A note on generalized trigonometric and hyperbolic functions
2014
In this paper, we prove that the function x → log (x/sinp(x))/ log (sinhp(x)/x) (p ∈ [2,∞)) is strictly increasing on (0,πp/2) , where πp/2 = ∫ 1 0 (1− t p)−1/pdt , and sinp(x) and sinhp(x) denote the generalized trigonometric sine and generalized hyperbolic sine functions, respectively. As application, a conjecture due to Klen, Vuorinen and Zhang [J. Math. Anal. Appl. 409 (2014), 521–529] is proved, and the best positive constants α and β such that ( sinhp(x) x )α < x sinp(x) < ( sinhp(x) x )β
Keywords:
- Mathematical analysis
- Mathematics
- Integration using Euler's formula
- Pythagorean trigonometric identity
- Hyperbolic angle
- Trigonometric integral
- Differentiation of trigonometric functions
- Inverse hyperbolic function
- Proofs of trigonometric identities
- Trigonometric substitution
- Hyperbolic function
- Trigonometric interpolation
- Combinatorics
- Trigonometric polynomial
- Correction
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