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4 Elementary FunctionsHyperbolic Functions

§4.32 Inequalities

For x real,

4.32.1 coshx(sinhxx)3,
4.32.2 sinxcosx<tanhx<x,
x>0,
4.32.3 |coshx-coshy||x-y|sinhxsinhy,
x>0, y>0,
4.32.4 arctanx12πtanhx,
x0.

For these and other inequalities involving hyperbolic functions see Mitrinović (1964, pp. 61, 76, 159) and Mitrinović (1970, p. 270).