Abstract

Using Hölder's inequality, the chain rule on time scales and the properties of geometrically convex and concave functions we prove some new dynamic inequalities and their converses on time scales. As a special case, we derive the classical Saitoh integral inequality.

Author information

Contact details are reproduced from the original publication and may be historical.

Donal O'Regan

School of Mathematics, National University of Ireland, Galway, Ireland

Samir H. Saker

Department of Mathematics, Faculty of Science, University of Mansoura, Egypt

S. S. Rabie

Department of Mathematics, Faculty of Science, University of Mansoura, Egypt

Ravi P. Agarwal

Department of Mathematics, Texas A&M University, Kingsville, TX 78363, U.S.A.

Ravi.Agarwal@tamuk.edu

Keywords

  • Opial's inequality
  • Saitoh's inequality
  • time scales
  • Hölder's inequality

Mathematics Subject Classification

26A15, 26D10, 26D15, 39A13, 34A40

References

  1. R. P. Agarwal, M. Bohner: Basic calculus on time scales and some of its applications, Results Math. 35 (1999) 3–22.https://doi.org/10.1007/BF03322019Google Scholar
  2. R. P. Agarwal, D. O’Regan, S. Saker: Dynamic Inequalities on Time Scales, Springer, Cham (2014).https://doi.org/10.1007/978-3-319-11002-8Google Scholar
  3. R. P. Agarwal, P. Y. H. Pang: Opial Inequalities with Applications in Differential and Difference Equations, Kluwer, Dordrecht (1995).https://doi.org/10.1007/978-94-015-8426-5Google Scholar
  4. G. Anastassiou: Time scales inequalities, Int. J. Diff. Equations 5 (2010) 1–23.https://doi.org/10.1155/2010/317416Google Scholar
  5. M. Bohner, B. Kaymakçalan: Opial inequalities on time scales, Annales Polon. Math. 77 (2001) 11–20.https://doi.org/10.4064/ap77-1-2Google Scholar
  6. M. Bohner, A. Peterson: Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston (2001).https://doi.org/10.1007/978-1-4612-0201-1Google Scholar
  7. M. Bohner, A. Peterson: Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston (2003).https://doi.org/10.1007/978-0-8176-8230-9Google Scholar
  8. G. Hardy, J. E. Littlewood, G. Pólya: Inequalities, 2nd ed., Cambridge University Press, Cambridge (1991).Google Scholar
  9. S. Hilger: Analysis on measure chains – a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56.https://doi.org/10.1007/BF03323153Google Scholar
  10. B. Karpuz, U. M. Öclan: Some generalizations for Opial’s inequality involving several functions and their derivatives of arbitrary order on arbitrary time scales, Math. Ineq. App. 14 (2011) 79–92.https://doi.org/10.7153/mia-14-07Google Scholar
  11. A. Lasota: A discrete boundary value problem, Annales Polon. Math. 20 (1968) 183–190.https://doi.org/10.4064/ap-20-2-183-190Google Scholar
  12. C. P. Niculescu: Convexity according to the geometric mean, Math. Ineq. Appl. 2 (2000) 155–167.https://doi.org/10.7153/mia-03-19Google Scholar
  13. C. Olech: A simple proof of a certain result of Z. Opial, Annales Polon. Math. 8 (1960) 61–63.https://doi.org/10.4064/ap-8-1-61-63Google Scholar
  14. Z. Opial: Sur uné inegalité, Annales Polon. Math. 8 (1960) 92–32.https://doi.org/10.4064/ap-8-1-29-32Google Scholar
  15. S. Saitoh: Natural norm inequalities in nonlinear transforms, in: General Inequalities 7, Oberwolfach 1995, Internat. Ser. Numer. Math. 123, Birkhäuser, Basel (1997) 39–52.https://doi.org/10.1007/978-3-0348-8942-1_4Google Scholar
  16. A. Yamada: Saitoh’s inequality and Opial’s inequality, Math. Ineq. Appl. 14 (2011) 523–528.https://doi.org/10.7153/mia-14-44Google Scholar
  17. X.-M. Zhang, T.-Q. Xu, L.-B. Situ: Geometric convexity of a function involving gamma function and applications to inequality theory, J. Ineq. Pure Appl. Math. 8 (2007), art. no. 17, 9 pp.Google Scholar

Suggested citation

D. O'Regan, S. H. Saker, S. S. Rabie, R. P. Agarwal. “Opial and Saitoh Type Inequalities on Time Scales.” Journal of Convex Analysis 27 (2020), No. 3, 1073–1090. https://doi.org/10.68381/jca27056

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2020. Rights now held by Banach Press.