Skip to main content
SpringerLink
Log in

Abel’s and Related Inequalities

  • Chapter
Classical and New Inequalities in Analysis

Part of the book series: Mathematics and Its Applications () ((MAEE,volume 61))

  • 1727 Accesses

Abstract

General linear inequalities are old inequalities. We are not sure who is the author of such inequalities. So we shall given here only some basic facts about such inequalities but only for monotonic functions and some related results.

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 709.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 899.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 899.99
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. STEFFENSEN, J. F., On a generalization of certain inequalities by Tchebychef and Jensen, Skandinavisk Aktuarietidskrift 1925, 137–147.

    Google Scholar 

  2. MARSHALL, A. W. and F. PROSHAN, Mean life of series and parallel systems, J. Appl. Probab. 7 (1970), 165–174.

    Article  MATH  Google Scholar 

  3. MARSHALL, A. W. and I. OLKIN, “Inequalities: Theory of Majorization and Its Applications”, New York/London/Toronto/Sydney/San Francisco, 1979.

    Google Scholar 

  4. BROMWICH, T. J. I’A., Various extensions of Abel’s Lemma, Proc. London Math. Soc. (2) 6 (1908), 58–76.

    Article  MathSciNet  MATH  Google Scholar 

  5. BROMWICH, T. J. I’A., “An Introduction to the Theory of Infinite Series”, 2nd ed., London, 1955, pp. 57–58, 252, 418–420, 473–474.

    Google Scholar 

  6. KARAMATA, J., “Teorija i Praksa Stieltjes-ova Integrala”, Beograd, 1949.

    Google Scholar 

  7. MARIK, J., Estimate of mean values of integrals and criteria for the convergence of improper integrals (Czech). Casopis Pest. Mat. Fys. 73 (1949), D49 - D52.

    MathSciNet  MATH  Google Scholar 

  8. MYKIS, A. D., Nekotorye neravenstva v teorii integrala Stil’t’esa, In: Teorija Lineinyh Differencial’nyh Uravneii s Zapazdyvajuséim Argumentom. Moskova, 1951.

    Google Scholar 

  9. MYSKIS, A. D. and A. Ja. LEPIN, Ob odnom neravenstve v teorii integrala Stil’t’esa, Ucen. Zap. Latv. Un-ta, Fiz.-mat. Nauki 1 (1952), 17–24.

    Google Scholar 

  10. POPOVICIU, T., On an inequality (in Romanian), Gaz. Mat. Fiz. Ser. A 8 (64) (1959), 451–461.

    MathSciNet  Google Scholar 

  11. KARLIN, S. and W. J. STUDDEN, “Tchebycheff Systems: with Applications in Analysis and Statistics”, New York, London, Sydney, 1966.

    MATH  Google Scholar 

  12. SIMEUNOVIC, D. M., Sur les limites des modules des zéros des polynômes, Mat. Vesnik 4 (19) (1967), 293–298.

    MathSciNet  Google Scholar 

  13. DARST, R. and H. POLLARD, An inequality for the Riemann-Stieltjes integral, Proc. Amer. Math. Soc. 25 (1970), 912–913.

    Article  MathSciNet  MATH  Google Scholar 

  14. BEESACK, P. R., Bounds for Riemann-Stieltjes integrals, R.cky Mount. J. Math. 5 (1975), 75–78.

    Article  MathSciNet  MATH  Google Scholar 

  15. MARIK, J., A remark on Perron integrable functions, unpublished private communication, June, 1974.

    Google Scholar 

  16. PECARIC, J. E., On an inequality of T. Popoviciu, Bul. $ti. Tehn. Inst. Politehn. Timisoara 2, 24 (38) (1979), 9–15.

    Google Scholar 

  17. PECARIC, J. E., On an inequality of T. Popoviciu II, Bul. Sti. Tehn. Inst. Politehn. Timisoara 2, 24 (38) (1979), 44 46.

    Google Scholar 

  18. PECARIC, J. E., A generalization of an inequality of Ky Fan, Univ. Beograd. Publ. E.ektrotehn. Fak. Ser. Mat. Fiz. No. 678–715 (1980), 75–84.

    Google Scholar 

  19. PECARIC, J. E., On some inequalities analogous to Grüss’ inequality, Mat. Vesnik 4 (17) (32) (1980), 197–202.

    Google Scholar 

  20. PECARIC, J. E. and B. D. CRSTICI, Generalizarea unei inegalitati a lui T. Popoviciu, Ses. “Ineg.matem.” Sibiu, 1981.

    Google Scholar 

  21. PECARIC, J. E., On some inequalities for quasi-monotone sequences, Publ. Inst. Math. (Beograd) N.S. 30 (44) (1981), 153–156.

