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Let $f,g:(0,\infty)\rightarrow (0,\infty)$ be right-continuous decreasing functions. Assume that $$\frac{\inf_{s>0} s+t \cdot f(s) }{\inf_{s>0}s+t\cdot g(s)}<1$$ for all $t>0$. Find a positive number $C$ such that $$f(s)<C\cdot g(s/C)$$ for all $s$.

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  • $\begingroup$ What are the infima being taken over? The entire numerator and the denominator? $\endgroup$
    – CJ Dowd
    Aug 6, 2017 at 3:22
  • $\begingroup$ @CJDowd There are two infimas. $\inf_{s>0} (s+t \cdot f(s)) $ and $\inf_{s>0}(s+t\cdot g(s))$. $\endgroup$
    – user92646
    Aug 6, 2017 at 5:35

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