Analogy of Classical and Dynamic Inequalities Merging on Time Scales

In this paper, we present analogues of Radon's inequality and Nesbitt's inequality on time scales. Furthermore, we find refinements of some classical inequalities such as Bergström's inequality, the weighted power mean inequality, Cauchy-Schwarz's inequality and Hölder's inequality. Our investigations unify and extend some continuous inequalities and their corresponding discrete analogues.

Full text (pdf)

1. R.P. Agarwal, D. O'Regan, S.H. Saker, Dynamic Inequalities on Time Scales, Springer International Publishing, Cham, Switzerland 2014.
2. D. Anderson, J. Bullock, L. Erbe, A. Peterson, H. Tran, Nabla dynamic equations on time scales, Pan{American Mathematical Journal 13 (1) (2003) 1-47.
3. D.M. Bătineţu-Giurgiu, N. Stanciu, New generalizations and new applications for Nesbitt's inequality, Journal of Science and Arts 12 (2012) no.4 (21) 425-430.
4. E.F. Beckenbach, R. Bellman, Inequalities, Springer, Berlin, Göttingen and Heidelberg, 1961.
5. R. Bellman, Notes on matrix theory-IV (An inequality due to Bergström), Amer. Math. Monthly 62 (1955) 172-173.
6. M. Bohner, A. Peterson, Dynamic Equations on Time Scales, Birkhäuser Boston, Inc., Boston, MA, 2001.
7. M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser Boston, Boston, MA, 2003.
8. H. Bergström, A triangle inequality for matrices, Den Elfte Skandinaviske Matematikerkongress, Trondheim (1949), Johan Grundt Tanums Forlag, Oslo (1952) 264-267.
9. Z. Cvetkovski, Inequalities. Theorems, Techniques and Selected Problems, Springer-Verlag Berlin Heidelberg, Heidelberg, 2012.
10. C. Dinu, Convex functions on time scales, Annals of the University of Craiova, Math. Comp. Sci. Ser. 35 (2008) 87-96.
11. G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 1934.
12. S. Hilger, Ein Maꞵkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Universität Würzburg, 1988.
13. L. Maligranda, Why Hölder's inequality should be called Rogers' inequality, Mathematical Inequalities & Applications 1 (1) (1998) 69-83.
14. D.S. Mitrinović, Analytic Inequalities, Springer-Verlag, Berlin, 1970.
15. D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis, Mathematics and Its Applications (East European Series), vol. 61, Kluwer Academic Publishers, Dordrecht, 1993.
16. M.J.S. Sahir, Hybridization of classical inequalities with equivalent dynamic inequalities on time scale calculus, The Teaching of Mathematics, XXI (2018) no.1 38-52.
17. M.J.S. Sahir, Formation of versions of some dynamic inequalities unified on time scale calculus, Ural Mathematical Journal 4 (2) (2018) 88-98.
18. M.J.S. Sahir, Symmetry of classical and extended dynamic inequalities unified on time scale calculus, Turkish J. Ineq. 2 (2) (2018) 11-22.
19. Q. Sheng, M. Fadag, J. Henderson, J.M. Davis, An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal. Real World Appl. 7 (3) (2006) 395-413.
20. F. Wei, S. Wu, Generalizations and analogues of the Nesbitt's inequality, Octogon Mathematical Magazine 17 (1) (2009) 215-220.
21. S. Wu, An exponential generalization of a Radon inequality, J. Huaqiao Univ. Nat. Sci. Ed. 24 (1) (2003) 109-112.
22. S. Wu, A result on extending Radon's inequality and its application, J. Guizhou Univ. Nat. Sci. Ed. 22 (1) (2004) 1-4.
23. S. Wu, A new generalization of the Radon inequality, Math. Practice Theory 35 (9) (2005) 134-139.
24. S. Wu, A class of new Radon type inequalities and their applications, Math. Practice Theory 36 (3) (2006) 217-224.