Skip to main content
SpringerLink
Log in

Norm inequalities for derivatives

  • Conference paper
  • First Online:
Ordinary and Partial Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 846))

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 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight 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. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.

    MATH  Google Scholar 

  2. V. V. Arestov, "Exact inequalities between norms of functions and their derivatives," Acta Scientiarum Mathematicarum, 33(1972), 243–267.

    MathSciNet  Google Scholar 

  3. _____, "The best approximation to differentiation operators," Matematicheskie Zametkie, 1(1967), part 2; 149–154.

    MathSciNet  MATH  Google Scholar 

  4. _____, "On the best approximation of the operators of differentiation and related questions," Approximation Theory, Proceeding of Conference in Poznan, Poland 1972. Reidel Publishing Co., Boston.

    Google Scholar 

  5. V. I. Berdyshev, "The best approximation in L(0, ∞) to the differentiation operator," Matematicheskie Zametki, 5(1971), 477–481.

    MATH  Google Scholar 

  6. Yu. G. Bosse (G. E. Shilov), "O neravenstvakh mezhdu proizovdnymi," Mosk. Univ. Sbornik raport nauchnykh studencheskikh kruzhkov, 1937, 17–27.

    Google Scholar 

  7. J. Bradley and W. N. Everitt, "On the inequality ‖f″‖2 ≤ K‖f‖ ‖f(4)‖," Quart. J. Math. 25(1974), 241–252.

    Article  MathSciNet  MATH  Google Scholar 

  8. __________, "Inequalities associated with regular and singular problems in the calculus of variations," Trans. Amer. Math. Soc. 182(1973), 303–321.

    Article  MathSciNet  MATH  Google Scholar 

  9. K. W. Brodlie and W. N. Everitt, "On an inequality of Hardy and Littlewood," Proc. Roy. Soc. Edinburgh A72, (1975), 179–186.

    MathSciNet  MATH  Google Scholar 

  10. A. S. Cavaretta, "An elementary proof of Kolmogorov's theorem," Amer. Math. Monthly 81(1974), 480–486.

    Article  MathSciNet  MATH  Google Scholar 

  11. _____, "One-sided inequalities for the successive derivatives of a function," Bull. Amer. Math. Soc. 82(1976), 303–305.

    Article  MathSciNet  MATH  Google Scholar 

  12. _____, "A refinement of Kolmogorov's inequality," MRC Technical Summary Report #1788.

    Google Scholar 

  13. H. Cartan, "On inequalities between the maxima of the successive derivatives of a function," Comptes Rendus Sci. Acad. 208(1939), 414–426.

    Google Scholar 

  14. M. W. Certain and T. G. Kurtz, "Landau-Kolmogorov inequalities for semigroups and groups," Proc. Amer. Math. Soc. 63(1977), 226–230

    Article  MathSciNet  MATH  Google Scholar 

  15. Ceisielski and J. Musielak, "Approximation Theory," Proceedings of the Conference jointly organized by the Mathematical Institute of the Polish Academy of Sciences and the Institute of Mathematics of the Adam Mickiewicz University held in Proznan 22–26 August, 1972. D. Reidel Publishing Co., Boston.

    Google Scholar 

  16. E. T. Copson, "On two integral inequalities," Proc. Roy. Soc. Edinburgh, 77A(1977), 325–328.

    Article  MathSciNet  MATH  Google Scholar 

  17. _____, "On two inequalities of Brodlie and Everitt," Proc. Roy. Soc. Edinburgh, 77A(1977), 329–333.

    Article  MathSciNet  MATH  Google Scholar 

  18. Z. Ditzian, "Some remarks on inequalities of Landau and Kolmogorov," Equ. Math. 12(1975), 145–151.

    MathSciNet  MATH  Google Scholar 

  19. _____, "Note on Hille's question," Aequationes Mathematicae 15(1977), 143–144.

    Article  MathSciNet  MATH  Google Scholar 

  20. W. D. Evans and A. Zettl, "Norm inequalities involving derivatives," Pro. Roy. Soc. Edinburgh, 82A(1978), 51–70.

    Article  MathSciNet  MATH  Google Scholar 

  21. W. N. Everitt, "On an extension to an integro-differential inequality of Hardy, Littlewood and Polya," Proc. Roy. Soc. Edinburgh (A) 69(1972), 295–333.

    MathSciNet  MATH  Google Scholar 

  22. W. N. Everitt and M. Giertz, "On the integro-differential inequality ‖f′¦ 22 ≤ K‖f‖p ‖f″‖q," J. Math. Anal. and Appl. 45(1974), 639–653.

