Proof of an inequality involiving hyperbolic function

Question

How can I get a proof of the following inequality: $$\frac{x}{\sinh^{-1}(x)}\lt\frac{\sinh(x)}{x}?$$ for $x\gt 0$ Thanks

score 1 · Accepted Answer · answered Oct 04 '13 at 15:21

1

Rewrite it as $\frac{\sinh(y)}{y}<\frac{\sinh(x)}{x}$ for $y=\sinh^{-1}(x)$

answered Oct 04 '13 at 15:21

Empy2

50,853

score 0 · Answer 2 · answered Oct 04 '13 at 15:20

You can bring everything to the left side and make common denominators. Replace the hyperbolic functions with their equivalencies, i.e. the e-power representation and the ln term for the hyperbolic sine and its inverse respectively. Since x>0 all original terms will be positive, so your inequality sign won't change. Now check out the numerator. It contains an x² term as well as an e-power, subject to a minus sign. If x goes to infinity, who is going to win? The quadratic term or the e-power? Oh yeah, the e-power is of course also subject to a multiplication with that ln term. But if x goes to infinity, what happens to that ln term? Can you finish it from here?

Proof of an inequality involiving hyperbolic function

2 Answers2