(1,\infty\right).$
Anyone has an idea to show one? Thank you.
Note that: if $f(x) = \ln(x)$ then $\dfrac{\left(f'(x)\right)^{2}-f''(x)}{f'(x)} = \dfrac{2}{x}$, which is not satisfied but almost!
Potentially useful: Note that since $$ \frac{\left( f'(x) \right)^2 - f''(x)}{f'(x)} = \frac{\mathrm{d}}{\mathrm{d}x} \left( f(x) - \ln f'(x) \right) \text{,} $$ we may instead consider $$ f(x) - \ln f'(x) \leq \ln x + C \text{,} $$ for some constant $C$. So $$ f(x) \leq \ln( x f'(x) ) + C \text{.} $$
(I'd keep at this, but I have a new rule about not trying to post correct math after midnight.)