Inequalities for eigenfunctions of the $p$-Laplacian

B. A. Bhayo ; M. Vuorinen

Problemy analiza, Tome 2 (2013) no. 1, pp. 14-37.

Résumé

Motivated by the work of P. Lindqvist, we study eigenfunctions of the one-dimensional $p$-Laplace operator, the $\sin_{p}$ functions, and prove several inequalities for these and $p$-analogues of other trigonometric functions and their inverse functions. Similar inequalities are given also for the $p$-analogues of the hyperbolic functions and their inverses.

Détail

Keywords: eigenfunctions of $p$-Laplacian; $\sin_p$; generalized trigonometric function.

Bibliographie

[1] Lindqvist P., “Some remarkable sine and cosine functions”, Ricerche di Matematica, 44 (1995), 269–290 | MR | Zbl

[2] Lindqvist P., Peetre J., “$p$-arclength of the $q$-circle”, The Mathematics Student., 72 (2003), 139–145 | MR | Zbl

[3] Biezuner R. J., Ercole G., Martins E. M., “Computing the first eigenvalue of the $p$-Laplacian via the inverse power method”, J. Funct. Anal., 257:1 (2009), 243–270 | DOI | MR | Zbl

[4] Biezuner R. J., Ercole G., Martins E. M., “Computing the $\sin_p$ function via the inverse power method”, Comput. Methods Appl. Math., 11:2 (2011), 129–140, arXiv: 1011.3486[math.CA] | DOI | MR | Zbl

[5] Drábek P., Manásevich R., “On the closed solution to some $p$-Laplacian nonhomogeneous eigenvalue problems”, Diff. and Int. Eqns., 12 (1999), 723–740

[6] Reichel W., Walter W., “Sturm-Liouville type problems for the $p$-Laplacian under asymptotic non-resonance conditions”, J. Differential Equations, 156:1 (1999), 50–70 | DOI | MR | Zbl

[7] Pino M. del, Elgueta M., Manasevich R., “A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u'|^{p-2} u')' + f (t,u) = 0$; $u(0) = u(T) = 0$; $p > 1$”, J. Differential Equations, 80 (1989), 1–13 | DOI | MR | Zbl

[8] Abramowitz M., Stegun I., Handbook of mathematical functions with formulas, graphs and mathematical tables, United states department of commerce, Washington: National Bureau Of Standards, 1964 | MR

[9] Bushell P. J., Edmunds D. E., “Remarks on generalised trigonometric functions”, Rocky Mountain J. Math., 42 (2012), 25–57 | DOI | MR | Zbl

[10] Takeuchi S., “Generalized Jacobian elliptic functions and their application to bifurcation problems associated with $p$-Laplacian”, J. Math. Anal. Appl., 385:1 (2012), 243–255 | DOI | MR

[11] Ruskeepää H., Mathematical\circledR Navigator, [S. l.], 3rd ed., Academic Press, 2009 | Zbl

[12] Neuman E., “Inequalities involving inverse circular and inverse hyperbolic functions”, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 18 (2006), 32–37 | MR

[13] Neuman E., “Inequalities involving a logarithmically convex function and their applications to special functions”, J. Inequal. Pure Appl. Math., 2006, 16 | MR

[14] Anderson G. D., Qiu S.-L., Vamanamurthy M. K., Vuorinen M., “Generalized elliptic integrals and modular equation”, Pacific J. Math., 192:1 (2000), 1–37 | DOI | MR | Zbl

[15] Anderson G. D., Vamanamurthy M. K., Vuorinen M., Conformal invariants, inequalities and quasiconformal maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley Sons Inc., New York, 1997 | MR

[16] Anderson G. D., Vamanamurthy M. K., Vuorinen M., “Generalized convexity and inequalities”, J. Math. Anal. Appl., 335 (2007), 1294–1308 | DOI | MR | Zbl

[17] Kuczma M., An introduction to the theory offunctional equations and inequalities. Cauchy's equation and Jensen's inequality, Uniwersytet Slaski, Warszawa; Krakow; Katowice, 1985 | MR

[18] Zhu L., “New inequalities of Shafer-Fink Type for arc hyperbolic sine”, J. Inequal. Appl., 2008, 368275, 5 pp. | DOI | MR | Zbl

[19] Klén R., Visuri M., Vuorinen M., “On Jordan type inequalities for hyperbolic functions”, J. Inequal. Appl., 2010, 362548, 14 pp. | DOI | MR | Zbl

[20] Brown B. M., Reichel W., “Eigenvalues of the radially symmetric $p$-Laplacian in $\mathbb{R}^n$”, J. London Math. Soc., 69:3 (2004), 657–675 | DOI | MR | Zbl