    Article  MathSciNet  MATH  Google Scholar 

  23. __________, "Some inequalities associated with certain ordinary differential operators," Math. Z. 126(1972), 308–326.

    Article  MathSciNet  MATH  Google Scholar 

  24. __________, "Inequalities and separation for certain ordinary differential operators," P. London Math. Soc. 3, 28, (1974), 352–372.

    Article  MathSciNet  MATH  Google Scholar 

  25. W. N. Everitt and D. S. Jones, "On an integral inequality," Proc. Roy. Soc. London (A), (to appear).

    Google Scholar 

  26. W. N. Everitt and A. Zettl, "On a class of integral inequalities," J. London Math. Soc. (2), 17(1978), 291–303.

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Friedman, "Partial Differential Equations," New York, 1969.

    Google Scholar 

  28. A. M. Fink, "Best possible approximation constants," Trans. Amer. Math. Soc. 226(1977), 243–255.

    Article  MathSciNet  MATH  Google Scholar 

  29. V. N. Gabushin, "Inequalities for norms of a function and its derivatives in Lp metrics," Mat. Zam. 1(1967), 291–8.

    MATH  Google Scholar 

  30. _____, "Exact constants in inequalities between norms of derivatives of functions," Mat. Zam. 4(1968), 221–32.

    MATH  Google Scholar 

  31. _____, "The best approximation for differentiation operators on the half-line," Mat. Zam. 6(1969), 573–82.

    Google Scholar 

  32. G. Gindler and J.A. Goldstein, "Dissipative operator versions of some classical inequalities," J. D'Analyse Math. 28(1975), 213–238.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Goldstein, "On improving the constants in the Kolmogorov inequalities," pre print.

    Google Scholar 

  34. A. Gorny, "Contributions to the study of differentiable functions of a real variable," Acta Math. 71(1939), 317–358.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. Hadamard, "Sur le module maximum d'une fonction et de ses derivées," C. R. des Séances de l'annee 1914. Soc. Math. de France (1914), 66–72.

    Google Scholar 

  36. G. H. Hardy and J. E. Littlewood, "Some integral inequalities connected with the calculus of variations," Quart. J. Math. Oxford Ser. 2, 3, (1932), 241–252.

    Article  MATH  Google Scholar 

  37. G. H. Hardy, J. E. Littlewood and G. Polya, "Inequalities, Cambridge, 1934.

    Google Scholar 

  38. E. Hille, "Generalizations of Landau's inequality to linear operators," Linear operators and approximation," (Edited by P.L. Butzer, et. al.) Birkhäuser Verlag, Basel and Stuttgart 1972.

    Google Scholar 

  39. _____, "Remark on the Landau-Kallman-Rota inequality," Aequationes Mat. 4(1970), 239–240.

    Google Scholar 

  40. _____, "On the Landau-Kallman-Rota inequality," J. of Approx. Theory 6(1972), 117–122.

    Article  MathSciNet  MATH  Google Scholar 

  41. E. Hille and R. S. Phillips, "Functional Analysis and Semigroups," Amer. Math. Soc. Coll. Publ. 31, Rev. ed. Providence, 1957.

    Google Scholar 

  42. J. A. R. Holbrook, "A Kallman-Rota-Kato inequality for nearly euclidean spaces," pre-print.

    Google Scholar 

  43. R. R. Kallman and G. C. Rota, "On the inequality ‖f′‖2 ≤ 4‖f‖ ‖f″‖," Inequalities II (O. Shisha, ed.), Academic Press, New York, 1970, 187–192.

    Google Scholar 

  44. T. Kato, "On an inequality of Hardy, Littlewood and Polya," Advances in Math. 7(1971), 217–218.

    Article  MathSciNet  MATH  Google Scholar 

  45. A. N. Kolmogorov, "On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval," Amer. Math. Soc. transl. (1) 2(1962), 233–243.

    Google Scholar 

  46. N. P. Kupcov, "Kolmogorov estimates for derivatives in L2(0, ∞)," Procedings of the Steklov Institute of Mathematics, 138(1975), AMS transl. 1977, 101–125.

    Google Scholar 

  47. S. Kurepa, "Remarks on the Landau inequality," Aequations Math. 4(1970), 240–241.

    MathSciNet  Google Scholar 

  48. M. K. Kwong and A. Zettl, "Remarks on best constants for norm inequalities smong powers of an operator," J. Approx. Theory, 26(1979), 249–258.

    Article  MathSciNet  MATH  Google Scholar 

  49. __________, "Norm inequalities for dissipative operators on inner product spaces," Houston J. Math. (to appear).

    Google Scholar 

  50. __________, "An extension of the Hardy-Littlewood inequality," Proc. Amer. Math. Soc. 77(1979), 117–118.

