Weighted integral inequalities related to Wirtinger’s result for p-norms with applications

Silvestru Sever Dragomir

Abstract


In this paper we establish several natural consequences of some Wirtinger type integral inequalities for p-norms. The corresponding weighted versions and applications related to the weighted trapezoid inequalities, to weighted Gruss’ type inequalities and reverses of Jensen’s inequality are also provided.

Keywords


Wirtinger’s inequality; trapezoid inequality; Gruss’ inequality; Jensen’s inequality

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References


Alomari, M. W., On Beesack–Wirtinger inequality, Results Math. 72 (2017), 1213–1225.

Beesack, P. R., Extensions of Wirtinger’s inequality, Trans. R. Soc. Can. 53 (1959), 21–30.

Cerone, P., Dragomir, S. S., A refinement of the Gruss inequality and applications, Tamkang J. Math. 38 (1) (2007), 37–49. (preprint RGMIA Res. Rep. Coll. 5 (2) (2002), Article 14. [Online http://rgmia.vu.edu.au/v5n2.html]).

Chebyshev, P. L., Sur les expressions approximatives des integrals definis par les outres prises entre les meme limites, Proc. Math. Soc. Charkov, 2 (1882), 93–98.

Diaz, J. B., Metcalf, F. T., Variations on Wirtinger’s inequality, in: Inequalities, Academic Press, New York, 1967, pp. 79–103.

Drabek, P., Manasevich, R., On the closed solution to some nonhomogeneous eigenvalue problems with p-Laplacian, Differential Integral Equations 12 (1999) 773–788.

Dragomir, S. S., A Gruss type inequality for isotonic linear functionals and applications, Demonstratio Math. 36 (3) (2003), 551–562 (preprint RGMIA Res. Rep. Coll. 5 (2002), Suplement, Art. 12 [Online http://rgmia.org/papers/v5e/GTIILFApp.pdf]).

Dragomir, S. S., Integral inequalities related to Wirtinger’s result, preprint RGMIA Res. Rep. Coll. 21 (2018), Art. 59, 16 pp. [Online https://rgmia.org/papers/v21/v21a59.pdf]

Fejer, L., Uber die Fourierreihen, II, Math. Naturwiss, Anz. Ungar. Akad. Wiss. 24 (1906), 369–390 (in Hungarian).

Giova, R., An estimate for the best constant in the (L_p)-Wirtinger inequality with weights, J. Func. Spaces Appl. 6 (1) (2008), 1–16.

Gruss, G., Uber das Maximum des absoluten Betrages von (frac{1}{b-a} int_a^b f(x)g(x)dx - frac{1}{(b-a)^2}int_a^b f(x)dx int_a^b g(x)dx), Math. Z. 39(1935), 215–226.

Jaros, J., On an integral inequality of the Wirtinger type, Appl. Math. Lett. 24 (2011) 1389–1392.

Lee, C. F., Yeh, C. C., Hong, C. H., Agarwal, R. P., Lyapunov and Wirtinger inequalities, Appl. Math. Lett. 17 (2004) 847–853.

Lupas, A., The best constant in an integral inequality, Mathematica (Cluj, Romania), 15(38) (2) (1973), 219–222.

Ostrowski, A. M., On an integral inequality, Aequat. Math. 4 (1970), 358–373.

Ricciardi, T., A sharp weighted Wirtinger inequality, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 8 (1) (2005), 259–267.

Swanson, C. A., Wirtinger’s inequality, SIAM J. Math. Anal. 9 (1978) 484–491.

Takahasi, S.-E., Miura, T., Hayata, T., On Wirtinger’s inequality and its elementary proof, Math. Inequal. Appl. 10 (2) (2007), 311–319.

Takeuchi, S., Generalized elliptic functions and their application to a nonlinear eigenvalue problem with p-Laplacian, (2010), pp. 1–17, arXiv:1001.0377v2.




DOI: http://dx.doi.org/10.17951/a.2021.75.1.15-36
Date of publication: 2021-07-24 12:06:59
Date of submission: 2021-07-21 20:22:33


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