Hyperbolic functions: Difference between revisions

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===Inequalities===

Below are several notable inequalities involving hyperbolic functions:

The following inequality is useful in statistics:<ref>{{cite news |last1=Audibert |first1=Jean-Yves |date=2009 |title=Fast learning rates in statistical inference through aggregation |publisher=The Annals of Statistics |page=1627}} [https://projecteuclid.org/download/pdfview_1/euclid.aos/1245332827]</ref>

<math display="block">\operatorname{cosh}(t) \leq e^{t^2 /2}.</math>

1. '''Inequality involving hyperbolic cosine and sine:'''

It can be proved by comparing the Taylor series of the two functions term by term.

<math>\cosh(x) \leq \left(\frac{\sinh(x)}{x}\right)^3,\quad x > 0.</math>

This relates the hyperbolic cosine and sine through a cubic expression.<ref>{{cite web |title=Hyperbolic functions |work=Wikipedia |url=https://en.wikipedia.org/wiki/Hyperbolic_functions}}</ref>

2. '''Inequality involving hyperbolic tangent and sine:'''

<math>\sin(x)\cos(x) < \tanh(x) < x,\quad x > 0.</math>

This provides bounds for the hyperbolic tangent function in terms of trigonometric and linear functions.<ref>{{cite web |title=Hyperbolic functions |work=Wikipedia |url=https://en.wikipedia.org/wiki/Hyperbolic_functions}}</ref>

3. '''Inequality involving hyperbolic cosine difference:'''

<math>|\,\cosh(x) - \cosh(y)\,| \geq |x - y|\sinh(x)\sinh(y),\quad x > 0,\; y > 0.</math>

This gives a lower bound for the absolute difference of hyperbolic cosines in terms of hyperbolic sines.<ref>{{cite web |title=Hyperbolic functions |work=Wikipedia |url=https://en.wikipedia.org/wiki/Hyperbolic_functions}}</ref>

4. '''Inequality involving arctangent and hyperbolic tangent:'''

<math>\arctan(x) \leq \frac{\pi}{2}\tanh(x),\quad x \geq 0.</math>

This relates the arctangent function to the hyperbolic tangent function, scaled by \(\frac{\pi}{2}\).<ref>{{cite web |title=Hyperbolic functions |work=Wikipedia |url=https://en.wikipedia.org/wiki/Hyperbolic_functions}}</ref>

5. '''Cusa-type inequality for hyperbolic functions:'''

<math>\frac{\sinh(x)}{x} > \cosh(x),\quad x > 0.</math>

This is a hyperbolic analogue of the classical Cusa-Huygens inequality.<ref>{{cite journal |last1=Zhu |first1=Ling |date=2010 |title=Inequalities for Hyperbolic Functions and Their Applications |journal=Journal of Inequalities and Applications |volume=2010 |page=130821 |url=https://journalofinequalitiesandapplications.springeropen.com/articles/10.1155/2010/130821}}</ref>

6. '''Wilker-type inequality for hyperbolic functions:'''

<math>\left(\frac{\sinh(x)}{x}\right)^2 + \frac{\tanh(x)}{x} > 2,\quad x > 0.</math>

This is an analogue of Wilker's inequality for the hyperbolic case.<ref>{{cite journal |last1=Zhu |first1=Ling |date=2010 |title=Inequalities for Hyperbolic Functions and Their Applications |journal=Journal of Inequalities and Applications |volume=2010 |page=130821 |url=https://journalofinequalitiesandapplications.springeropen.com/articles/10.1155/2010/130821}}</ref>

7. '''Shafer-Fink-type inequality for hyperbolic functions:'''

<math>\tanh(x) > \frac{2x}{2 + x^2},\quad x > 0.</math>

This provides a lower bound for the hyperbolic tangent function.<ref>{{cite journal |last1=Zhu |first1=Ling |date=2010 |title=Inequalities for Hyperbolic Functions and Their Applications |journal=Journal of Inequalities and Applications |volume=2010 |page=130821 |url=https://journalofinequalitiesandapplications.springeropen.com/articles/10.1155/2010/130821}}</ref>

8. '''Inequality involving hyperbolic sine and the exponential function:'''

<math>\sinh(x) < \frac{e^x - 1}{2},\quad x > 0.</math>

This shows that \(\sinh(x)\) is bounded above by an exponential-based function.<ref>{{cite journal |last1=Zhu |first1=Ling |date=2012 |title=New inequalities for hyperbolic functions and their applications |journal=Journal of Inequalities and Applications |volume=2012 |page=303 |url=https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/1029-242X-2012-303}}</ref>

9. '''Inequality involving hyperbolic cosine and the exponential function:'''

<math>\cosh(x) < \frac{e^x + 1}{2},\quad x > 0.</math>

This provides an upper bound for \(\cosh(x)\) in terms of the exponential function.<ref>{{cite journal |last1=Zhu |first1=Ling |date=2012 |title=New inequalities for hyperbolic functions and their applications |journal=Journal of Inequalities and Applications |volume=2012 |page=303 |url=https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/1029-242X-2012-303}}</ref>

10. '''Inequality involving hyperbolic sine and cosine:'''

<math>\sinh(x) < x\cosh(x),\quad x > 0.</math>

This shows a relationship between \(\sinh(x)\) and \(\cosh(x)\), comparing the former to the product of \(x\) and \(\cosh(x)\).<ref>{{cite journal |last1=Zhu |first1=Ling |date=2012 |title=New inequalities for hyperbolic functions and their applications |journal=Journal of Inequalities and Applications |volume=2012 |page=303 |url=https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/1029-242X-2012-303}}</ref>

==Inverse functions as logarithms==