Weighted integral inequalities related to Wirtinger’s result for p-norms with applications
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 75 (2021) no. 1, pp. 37-51.

Voir la notice de l'article provenant de la source Library of Science

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
@article{AUM_2021_75_1_a2,
     author = {Dragomir, Silvestru Sever},
     title = {Weighted integral inequalities related to {Wirtinger{\textquoteright}s} result for p-norms with applications},
     journal = {Annales Universitatis Mariae Curie-Sk{\l}odowska. Mathematica },
     pages = {37--51},
     publisher = {mathdoc},
     volume = {75},
     number = {1},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AUM_2021_75_1_a2/}
}
TY  - JOUR
AU  - Dragomir, Silvestru Sever
TI  - Weighted integral inequalities related to Wirtinger’s result for p-norms with applications
JO  - Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
PY  - 2021
SP  - 37
EP  - 51
VL  - 75
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AUM_2021_75_1_a2/
LA  - en
ID  - AUM_2021_75_1_a2
ER  - 
%0 Journal Article
%A Dragomir, Silvestru Sever
%T Weighted integral inequalities related to Wirtinger’s result for p-norms with applications
%J Annales Universitatis Mariae Curie-Skłodowska. Mathematica 
%D 2021
%P 37-51
%V 75
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AUM_2021_75_1_a2/
%G en
%F AUM_2021_75_1_a2
Dragomir, Silvestru Sever. Weighted integral inequalities related to Wirtinger’s result for p-norms with applications. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 75 (2021) no. 1, pp. 37-51. http://geodesic.mathdoc.fr/item/AUM_2021_75_1_a2/

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

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

[3] 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]).

[4] 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.

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

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

[7] 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]).

[8] 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]

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

[10] 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.

[11] 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.

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

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

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

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

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

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

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

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