×
Miyakawa, Hiroki; Takeuchi, Shingo

Applications of a duality between generalized trigonometric and hyperbolic functions. II. (English) Zbl 1529.33003

J. Math. Inequal. 16, No. 4, 1571-1585 (2022).
Summary: Generalized trigonometric functions and generalized hyperbolic functions can be converted to each other by the duality formulas previously discovered by the authors. In this paper, we apply the duality formulas to prove dual pairs of Wilker-type inequalities, Huygens-type inequalities, and (relaxed) Cusa-Huygens-type inequalities for the generalized functions. In addition, multiple- and double-angle formulas not previously obtained are also given.
For Part I, see [the authors, J. Math. Anal. Appl. 502, No. 1, Article ID 125241, 17 p. (2021; Zbl 1472.33002)].

MSC:

33B10 Exponential and trigonometric functions
26D05 Inequalities for trigonometric functions and polynomials
26D07 Inequalities involving other types of functions
31C45 Other generalizations (nonlinear potential theory, etc.)
34A34 Nonlinear ordinary differential equations and systems

Keywords:

generalized trigonometric function; generalized hyperbolic function; Mitrinović-Adamović inequality; Wilker inequality; Huygens inequality; Cusa-Huygens inequality; multiple-angle formula; double-angle formula; \(p\)-Laplacian

Citations:

Zbl 1472.33002
Cite Review PDF
Full Text: DOI arXiv

References:

[1] DAVIDA. COX ANDJERRYSHURMAN,Geometry and number theory on clovers, Amer. Math. Monthly112(2005), no. 8, 682-704,doi:10.2307/30037571. · Zbl 1107.51007
[2] A. C. DIXON,On the doubly periodic functions arising out of the curve x3+y3−3αxy=1 , The Quarterly Journal of Pure and Applied Mathematics24(1890), 167-233. · JFM 21.0490.01
[3] ONDREJˇDOˇSLY AND´PAVELRˇEHAK´,Half-linear differential equations, North-Holland Mathematics Studies, vol. 202, Elsevier Science B.V., Amsterdam, 2005. · Zbl 1090.34001
[4] DAVIDE. EDMUNDS, PETRGURKA,ANDJANLANG,Properties of generalized trigonometric functions, J. Approx. Theory164(2012), no. 1, 47-56,doi:10.1016/j.jat.2011.09.004. · Zbl 1241.42019
[5] RIKUKLEN´, MATTIVUORINEN,ANDXIAOHUIZHANG,Inequalities for the generalized trigonometric and hyperbolic functions, J. Math. Anal. Appl.409(2014), no. 1, 521-529. · Zbl 1306.33013
[6] JANLANG ANDDAVIDEDMUNDS,Eigenvalues, embeddings and generalised trigonometric functions, Lecture Notes in Mathematics2016, Springer, Heidelberg, 2011. · Zbl 1220.47001
[7] XIAOYANMA, XIANGBINSI, GENHONGZHONG,ANDJIANHUIHE,Inequalities for the generalized trigonometric and hyperbolic functions, Open Math.18(2020), no. 1, 1580-1589, doi:10.1515/math-2020-0096. · Zbl 1481.26014
[8] HIROKIMIYAKAWA ANDSHINGOTAKEUCHI,Applications of a duality between generalized trigonometric and hyperbolic functions, J. Math. Anal. Appl.502(2021), no. 1, Paper No. 125241, 17, doi:10.1016/j.jmaa.2021.125241. · Zbl 1472.33002
[9] EDWARDNEUMAN,Inequalities for the generalized trigonometric, hyperbolic and Jacobian elliptic functions, J. Math. Inequal.9(2015), no. 3, 709-726,doi:10.7153/jmi-09-59. · Zbl 1333.26015
[10] SHOTASATO ANDSHINGOTAKEUCHI,Two double-angle formulas of generalized trigonometric functions, J. Approx. Theory250(2020), 105322, 5,doi:10.1016/j.jat.2019.105322. · Zbl 1431.33022
[11] KAZUNORISHINOHARA,Addition formulas of leaf functions according to integral root of polynomial based on analogies of inverse trigonometric functions and inverse lemniscate functions, Applied Mathematical Sciences11(2017), no. 52, 2561-2577.
[12] SHINGOTAKEUCHI,Multiple-angle formulas of generalized trigonometric functions with two parameters, J. Math. Anal. Appl.444(2016), no. 2, 1000-1014,doi:10.1016/j.jmaa.2016.06.074. · Zbl 1350.33040
[13] SHINGOTAKEUCHI,Some double-angle formulas related to a generalized lemniscate function, Ramanujan J.56(2021), no. 2, 753-761,doi:10.1007/s11139-021-00395-x. · Zbl 1502.33011
[14] LIYIN, LIGUOHUANG,ANDFENGQI,Some inequalities for the generalized trigonometric and hyperbolic functions, Thurkish Journal of Analysis and Number Theory2(2014), no. 3, 96-101, doi:10.12691/tjant-2-3-8.
[15] LIYIN, LIGUOHUANG, YONGLIWANG,ANDXIULILIN,A survey for generalized trigonometric and hyperbolic functions, Journal of Mathematical Inequalities13(2019), no. 3, 833-854 · Zbl 1425.33001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.