Showing content from https://dlmf.nist.gov/4.32 below:
§4.32 Inequalities ⣠Hyperbolic Functions ⣠Chapter 4 Elementary Functions
§4.32 Inequalities â
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Keywords:
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hyperbolic functions, inequalities
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http://dlmf.nist.gov/4.32
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Annotations for Ch.4
For x real,
4.32.1 cosh â¡ x ⤠( sinh â¡ x x ) 3 , â
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Symbols:
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cosh â¡ z : hyperbolic cosine function, sinh â¡ z : hyperbolic sine function and x : real variable
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http://dlmf.nist.gov/4.32.E1
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Annotations for §4.32 and Ch.4
4.32.2 sin â¡ x ⢠cos â¡ x < tanh â¡ x < x , x > 0 , â
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cos â¡ z : cosine function, tanh â¡ z : hyperbolic tangent function, sin â¡ z : sine function and x : real variable
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http://dlmf.nist.gov/4.32.E2
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Annotations for §4.32 and Ch.4
4.32.3 | cosh â¡ x â cosh â¡ y | ⥠| x â y | ⢠sinh â¡ x ⢠sinh â¡ y , x > 0 , y > 0 , â
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cosh â¡ z : hyperbolic cosine function, sinh â¡ z : hyperbolic sine function, x : real variable and y : real variable
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http://dlmf.nist.gov/4.32.E3
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Annotations for §4.32 and Ch.4
4.32.4 arctan â¡ x ⤠1 2 â¢ Ï â¢ tanh â¡ x , x ⥠0 . â
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Ï : the ratio of the circumference of a circle to its diameter, tanh â¡ z : hyperbolic tangent function, arctan â¡ z : arctangent function and x : real variable
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http://dlmf.nist.gov/4.32.E4
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Annotations for §4.32 and Ch.4
For these and other inequalities involving hyperbolic functions see MitrinoviÄ (1964, pp. 61, 76, 159) and MitrinoviÄ (1970, p. 270).
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