Abstract
We study the power mean inequality for generalized trigonometric and hyperbolic functions with two parameters. The generalized p-trigonometric and (p, q)-trigonometric functions were introduced by Lindqvist and Takeuchi, respectively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I., eds.: Handbook of mathematical functions with formulas, graphs and mathematical tables. National Bureau of Standards (Russian translation, Nauka 1979) (1964)
Anderson G.D., Vamanamurthy M.K., Vuorinen M.: Conformal invariants, inequalities and quasiconformal maps. Wiley, New York (1997)
Baricz Á.: Geometrically concave univariate distributions. J. Math. Anal. Appl. 363(1), 182–196 (2010)
Baricz, Á., Bhayo, B.A., Vuorinen, M.: Turán type inequalities for generalized inverse trigonometric functions (arXiv:1305.0938)
Bhayo, B.A., Vuorinen, M.: Inequalities for eigenfunctions of the p-Laplacian (arXiv.1101.3911)
Bhayo B.A., Vuorinen M.: On generalized trigonometric functions with two parameters. J. Approx. Theory 164, 1415–1426 (2012)
Bhayo, B.A., Vuorinen, M.: Power mean inequality of generalized trigonometric functions (arXiv.1209.0873)
Biezuner R.J., Ercole G., Martins E.M.: Computing the first eigenvalue of the p-Laplacian via the inverse power method. J. Funct. Anal. 257, 243–270 (2009)
Bushell P.J., Edmunds D.E.: Remarks on generalised trigonometric functions. Rocky Mt. J. Math. 42, 25–57 (2012)
Chu, Y.-M., Jiang, Y.-P., Wang, M.-K.: Inequalities for generalized trigonometric and hyperbolic sine functions (arXiv.1212.4681)
Drábek P., Manásevich R.: On the closed solution to some p−Laplacian nonhomogeneous eigenvalue problems. Differ. Integr. Equ. 12, 773–788 (1999)
Edmunds D.E., Gurka P., Lang J.: Properties of generalized trigonometric functions. J. Approx. Theory 164, 47–56 (2012)
Jiang, W.-D., Qi, F.: Geometric convexity of the generalized sine and the generalized hyperbolic sine (arXiv.1301.3264)
Klén, R., Manojlović, V., Simić, S., Vuorinen, M.: Bernoulli inequality and hypergeometric functions. Proc. Am. Math. Soc. (2013, in press)
Kuczma, M.: An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality, With a Polish summary. Prace Naukowe Uniwersytetu Ślaskiego w Katowicach [Scientific Publications of the University of Silesia], 489. Uniwersytet Ślaski, Katowice; Państwowe Wydawnictwo Naukowe (PWN), Warsaw (1985)
Lang J., Edmunds D.E.: Eigenvalues, Embeddings and Generalised Trigonometric Functions, Lecture Notes in Mathematics, vol. 2016. Springer, Berlin (2011)
Lindqvist P.: Some remarkable sine and cosine functions. Ricerche di Matematica 44, 269–290 (1995)
Lindqvist P., Peetre J.: Two remarkable identities, called twos, for inverses to some Abelian integrals. Amer. Math. Mon. 108(5), 403–410 (2001)
Mršević M.: Convexity of the inverse function. Teach. Math. 11, 21–24 (2008)
Takeuchi S.: Generalized Jacobian elliptic functions and their application to bifurcation problems associated with p-Laplacian. J. Math. Anal. Appl. 385, 24–35 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Baricz, Á., Bhayo, B.A. & Klén, R. Convexity properties of generalized trigonometric and hyperbolic functions. Aequat. Math. 89, 473–484 (2015). https://doi.org/10.1007/s00010-013-0222-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00010-013-0222-x