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I'm looking for a continuous, strictly increasing, strictly convex function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$, with $f(0)=0$, and such that

$$ \lim_{x \rightarrow\infty} \frac{f(x)}{x} \leq c $$

for some $c \in \mathbb{R}_\geq 0$.

Suggestions?

  • 2
    $f(x) = c\cdot x$ or what am I missing?2012-05-12
  • 1
    Are you looking for a *strictly* convex function?2012-05-12
  • 0
    Sorry, something is missing. Yes, I mean a strictly convex function.2012-05-12
  • 1
    $f(x)=x+1-\sqrt{x+1}$.2012-05-12

2 Answers 2