Skip to main content
SpringerLink
Log in

Discrete Inequalities of Wirtinger’s Type

  • Chapter
Recent Progress in Inequalities

Part of the book series: Mathematics and Its Applications ((MAIA,volume 430))

Abstract

Various discrete versions of Wirtinger’s type inequalities are considered. A short account on the first results in this field given by Fan, Taussky and Todd [10] as well as some generalisations of these discrete inequalities are done. Also, a general method for finding the best possible constants A n and B n in inequalities of the form

$$A_{n}\sum_{k=1}^{n}p_kx_{k}^{2}\leq\sum_{k=0}^{n}r_{k}(x_{k}-x_{k+1})^{2}\leq B_{n}\sum_{k=1}^{n}p_kx_{k}^{2}$$

where p = (p k ) and r = (r k ) are given weight sequences and x = (x k ) is an arbitrary sequence of the real numbers, is presented. Two types of problems are investigated and several corollaries of the basic results are obtained. Further generalisations of discrete inequalities of Wirtinger’s type for higher differences are also treated.

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

Access this chapter

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. P. Agarwal, Difference Equations and Inequalities — Theory, Methods, and Applications, Marcel Dekker, New York — Basel — Hong Kong, 1992.

    MATH  Google Scholar 

  2. H. Alzer Converses of two inequalities by Ky Fan, O. Taussky, and J. Todd, J. Math. Anal. Appl. 161 (1991), 142–147.

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Alzer Note on a discrete Opial-type inequality, Arch. Math. 65 (1995), 267–270.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. F. Beckenbach and R. Bellman, Inequalities, Springer Verlag, Berlin — Heidelberg — New York, 1971.

    MATH  Google Scholar 

  5. W. Blaschke, Kreis und Kugel, Veit u. Co., Leipzig, 1916.

    MATH  Google Scholar 

  6. H. D. Block, Discrete analogues of certain integral inequalities, Proc. Amer. Math. Soc. 8 (1957), 852–859.

    Article  MathSciNet  MATH  Google Scholar 

  7. W. Chen On a question of H. Alzer, Arch. Math. 62 (1994), 315–320.

    Article  MATH  Google Scholar 

  8. S.-S. Cheng Discrete quadratic Wirtinger’s inequalities, Linear Algebra Appl. 85 (1987), 57–73.

    Article  MathSciNet  MATH  Google Scholar 

  9. E. Egerváry and O. Szász Einige Extremalprobleme im Bereiche der trigonometrischen Polynome, Math. Z. 27 (1928), 641–692.

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Fan, O. Taussky and J. Todd Discrete analogs of inequalities of Wirtinger, Monatsh. Math. 59 (1955), 73–90.

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Fejér Über trigonometrische Polynome, J. Reine Angew. Math. 146 (1915), 53–82.

    MATH  Google Scholar 

  12. A. M. Fink Discrete inequalities of generalized Wirtinger type, Aequationes Math. 11 (1974), 31–39.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd Edition, Univ. Press, Cambridge, 1952.

    MATH  Google Scholar 

  14. C.-M. Lee On a discrete analogue of inequalities of Opial and Yang, Canad. Math. Bull. 11 (1968), 73–77.

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Losonczi, On some discrete quadratic inequalities, General Inequalities 5 (Oberwolfach, 1986) (W. Walter, ed.), ISNM Vol. 80, Birkhäuser Verlag, Basel, 1987, pp. 73–85.

    Chapter  Google Scholar 

  16. G. Lunter New proofs and a generalisation of inequalities of Fan, Taussky, and Todd, J. Math. Anal. Appl. 185 (1994), 464–476.

    Article  MathSciNet  MATH  Google Scholar 

  17. G. V. Milovanović, Numerical Analysis, Part I, 3rd Edition, Naučna Knjiga, Belgrade, 1991. (Serbian)

    Google Scholar 

  18. G. V. Milovanović, Numerical Analysis, Part II, 3rd Edition, Naučna Knjiga, Belgrade, 1981. (Serbian)

    Google Scholar 

  19. G. V. Milovanović and I. Ž. Milovanović On discrete inequalities of Wirtinger’s type, J. Math. Anal. Appl. 88 (1982), 378–387.

