The following inequality
$$(*) \quad \frac{x}{\sinh(x)} \leq e^{-x}, \quad \forall x>0$$ or $$(**) \quad \frac{x}{\sinh(x)} \leq e^{x}, \quad \forall x>0$$ are true ?
thank you in advance
The inequality is true?
0
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calculus
real-analysis
inequality
hyperbolic-functions
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1The latter is true. – 2017-02-05
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0Ok thanks @ Akiva Weinberger – 2017-02-05
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0$$\frac{x}{\sinh x}\leq\frac{6}{6+x^2}$$ is true. – 2017-02-05
2 Answers
3
Seeing as you provide no context, I'll simply answer the question: no, it's not. Try substituting $x=1$ and the inequality won't be satisfied.
https://www.wolframalpha.com/input/?i=1+%2F+(sinh(1))+-+e%5E(-1)
1
If $x>0$ and $x/\sinh x\leq e^{-x}$ then $x\leq (1-e^{-2x})/2<1/ 2,$ but this is clearly false when $x\geq 1/2.$