Some Improvements of Hölder's Inequality on Time Scales

Dinu, Cristian; Stancu, Mihai; Dănciulescu, Daniela

doi:10.1515/auom-2017-0037

Abstract

The theory and applications of dynamic derivatives on time scales have recently received considerable interest. In this paper, we define a function using the combined diamond- α integral, investigate its monotonicity and give some refinements for Hölder's inequality. We present some applications for ℤ and ℝ.

References

[1] R. P. Agarwal, M. Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1999), 3-22.10.1007/BF03322019
Search in Google Scholar Back to article
[2] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An introduction with Applications, Birkhäuser, Boston, 2001.10.1007/978-1-4612-0201-1
Search in Google Scholar Back to article
[3] C. Dinu, A weighted Hermite-Hadamard inequality for Steffensen- Popoviciu and Hermite-Hadamard weights on time scales, Analele Ştiinţifice ale Universităţii \Ovidius", Constanţa, Ser. Mat., 17 (2009).
Search in Google Scholar Back to article
[4] C. Dinu, \Hermite-Hadamard Inequality on Time Scales", Journal of Inequalities and Applications, vol. 2008, Article ID 287947, 24 pages, 2008.10.1155/2008/287947
Open DOI Search in Google Scholar Back to article
[5] S. Hilger, \Analysis on measure chains-a uni_ed approach to continuous and discrete calculus", Results Math. 35 (1990), pp. 18-56.10.1007/BF03323153
Open DOI Search in Google Scholar Back to article
[6] S. Hussain and M. Anwar, \On certain inequalities improving the Hermite-Hadamard inequality", J. Inequal. Pure and Appl. Math. (JIPAM) 8 (2007), Issue 2, Article no. 60.
Search in Google Scholar Back to article
[7] C. P. Niculescu and L.-E. Persson, Convex functions and their applications. A contemporary approach, CMS Books in Mathematics vol. 23, Springer-Verlag, New York, 2006.10.1007/0-387-31077-0
Open DOI Search in Google Scholar Back to article
[8] M. R. S. Ammi, Rui A. C. Ferreira, and Del_m F. M. Torres, \Diamond- Jensen's Inequality on Time Scales and Applications", Journal of Inequalities and Applications, vol. 2008, Article ID 576876, 13 pages, 2008.10.1155/2008/576876
Search in Google Scholar Back to article
[9] T. Popoviciu, \Notes sur les fonctions convexes d'ordre superieur" (IX), Bull. Math. Soc. Roum. Sci., 43 (1941), 85-141.
Search in Google Scholar Back to article
[10] J. W. Rogers Jr. and Q. Sheng, \Notes on the diamond- dynamic derivative on time scales", J. Math. Anal. Appl., 326 (2007), no. 1, 228-241.10.1016/j.jmaa.2006.03.004
Search in Google Scholar Back to article
[11] Q. Sheng, M. Fadag, J. Henderson and J. M. Davis, \An exploration of combined dynamic derivatives on time scales and their applications", Nonlinear Anal. Real World Appl., 7 (2006), no. 3, 395-413.10.1016/j.nonrwa.2005.03.008
Search in Google Scholar Back to article
[12] Liang-Cheng Wang, “Two Mappings Related to Hölder's Inequality”, Univ. Beograd. Publ. Elektrotehn. Fak., Ser. Mat. 15 (2004), 91-96.10.2298/PETF0415091W
Search in Google Scholar Back to article

Some Improvements of Hölder's Inequality on Time Scales

Abstract

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