    Article  MathSciNet  MATH  Google Scholar 

  20. G. V. Milovanović and I. Ž. Milovanović Some discrete inequalities of OpiaVs type, Acta Sci. Math. (Szeged) 47 (1984), 413–417.

    MathSciNet  MATH  Google Scholar 

  21. ——, Discrete inequalities of Wirtinger’s type for higher differences, J. Ineq. Appl. 1 (1997) (to appear).

    Google Scholar 

  22. G. V. Milovanovič, I. Ž. Milovanović and L. Z. Marinkovic, Extremal problems for polynomials and their coefficients, Topics in Polynomials of One and Several Variables and Their Applications (Th. M. Rassias, H. M. Srivastava and A. Yanushauskas, eds.), World Scientific, Singapore — New Jersey — London — Hong Kong, 1993, pp. 435–455.

    Google Scholar 

  23. G. V. Milovanovič, D. S. Mitrinović and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore — New Jersey — London — Hong Kong, 1994.

    MATH  Google Scholar 

  24. D. S. Mitrinović and P. M. Vasić An inequality ascribed to Wirtinger and its variations and generalization, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No 247 — No 273 (1969), 157–170.

    Google Scholar 

  25. D. S. Mitrinović (with P. M. Vasić), Analytic Inequalities, Springer Verlag, Berlin — Heidelberg — New York, 1970.

    MATH  Google Scholar 

  26. D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer, Dordrecht — Boston — London, 1991.

    Book  MATH  Google Scholar 

  27. J. Novotná Variations of discrete analogues of Wirtinger’s inequality, Časopis Pest. Mat. 105 (1980), 278–285.

    MATH  Google Scholar 

  28. J. Novotná Discrete analogues of Wirtinger’s inequality for a two-dimensional array, Časopis Pest. Mat. 105 (1980), 354–362.

    MATH  Google Scholar 

  29. J. Novotná A sharpening of discrete analogues of Wirtinger’s inequality, časopis Pest. Mat. 108 (1983), 70–77.

    MATH  Google Scholar 

  30. A. M. Pfeffer, On certain discrete inequalities and their continuous analogs, J. Res. Nat. Bur. Standards Sect. B 70B (1966), 221–231.

    Article  MathSciNet  Google Scholar 

  31. I. J. Schoenberg The finite Fourier series and elementary geometry, Amer. Math. Monthly 57 (1950), 390–404.

    Article  MathSciNet  MATH  Google Scholar 

  32. O. Shisha On the discrete version of Wirtinger’s inequality, Amer. Math. Monthly 80 (1973), 755–760.

    Article  MathSciNet  MATH  Google Scholar 

  33. G. Szegö, Koeffizientenabschätzungen bei ebenen und räumlichen harmonischen Entwicklungen, Math. Ann. 96 (1926/27), 601–632.

    Article  Google Scholar 

  34. J. S. W. Wong A discrete analogue of Opial’s inequality, Canad. Math. Bull. 10 (1967), 115–118.

    Article  MathSciNet  MATH  Google Scholar 

  35. G.-S. Yang and C.-D. You A note on discrete Opial’s inequality, Tamking J. Math. 23 (1992), 67–78.

    MathSciNet  MATH  Google Scholar 

  36. X.-R. Yin A converse inequality of Fan, Taussky, and Todd, J. Math. Anal. Appl. 182 (1994), 654–657.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Electronic Engineering, P. O. Box 73, 18000, Niš, Yugoslavia

    Gradimir V. Milovanović & Igor Ž. Milovanović

Authors
  1. Gradimir V. Milovanović
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor Ž. Milovanović
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Niš, Niš, Yugoslavia

    G. V. Milovanović (Faculty of Electronic Engineering) (Faculty of Electronic Engineering)

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer Science+Business Media New York

About this chapter

Cite this chapter

Milovanović, G.V., Milovanović, I.Ž. (1998). Discrete Inequalities of Wirtinger’s Type. In: Milovanović, G.V. (eds) Recent Progress in Inequalities. Mathematics and Its Applications, vol 430. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9086-0_16

Download citation

  • DOI: https://doi.org/10.1007/978-94-015-9086-0_16

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4945-2

  • Online ISBN: 978-94-015-9086-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation