Bhayo, Barkat Ali; Yin, Li Logarithmic mean inequality for generalized trigonometric and hyperbolic functions. (English) Zbl 1315.26036 Acta Univ. Sapientiae, Math. 6, No. 2, 135-145 (2014). Summary: In this paper we study the convexity and concavity properties of generalized trigonometric and hyperbolic functions in case of logarithmic mean. Cited in 4 Documents MSC: 26E60 Means 26D07 Inequalities involving other types of functions 33B10 Exponential and trigonometric functions Keywords:logarithmic mean; generalized trigonometric and hyperbolic functions; inequalities; generalized convexity PDFBibTeX XMLCite \textit{B. A. Bhayo} and \textit{L. Yin}, Acta Univ. Sapientiae, Math. 6, No. 2, 135--145 (2014; Zbl 1315.26036) Full Text: DOI arXiv OA License References: [1] M. Abramowitz, I. Stegun, eds., Handbook of mathematical functions with formulas, graphs and mathematical tables, National Bureau of Standards, Dover, New York, 1965.; [2] G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen, Genenalized convexity and inequalities, J. Math. Anal. Appl. 335 (2007), 1294{1308.;} · Zbl 1125.26017 [3] H. Alzer, S.-L Qiu, Inequalities for means in two variables, Arch. Math. 80 (2003), 201{205.;} · Zbl 1020.26011 [4] Á. Baricz, Geometrically concave univariate distributions, J. Math. Anal. Appl. 363 (1) (2010), 182{196.;} · Zbl 1185.60011 [5] Á. Baricz, B. A. Bhayo, R. Kl_en, Convexity properties of generalized trigonometric and hyperbolic functions, Aequat. Math. DOI: 10.1007/s00010-013-0222-x.; · Zbl 1394.33006 [6] Á. Baricz, B. A. Bhayo, M. Vuorinen, Tur_an type inequalities for generalized inverse trigonometric functions, available online at http://arxiv. org/abs/1209.1696.; · Zbl 1474.33001 [7] B. A. Bhayo, Power mean inequality of generalized trigonometric functions, Mat. Vesnik, (to appear) http://mv.mi.sanu.ac.rs/Papers/ MV2013_033.pdf.; · Zbl 1474.33002 [8] B. A. Bhayo, M. Vuorinen, On generalized trigonometric functions with two parameters, J. Approx. Theory, 164 (10) (2012),1415{1426.;} · Zbl 1257.33052 [9] B. A. Bhayo, M. Vuorinen, Inequalities for eigenfunctions of the p- Laplacian, Issues of Analysis 2 (20), No 1, (2013), http://arxiv.org/ abs/1101.3911; · Zbl 1294.33001 [10] P. J. Bushell, D. E. Edmunds, Remarks on generalised trigonometric functions, Rocky Mountain J. Math., 42 (1) (2012), 25{57.;} · Zbl 1246.33001 [11] B. C. Carlson, Some inequalities for hypergeometric functions, Proc. Amer. Math. Soc., 17 (1), (1966), 32{39.;} · Zbl 0137.26803 [12] P. Drábek, R. Man_asevich, On the closed solution to some p-Laplacian nonhomogeneous eigenvalue problems, Di_. and Int. Eqns., 12 (1999), 723{740.;} [13] D. E. Edmunds, P. Gurka, J. Lang, Properties of generalized trigonometric functions, J. Approx. Theory, 164 (2012) 47{56, doi:10.1016/j.jat.2011.09.004.;} · Zbl 1241.42019 [14] W.-D. Jiang, M.-K. Wang, Y.-M. Chu, Y.-P. Jiang, F. Qi,Convexity of the generalized sine function and the generalized hyperbolic sine function, J. Approx. Theory, 174 (2013), 1{9.;} · Zbl 1292.33003 [15] D. B. Karp, E. G. Prilepkina, Parameter convexity and concavity of generalized trigonometric functions, arXiv:1402.3357[math.CA]; · Zbl 1298.26037 [16] R. Klén, M. Visuri, M. Vuorinen, On Jordan type inequalities for hyperbolic functions, J. Ineq. Appl., vol. 2010, pp. 14.; · Zbl 1211.26015 [17] J.-C. Kuang, Applied inequalities (Second edition), Shan Dong Science and Technology Press. Jinan, 2002.; [18] R. Klén, M. Vuorinen, X.-H. Zhang, Inequalities for the generalized trigonometric and hyperbolic functions, J. Math. Anal. Appl., 409 (1) (2014), 521-29.; · Zbl 1306.33013 [19] P. Lindqvist, Some remarkable sine and cosine functions, Ricerche di Matematica, Vol. XLIV (1995), 269{290.;} · Zbl 0944.33002 [20] D. S. Mitrinović, Analytic Inequalities, Springer, New York, USA, 1970.; · Zbl 0199.38101 [21] E. Neuman, J. S_andor, Optimal inequalities for hyperbolic and trigonometric functions, Bull. Math. Anal. Appl, 3(3), (2011), 177{181. http://www.emis.de/journals/BMAA/repository/docs/BMAA3_3_20.pdf.;} · Zbl 1314.26019 [22] F. Qi, Z. Huang, Inequalities of the complete elliptic integrals, Tamkang J. Math, 29 (3) (1998), 165{169.;} · Zbl 0912.33012 [23] S. Takeuchi, Generalized Jacobian elliptic functions and their application to bifurcation problems associated with p-Laplacian, J. Math. Anal. Appl. 385 (2012) 24{35.;} · Zbl 1232.34029 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.