Skip to main content
SpringerLink
Log in
Book cover

Modern Discrete Mathematics and Analysis pp 117–172Cite as

Recent Developments of Discrete Inequalities for Convex Functions Defined on Linear Spaces with Applications

  • Chapter
  • First Online:

  • 1502 Accesses

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 131))

Abstract

In this paper we survey some recent discrete inequalities for functions defined on convex subsets of general linear spaces. Various refinements and reverses of Jensen’s discrete inequality are presented. The Slater inequality version for these functions is outlined. As applications, we establish several bounds for the mean f-deviation of an n-tuple of vectors as well as for the f-divergence of an n -tuple of vectors given a discrete probability distribution. Examples for the K. Pearson χ 2 -divergence, theKullback-Leibler divergence, the Jeffreys divergence, the total variation distance and other divergence measures are also provided.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Cerone, P., Dragomir, S.S.: A refinement of the Grüss inequality and applications. Tamkang J. Math. 38(1), 37–49 (2007). Preprint RGMIA Res. Rep. Coll. 5(2) (2002). Article 14. http://rgmia.org/papers/v5n2/RGIApp.pdf

  2. Dragomir, S.S.: An improvement of Jensen’s inequality. Bull. Math. Soc. Sci. Math. Roum. 34(82)(4), 291–296 (1990)

    Google Scholar 

  3. Dragomir, S.S.: Some refinements of Ky Fan’s inequality. J. Math. Anal. Appl. 163(2), 317–321 (1992)

    Article  MathSciNet  Google Scholar 

  4. Dragomir, S.S.: Some refinements of Jensen’s inequality. J. Math. Anal. Appl. 168(2), 518–522 (1992)

    Article  MathSciNet  Google Scholar 

  5. Dragomir, S.S.: A further improvement of Jensen’s inequality. Tamkang J. Math. 25(1), 29–36 (1994)

    MathSciNet  MATH  Google Scholar 

  6. Dragomir, S.S.: A new improvement of Jensen’s inequality. Indian J. Pure Appl. Math. 26(10), 959–968 (1995)

    MathSciNet  MATH  Google Scholar 

  7. Dragomir, S.S.: New estimation of the remainder in Taylor’s formula using Grüss’ type inequalities and applications. Math. Inequal. Appl. 2(2), 183–193 (1999)

    MathSciNet  MATH  Google Scholar 

  8. Dragomir, S.S.: On the Ostrowski’s integral inequality for mappings of bounded variation and applications. RGMIA Res. Rep. Coll. 2(1), 63–70 (1999). http://rgmia.org/papers/v2n1/v2n1-7.pdf

    MathSciNet  Google Scholar 

  9. Dragomir, S.S.: An improvement of the remainder estimate in the generalised Taylor’s formula. RGMIA Res. Rep. Coll. 3(1) (2000). Article 1. http://rgmia.org/papers/v3n1/IREGTF.pdf

  10. Dragomir, S.S.: Other inequalities for Csiszár divergence and applications. Preprint, RGMIA Monographs, Victoria University (2000). http://rgmia.org/papers/Csiszar/ICDApp.pdf

    Google Scholar 

  11. Dragomir, S.S.: Some inequalities for \(\left (m,M\right ) \)-convex mappings and applications for the Csiszár f -divergence in Information Theory. Math. J. Ibaraki Univ. 33, 35–50 (2001). http://rgmia.org/papers/Csiszar/ImMCMACFDIT.pdf

    Article  MathSciNet  Google Scholar 

  12. Dragomir, S.S.: Some inequalities for two Csiszár divergences and applications. Mat. Bilten (Macedonia) 25, 73–90 (2001). http://rgmia.org/papers/Csiszar/In2CsisDivApp.pdf

    MathSciNet  MATH  Google Scholar 

  13. Dragomir, S.S.: On a reverse of Jessen’s inequality for isotonic linear functionals. J. Inequal. Pure Appl. Math. 2(3) (2001). Article 36

    Google Scholar 

  14. Dragomir, S.S.: A Grüss type inequality for isotonic linear functionals and applications. Demonstratio Math. 36(3), 551–562 (2003). Preprint RGMIA Res. Rep. Coll. 5(2002). Supplement, Article 12. http://rgmia.org/papers/v5e/GTIILFApp.pdf

  15. Dragomir, S.S.: A converse inequality for the Csiszár Φ-divergence. Tamsui Oxf. J. Math. Sci. 20(1), 35–53 (2004). Preprint in Dragomir, S.S. (ed.) Inequalities for Csiszár f-Divergence in Information Theory, RGMIA Monographs. Victoria University (2000). http://rgmia.org/papers/Csiszar/Csiszar.pdf

  16. Dragomir, S.S.: Semi-inner Products and Applications. Nova Science Publishers Inc., New York (2004)

    MATH  Google Scholar 

  17. Dragomir, S.S.: Discrete Inequalities of the Cauchy-Bunyakovsky-Schwarz Type. Nova Science Publishers, New York (2004)

    MATH  Google Scholar 

  18. Dragomir, S.S.: Bounds for the normalized Jensen functional. Bull. Aust. Math. Soc. 74(3), 471–476 (2006)

    Article  MathSciNet  Google Scholar 

  19. Dragomir, S.S.: A refinement of Jensen’s inequality with applications for f-divergence measures. Taiwanese J. Math. 14(1), 153–164 (2010)

    Article  MathSciNet  Google Scholar 

  20. Dragomir, S.S.: A new refinement of Jensen’s inequality in linear spaces with applications. Math. Comput. Model. 52(9–10), 1497–1505 (2010)

    Article  MathSciNet  Google Scholar 

  21. Dragomir, S.S.: Inequalities in terms of the Gâteaux derivatives for convex functions on linear spaces with applications. Bull. Aust. Math. Soc. 83(3), 500–517 (2011)

    Article  MathSciNet  Google Scholar 

  22. Dragomir, S.S.: A refinement and a divided difference reverse of Jensen’s inequality with applications. Rev. Colomb. Mat. 50(1), 17–39 (2016). Preprint RGMIA Res. Rep. Coll. 14 (2011). Article 74. http://rgmia.org/papers/v14/v14a74.pdf

    Article  MathSciNet  Google Scholar 

  23. Dragomir, S.S.: Some Slater’s type inequalities for convex functions defined on linear spaces and applications. Abstr. Appl. Anal. 2012, 16 pp. (2012). Article ID 168405

    Google Scholar 

  24. Dragomir, S.S.: Some reverses of the Jensen inequality with applications. Bull. Aust. Math. Soc. 87(2), 177–194 (2013)

    Article  MathSciNet  Google Scholar 

  25. Dragomir, S.S.: Reverses of the Jensen inequality in terms of first derivative and applications. Acta Math. Vietnam. 38(3), 429–446 (2013)

    Article  MathSciNet  Google Scholar 

  26. Dragomir, S.S., Goh, C.J.: A counterpart of Jensen’s discrete inequality for differentiable convex mappings and applications in information theory. Math. Comput. Model. 24(2), 1–11 (1996)

    Article  MathSciNet  Google Scholar 

  27. Dragomir, S.S., Goh, C.J.: Some bounds on entropy measures in information theory. Appl. Math. Lett., 10(3), 23–28 (1997)

    Article  MathSciNet  Google Scholar 

  28. Dragomir, S.S., Ionescu, N.M.: Some converse of Jensen’s inequality and applications. Rev. Anal. Numér. Théor. Approx. 23(1), 71–78 (1994)

    MathSciNet  MATH  Google Scholar 

  29. Dragomir, S.S., Pečarić, J., Persson, L.E.: Properties of some functionals related to Jensen’s inequality. Acta Math. Hung. 70(1–2), 129–143 (1996)

    Article  MathSciNet  Google Scholar 

  30. Pečarić, J., Dragomir, S.S.: A refinements of Jensen inequality and applications. Studia Univ. Babeş-Bolyai, Math. 24(1), 15–19 (1989)

    Google Scholar 

  31. Simić, S.: On a global upper bound for Jensen’s inequality. J. Math. Anal. Appl. 343, 414–419 (2008)

    Article  MathSciNet  Google Scholar 

  32. Slater, M.S.: A companion inequality to Jensen’s inequality. J. Approx. Theory 32, 160–166 (1981)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics, College of Engineering & Science, Victoria University, Melbourne City, MC, Australia

    Silvestru Sever Dragomir

  2. DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science & Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa

    Silvestru Sever Dragomir

Authors
  1. Silvestru Sever Dragomir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Silvestru Sever Dragomir .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Hellenic Military Academy, Vari, Greece

    Nicholas J. Daras

  2. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Dragomir, S.S. (2018). Recent Developments of Discrete Inequalities for Convex Functions Defined on Linear Spaces with Applications. In: Daras, N., Rassias, T. (eds) Modern Discrete Mathematics and Analysis . Springer Optimization and Its Applications, vol 131. Springer, Cham. https://doi.org/10.1007/978-3-319-74325-7_6

Download citation

Publish with us

Policies and ethics

Search

Navigation