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Assume that $|f'(x)|\leq Cx^{-1-\epsilon}$, where $x\in(1,\infty)$. I am trying to find the minimal conditions for the following to hold

$$|f(x)|\leq C_1x^{-\epsilon}+C_2,$$

where $\epsilon>0$, $x\in(1,\infty)$.

For now I require that $f'(x)$ was absolutely continious. However, maybe there is a way to weaken the conditions?

TheGrandDuke
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  • Note that you can “integrate” the inequality without additional conditions on the derivative, compare https://math.stackexchange.com/q/3776825/42969. – Martin R Jan 18 '21 at 13:09
  • Fundamental theorem of calculus applies to fixed bounds $[a,b]$. Here upper bound is $x$. – TheGrandDuke Jan 18 '21 at 13:16
  • You can apply it to $[1, x]$. – Martin R Jan 18 '21 at 13:18
  • $x$ is not fixed. – TheGrandDuke Jan 18 '21 at 13:22
  • For $1 < x_0 < x$ you have $|f(x) - f(x_0)| \le \int_{x_0}^x C t^{-1-\epsilon} dt = \frac{C}{\epsilon}(1- x^{-\epsilon})$. Is that what you are looking for? – My point is that you don't need the derivative to be absolutely continuous for such an estimate. – Martin R Jan 18 '21 at 13:25

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