Parameter convexity and concavity of generalized trigonometric functions

We study the convexity properties of the generalized trigonometric functions viewed as functions of the parameter. We show that p→sinp⁡(y) and p→cosp⁡(y) are log-concave on the appropriate intervals while p→tanp⁡(y) is log-convex. We also prove similar facts about the generalized hyperbolic function...

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Veröffentlicht in:Journal of mathematical analysis and applications Jg. 421; H. 1; S. 370 - 382
Hauptverfasser: Karp, D.B., Prilepkina, E.G.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 01.01.2015
Schlagworte:
Eigenfunctions of p-Laplacian
Generalized hyperbolic functions
Generalized trigonometric functions
Log-concavity
Log-convexity
Turán type inequality
Log-concavity
Generalized trigonometric functions
Generalized hyperbolic functions
Eigenfunctions of p-Laplacian
Turán type inequality
Log-convexity
ISSN:0022-247X, 1096-0813
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Beschreibung
Zusammenfassung:We study the convexity properties of the generalized trigonometric functions viewed as functions of the parameter. We show that p→sinp⁡(y) and p→cosp⁡(y) are log-concave on the appropriate intervals while p→tanp⁡(y) is log-convex. We also prove similar facts about the generalized hyperbolic functions. In particular, our results settle a major part of the conjecture recently put forward in [4].
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2014.07.017