|
===Inequalities===
Below are several inequalities involving hyperbolic functions, arranged from more conceptually interesting to more basic, simplified inequalities. References are preserved in the original format. Some inequalities that relate hyperbolic functions to the exponential function or provide simpler upper bounds are grouped together.
Below are several notable inequalities involving hyperbolic functions:
1. '''Inequality involvingCusa-type hyperbolic cosine and sineinequality:'''
<math display="block">\cosh(x) \leq \left(\frac{\operatorname{sinh}(x)}{x} > \rightoperatorname{cosh}(x)^3, \quad x > 0.</math>
This relatesgives thea hyperbolic cosineanalogue andof sinethe throughclassical aCusa-Huygens cubic expressioninequality. <ref>{{cite webjournal |last1=Zhu |first1=Ling |date=2010 |title=Inequalities for Hyperbolic functionsFunctions and Their Applications |journal=Journal of Inequalities and Applications |volume=2010 |workpage=Wikipedia130821 |url=https://enjournalofinequalitiesandapplications.wikipediaspringeropen.com/articles/10.org1155/wiki2010/Hyperbolic_functions130821}}</ref>
2. '''Inequality involvingWilker-type hyperbolic tangent and sineinequality:'''
<math display="block">\sinleft(x) \cosfrac{\operatorname{sinh}(x)}{x} <\right)^2 + \frac{\operatorname{tanh}(x)}{x} <> x2, \quad x > 0.</math>
This is the Thishyperbolic showsanalogue thatof \(\sinh(x)\)Wilker’s isinequality, boundedgiving abovea bytight anlower exponential-basedbound functionthat mirrors classical results in trigonometry. <ref>{{cite journal |last1=Zhu |first1=Ling |date= 20122010 |title= New inequalitiesInequalities for hyperbolicHyperbolic functionsFunctions and theirTheir applicationsApplications |journal=Journal of Inequalities and Applications |volume= 20122010 |page= 303130821 |url=https://journalofinequalitiesandapplications.springeropen.com/articles/10. 11861155/2010/ 1029-242X-2012-303130821}}</ref> ▼
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 involvingShafer-Fink-type hyperbolic cosine differenceinequality:'''
<math display="block">|\,\coshoperatorname{tanh}(x) -> \cosh(y)\,|frac{2x}{2 \geq+ |x^2}, - y|\sinh(x)\sinh(y),\quad x > 0,\; y > 0.</math>
This givesprovides a neat rational lower bound for the absolute difference of hyperbolic cosines in terms of hyperbolic sinestangent. <ref>{{cite webjournal |last1=Zhu |first1=Ling |date=2010 |title=Inequalities for Hyperbolic functionsFunctions and Their Applications |journal=Journal of Inequalities and Applications |volume=2010 |workpage=Wikipedia130821 |url=https://enjournalofinequalitiesandapplications.wikipediaspringeropen.com/articles/10.org1155/wiki2010/Hyperbolic_functions130821}}</ref>
4. '''Arctangent–hyperbolic tangent comparison:'''
4. '''Inequality involving arctangent and hyperbolic tangent:'''
<math display="block">\operatorname{arctan}(x) \leq \frac{\pi}{2} \cdot \operatorname{tanh}(x), \quad x \geq 0.</math>
This relatescompares thean arctangentinverse trigonometric function towith thea scaled hyperbolic tangent function, scaled by \(\frac{\pi}{2}\). <ref>{{cite web |title=§4.32 Inequalities ‣ Hyperbolic functionsFunctions |work=WikipediaNIST Digital Library of Mathematical Functions |url=https://endlmf.wikipedianist.orggov/wiki/Hyperbolic_functions4.32}}</ref>
5. '''Cusa-typeHyperbolic inequalitycosine fordifference hyperbolic functionsinequality:'''
<math display="block">|\fracoperatorname{\sinhcosh}(x)} - \operatorname{xcosh}(y)| >\geq |x - y| \coshcdot \operatorname{sinh}(x) \cdot \operatorname{sinh}(y), \quad x > 0, \; y > 0.</math>
This gives a Thisnontrivial islower abound hyperbolicon analoguethe difference of thehyperbolic classicalcosines Cusa-Huygensusing inequalityhyperbolic sines. <ref>{{cite journalweb |last1title=Zhu§4.32 |first1=Ling |date=2010 |title=Inequalities for‣ Hyperbolic Functions and Their Applications |journalwork=JournalNIST ofDigital InequalitiesLibrary andof ApplicationsMathematical |volume=2010 |page=130821Functions |url=https://journalofinequalitiesandapplicationsdlmf.springeropennist.comgov/articles/104.1155/2010/13082132}}</ref>
6. '''Wilker-typeDouble inequality forbounding hyperbolic functionstangent:'''
<math display="block">\leftsin(x) \frac{cdot \sinhcos(x)}{x}\right)^2 +< \fracoperatorname{\tanh}(x)}{x} >< 2x, \quad x > 0.</math>
This places the Thishyperbolic istangent anbetween analoguea oftrigonometric Wilker'sproduct inequalityand fora thelinear hyperbolicfunction. case.<ref>{{cite journalweb |last1title=Zhu§4.32 |first1=Ling |date=2010 |title=Inequalities for‣ Hyperbolic Functions and Their Applications |journalwork=JournalNIST ofDigital InequalitiesLibrary andof ApplicationsMathematical |volume=2010 |page=130821Functions |url=https://journalofinequalitiesandapplicationsdlmf.springeropennist.comgov/articles/104.1155/2010/13082132}}</ref>
7. '''ShaferHyperbolic cosine-Fink-typesine cubic inequality for hyperbolic functions:'''
<math display="block">\tanhoperatorname{cosh}(x) >\leq \left( \frac{2x\operatorname{sinh}(x)}{2 + x^2} \right)^3, \quad x > 0.</math>
This relates hyperbolic Thiscosine providesto a lowercubic boundexpression for thein hyperbolic tangentsine. function.<ref>{{cite journalweb |last1title=Zhu§4.32 |first1=Ling |date=2010 |title=Inequalities for‣ Hyperbolic Functions and Their Applications |journalwork=JournalNIST ofDigital InequalitiesLibrary andof ApplicationsMathematical |volume=2010 |page=130821Functions |url=https://journalofinequalitiesandapplicationsdlmf.springeropennist.comgov/articles/104.1155/2010/13082132}}</ref>
8.===Further '''Inequalityinequalities involvingcomparing hyperbolic sine and the exponential function:'''functions, and simple bounds===
<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>
8. '''Hyperbolic sine-exponential upper bound:'''
9. '''Inequality involving hyperbolic cosine and the exponential function:'''
<math display="block">\coshoperatorname{sinh}(x) < \frac{e^x +- 1}{2}, \quad x > 0.</math>
This providesbounds anhyperbolic uppersine boundabove forby \(\cosh(x)\)a in terms of thesimple 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. '''Hyperbolic cosine-exponential upper bound:'''
10. '''Inequality involving hyperbolic sine and cosine:'''
<math display="block">\sinhoperatorname{cosh}(x) < x\cosh(frac{e^x) + 1}{2}, \quad x > 0.</math>
This showsbounds ahyperbolic relationshipcosine betweenabove \(\sinh(x)\)by andanother \(\cosh(x)\),simple comparingexponential-based thefunction. 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>
10. '''Hyperbolic sine-cosh linear inequality:'''
<math display="block">\operatorname{sinh}(x)/x < \cdot \operatorname{cosh}(x).</math>
This shows that the hyperbolic sine grows slower than a linear-cosh combination. <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==
| 