    MathSciNet  Google Scholar 

  22. LACKOVIC, I. B. and Lj. M. KOCIC, On some linear transformations of quasi-monotone sequences, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 634–677 (1979), 208–213.

    MathSciNet  Google Scholar 

  23. LACKOVIC, I. B., On positivity of linear continuous operators defined on the set of monotonic and continuous functions, Univ. B.ograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 735–762 (1982), 87–95.

    MathSciNet  Google Scholar 

  24. KOVACEC, A., A note on Popoviciu’s inequality for bilinear forms, A.al. Num. Th. Approx. 12 (2) (1983), 151–155.

    MathSciNet  MATH  Google Scholar 

  25. KOVACEC, A., On positive definite differences of multilinear forms and their representation, Proc. Roy. Soc. Edinburgh 96A (1984), 97–108.

    Article  MathSciNet  MATH  Google Scholar 

  26. PECARIC, J. E., On the Ostrowski generalization of Cebysev’s inequality, J. Math. Anal. Appl. 102 (1984), 385–392.

    Article  MathSciNet  MATH  Google Scholar 

  27. PECARIC, J. E. and B. SAVIC, O generalizacijama nekih resultata J. Karamate i nekim primjenama, Zbornik Radova AKoV (Beograd), 1984, 245–268.

    Google Scholar 

  28. KOCIC, Lj. M. and I. B. LACKOVIC, Generalized monotonicity and approximations by linear operators, Num. Meth. Approx. Th., Novi Sad, 1985, 1–6.

    Google Scholar 

  29. PECARIC, J. E., B. A. MESIHOVIC, I. Z. MILOVANOVIC and N. M. STOJANOVIC, On some inequalities for convex and V-convex sequences of higher order I, Period. Math. Hungar. 17 (1986), 235–239.

    Article  MathSciNet  MATH  Google Scholar 

  30. PECARIC, J. E., B. A. MESIHOVIC, I. Z. MILOVANOVIC and N. M. STOJANOVIC, On some inequalities for convex and -convex sequences of higher order II, Period Math. Hungar. 17 (1986), 313–320.

    Article  MathSciNet  MATH  Google Scholar 

  31. PECARIC, J. E., Some further remarks on the Ostrowski generalization of Cebysev’s inequality, J. Math. Anal. Appl. 123 (1987), 18–33.

    Article  MathSciNet  MATH  Google Scholar 

  32. FINK, A. M., Majorization for functions with monotone nth derivatives, in manuscript.

    Google Scholar 

  33. BRUNK, H. D., Integral inequalities for functions with nondecreasing increments, Pacific. J. Math. 14 (1964), 783–793.

    Article  MathSciNet  MATH  Google Scholar 

  34. CIESIELSKI, Z., A note on some inequalities of Jensen’s type,Annales Polonici Mathematici 4 (1957–58), 269–274.

    Google Scholar 

  35. PECARIC, J. E., On some inequalities for functions with nondecreasing increments, J. Math. Anal. Appl. 98 (1984), 188–197.

    Article  MathSciNet  MATH  Google Scholar 

  36. PECARIC, J. E., Generalization of some results of H. Burkill and L. Mirsky and some related results, Periodica Math. Hung. 15 (3) (1984), 241–247.

    Article  MATH  Google Scholar 

  37. PECARIC, J. E., On an inequality of G. Szegô, J. Math. Anal. Appl., to appear.

    Google Scholar 

  38. VASIC, P. M. and J. E. PECARIC, Some inequalities for Wright convex functions, to appear.

    Google Scholar 

  39. Two inequalities for functions with nondecreasing increments, to appear.

    Google Scholar 

  40. PECARIC, J. E., Notes on convex functions, General Inequalities 6, Oberwolfach, 1992, 449–454.

    Google Scholar 

  41. PECARIC, J. E, Some inequalities for generalizaed convex functions of several variables, Periodica Math. Hungar. 22 (1991), 83–90.

    Article  MATH  Google Scholar 

  42. PECARIC, J. E. and D. SVRTAN, New refinements of the Jensen inequalities based on samples with repetitions, to appear.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Belgrade, Belgrade, Yugoslavia

    D. S. Mitrinović

  2. University of Zagreb, Zagreb, Croatia

    J. E. Pečarić

  3. Department of Mathematics, Iowa State University of Science and Technology, Ames, Iowa, USA

    A. M. Fink

Authors
  1. D. S. Mitrinović
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. E. Pečarić
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. M. Fink
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Mitrinović, D.S., Pečarić, J.E., Fink, A.M. (1993). Abel’s and Related Inequalities. In: Classical and New Inequalities in Analysis. Mathematics and Its Applications (East European Series), vol 61. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1043-5_12

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-1043-5_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4225-5

  • Online ISBN: 978-94-017-1043-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation