Multivariate  Opial-type Inequalities on Time Scales

Yisa Oluwatoyin  Anthonio; Kamilu  Rauf; Abdullai Ayinla  Abdurasid; Oluwaseun Raphael  Aderele

doi:10.34198/ejms.12123.1326

Yisa Oluwatoyin Anthonio Department of Mathematics and Statistics, Lagos State University of Science and Technology, Ikorodu, Lagos, Nigeria
Kamilu Rauf Department of Mathematics, University of Ilorin, Ilorin, Nigeria
Abdullai Ayinla Abdurasid Department of Mathematics and Statistics, Lagos State University of Science and Technology, Ikorodu, Lagos, Nigeria
Oluwaseun Raphael Aderele Department of Mathematics and Statistics, Lagos State University of Science and Technology, Ikorodu, Lagos, Nigeria

DOI: https://doi.org/10.34198/ejms.12123.1326

Keywords: Opial-type inequalities, convex functions, refined Jensen inequality, time scale

Abstract

Opial inequality was developed to provide bounds for integral of functions and their derivatives. It has become an indispensable tool in the theory of mathematical analysis due to its usefulness. A refined Jensen inequality for multivariate functions is employed to establish new Opial-type inequalities for convex functions of several variables on time scale.

References

R. Agarwal, D. O'Regan and S. Saker, Dynamic Inequalities on Time Scales, Springer International Publishing Switzerland, 2014.

M. Anwar, R. Bibi, M. Bohner and J. Pečarić, Jensen functionals on time scales for several variables, Int. J. Anal. 2014, Art. ID 126797, 14 pp. https://doi.org/10.1155/2014/126797

P. R. Beesack, Elementary proofs of some Opial-type integral inequalities, J. Anal. Math. 36 (1979), 1-14. https://doi.org/10.1007/bf02798763

M. Bohner and B. Kaymakcalan, Opial inequalities on time scales, Ann. Polon. Math. 77(1) (2001), 11-20. https://doi.org/10.4064/ap77-1-2

J. Calvert, Some generalizations of Opial's inequality, Proc. Amer. Math. Soc. 18 (1967), 72-75. https://doi.org/10.1090/s0002-9939-1967-0204594-1

E. K. Godunova and V. I. Levin, O niekotoryh integralnyh nerabenstvah, soderzascih proizvodnye (Russian), Izv. vuzov, Matem. 91 (1969), 20-24.

S. Hilger, Ein Makettenkalkl mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Universtat Wurzburg, 1988.

S. Hilger, Differential and difference calculus-unified!, Proceedings of the Second World Congress of Nonlinear Analysts, Part 5 (Athens, 1996), Nonlinear Anal. 30(5) (1997), 2683-2694. https://doi.org/10.1016/s0362-546x(96)00204-0

Hüseyin Budak, Mehmet Zeki Sarikaya and Artion Kashuri, On weighted generalization of Opial type inequalities in two variables, Korean J. Math. 28(4) (2020), 717-737.

Josip Pečarić and Ilko Brnetić, Note on generalization of Godunova Levin-Opial Inequality, Demonstratio Math. 30(3) (1997), 545-549. https://doi.org/10.1515/dema-1997-0310

Z. Opial, Sur une inégalité, Ann. Polon. Math. 8 (1960), 29-32. https://doi.org/10.4064/ap-8-1-29-32

Z. Qi, Further generalization of Opial's inequality, Acta Math. Sinica (N.S.) 1(3) (1985), 196-200.

K. Rauf and Y. O. Anthonio, Results on an integral inequality of the Opial-type, Glob. J. Pure Appl. Sci. 23 (2017), 151-156. https://doi.org/10.4314/gjpas.v23i1.15

G. I. Rozanova, Integral inequalities with derivatives and with arbitrary convex functions (Russian), Moskov. Gos. Ped. Inst. Vcen Zap. 460 (1972), 58-65.

S. H. Saker, Some Opial-type inequalities on time scales. Abstr. Appl. Anal. 2011 (2011), Article ID 265316, 19 pp. https://doi.org/10.1155/2011/265316

Chang-Jian Zhao and Wing-Sum Cheung, On some Opial-type inequalities, J. Inequal. Appl. 2011 (2011), 7. https://doi.org/10.1186/1029-242x-2011-7

Multivariate Opial-type Inequalities on Time Scales

Abstract

References

E-ISSN : 2581-8147