    Article  MathSciNet  MATH  Google Scholar 

  51. __________, "Ramifications of Landau's inequality," Proc. Roy. Soc. Edinburgh. (to appear).

    Google Scholar 

  52. __________, "Landau's inequality," Rocky Mountain J. Math. (to appear).

    Google Scholar 

  53. __________, "Norm inequalities of product form in weighted LP spaces."

    Google Scholar 

  54. E. Landau, "Einige Ungleichungen für zweimal differenzierbare Funktionen," Proc. London Math. 13(1913), 43–49.

    MATH  Google Scholar 

  55. Ju. I. Ljubić (or Yu Lyubich), "On inequalities between the powers of a linear operator," Transl. Amer. Math. Soc. Ser. (2) 40(1964), 39–84; translated from Irv. Akad. Nauk. SSSR Ser. Mat. 24(1960), 825–864.

    Article  Google Scholar 

  56. A. P. Matorin, "Inequalities between the maximum absolute values of a function and its derivatives on the half-line," Ukrain. Mat. Zh. 7(1955), 262–266.

    MathSciNet  Google Scholar 

  57. D. S. Mitrinovic, "Analytic Inequalities," Springer-Verlag, Berlin, 1970.

    Book  MATH  Google Scholar 

  58. L. Nirenberg, "Remarks on strongly elliptic partial differential equations," Comm. Pure Appl. Math., 8(1955), 648–674.

    Article  MathSciNet  MATH  Google Scholar 

  59. B. Sz. Nagy, "Über Integralungleichungen zwischen einer Funktion und ihrer Ableitung," Acta. Sci. Math. 10(1941), 64–74.

    MATH  Google Scholar 

  60. B. Neta, "On determination of best possible constants in integral inequalities involving derivatives," pre-print.

    Google Scholar 

  61. R. M. Redheffer, "Über eine beste Ungleichung zwischen den Normen von f, f′, f″," Math. Zeitschr. 80(1963), 390–397.

    Article  MathSciNet  MATH  Google Scholar 

  62. I. J. Schönberg, "The elementary cases of Landau's problem of inequalities between derivatives," Amer. Math. Monthly, 80(1973), 121–158.

    Article  MathSciNet  Google Scholar 

  63. I. J. Schönberg, and A. Cavaretta, "Solution of Landau's problem concerning higher derivatives on the half line," MRCT. S. R. 1050, Madison, Wisconsin, 1970.

    Google Scholar 

  64. G. E. Shilov, "O neravenstvah merzdu proizvodnymi," Sbornik Rabot Studenceskih Naucnyh Kruzkov Moskovskogo Gosudarstvennogo Universiteta, 1937, 17–27.

    Google Scholar 

  65. S. B. Stechkin, "Inequalities between norms of derivatives of an arbitrary functions," Acta. Sci. Math. 26(1965), 225–230.

    MathSciNet  Google Scholar 

  66. _____, "The Inequalities between upper bounds for the derivatives of an arbitrary function on the half-line," Mat. Zametki I V. 6(1967), 665–674.

    MathSciNet  MATH  Google Scholar 

  67. E. M. Stein, "Functions of exponential type," Annals of Math. (2), 65(1957), 582–592.

    Article  MathSciNet  MATH  Google Scholar 

  68. W. Trebels and V. I. Westphal, "A note on the Landau-Kallman-Rota-Hille inequality," Linear Operators and Approximation (Edited by P.L. Butzer et. al). Birkhäuser Verlag, Basel and Stuttgart, 1972.

    Google Scholar 

  69. L. V. Taikov, "Inequalities of Kolmogorov type and the best formulae for numerical differentiation," R. Mat. Zam. 4(1968), 233–238.

    MathSciNet  Google Scholar 

  70. V. G. Solyer, "On an inequality between the norms of a function and its derivative," Izvestia Vysshikh Ucbebaykh Zavedemy Matematika 2 (165), 1976.

    Google Scholar 

Download references

Authors
  1. W. N. Everitt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. D. Sleeman
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

W. N. Everitt B. D. Sleeman

Rights and permissions

Reprints and permissions

Copyright information

© 1981 Springer-Verlag

About this paper

Cite this paper

Everitt, W.N., Sleeman, B.D. (1981). Norm inequalities for derivatives. In: Everitt, W.N., Sleeman, B.D. (eds) Ordinary and Partial Differential Equations. Lecture Notes in Mathematics, vol 846. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089840

Download citation

  • DOI: https://doi.org/10.1007/BFb0089840

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10569-5

  • Online ISBN: 978-3-540-38538-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation