Skip to main content
SpringerLink
Log in

Convolution, Rearrangement and Related Inequalities

  • Chapter
Inequalities Involving Functions and Their Integrals and Derivatives

Part of the book series: Mathematics and Its Applications ((MAEE,volume 53))

  • 1300 Accesses

Abstract

If \(\lambda > 0,v > 0,\lambda + v < 1\), and z is defined by \({z_n} = \sum\limits_{i + j = n} {{x_i}{y_j},{_r}(x) = {{\left( {\sum {x_i^r} } \right)}^{1/r}},}\) then

$${_{\frac{1}{{1 - \lambda - v}}}}(z) \leqslant {_{\frac{1}{{1 - \lambda }}}}(x){_{\frac{1}{{1 - v}}}}(y),$$
(1)

with equality only if all the x, or all the y, or all the x but one and all the y but one, are zero.

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. Young, W. H., On the multiplication of successions of Fourier constants, Proc. Royal Soc. (A) 87 (1912), 331–339.

    Article  MATH  Google Scholar 

  2. Young, W. H. —, Sur la généralisation du théorème de Parseval, C. R. Acad. Sci. Paris 155 (1912), 30–33.

    MATH  Google Scholar 

  3. Young, W. H. —, On the determination of the summability of a function by means of its Fourier constant, Proc. London Math. Soc. 122 (1913), 71–88.

    Article  MATH  Google Scholar 

  4. Hardy, G. H., J. E. Littlewood and G. Pólya, The maximum of a certain bilinear form, Proc. Lond. Math. Soc. 252 (1926), 265–282.

    Article  MATH  Google Scholar 

  5. Beckner, H., Inequalities in Fourier analysis, Ann. of Math. 102 (1975), 159–182.

    Article  MathSciNet  MATH  Google Scholar 

  6. Brascamp, H. J. and E. H. Lieb, Best constants in Young’s inequality, its converse, and its generlaization to more than three functions, Advances Math. 20 (1976), 151–173.

    Article  MathSciNet  MATH  Google Scholar 

  7. Brascamp, H. J., E. H. Lieb and J. M. Luttinger, A general rearrangement inequality for multiple integrals, J. Func. Anal. 17 (1974), 227–237.

    Article  MathSciNet  MATH  Google Scholar 

  8. Subrahmanyam, M. B., An extremal problem for convolution inequalities, J. Math. Anal. Appl. 87 (1982), 509–516.

    Article  MathSciNet  MATH  Google Scholar 

  9. Boas, P. R. and C. O. Imoru, Elementary convolution inequalities, SIAM J. Math. Anal. 6 (1975), 457–471.

    Article  MathSciNet  MATH  Google Scholar 

  10. Beesack, P. R. and J. E. PeČariĆ, Inequalities involving nth-order means, J. Math. Anal. Appl. 99 (1984), 127–149.

    Article  MathSciNet  MATH  Google Scholar 

  11. Beesack, P. R., Inequalities involving iterated kernels and convolutions, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 247–301 (1969), 11–16.

    MathSciNet  Google Scholar 

  12. PeČariĆ, J. E., On some inequalities analogous to Grüss’ inequality, Mat. Vesnik 417–32 (1980), 197–202.

    MathSciNet  Google Scholar 

  13. Ruderman, H. D., Two new inequalities, Amer. Math. Monthly 59 (1952), 29–32.

    Article  MathSciNet  MATH  Google Scholar 

  14. Lorentz, G. G., An inequality for rearrangements, Amer. Math. Monthly 60 (1953), 176–179.

    Article  MathSciNet  MATH  Google Scholar 

  15. Day, P. W., Rearrangement inequalities, Canad. J. Math. 24 (1972), 930–943.

    Article  MathSciNet  MATH  Google Scholar 

  16. Beck, E., Ein Satz über umerdnungs— Ungleichungen, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 388–409 (1972), 31–35.

    Google Scholar 

  17. Derman, C., G. J. Lieberman and S. M. Ross, Assembly of systems having maximum reliability, Naval. Res. Logist. Quart. 21 (1974), 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  18. Rinott, Y., Multivariate majorization and rearrangement inequalities with some applications to probability and statistics, Israel J. Math. 15 (1973), 60–77.

    Article  MathSciNet  MATH  Google Scholar 

  19. Jurkat, W. B. and H. J. Ryser, Term ranks and permanents of non-negative matrices, J. Algebra 5 (1967), 342–357.

    Article  MathSciNet  MATH  Google Scholar 

  20. Minc, H. Rearrangements, Trans. Amer. Math. Soc. 159 (1971), 497–504.

    Article  MathSciNet  MATH  Google Scholar 

  21. London, D., Rearrangement inequalities involving convex function, Pacific J. Math. 34 (1970), 749–753.

    MathSciNet  MATH  Google Scholar 

  22. Borell, G., A note on an inequality for rearrangements, Pacific J. Math. 47 (1973), 39–41.

    MathSciNet  MATH  Google Scholar 

  23. Abramovich, S., The increase of sums and products dependent on (y1,...,y2) by rearangement of this set, Israel J. Math. 5 (1967), 177–181.

    Article  MathSciNet  MATH  Google Scholar 

  24. Oppenheim, A., Inequalities connected with definite Hermitian forms, II, Amer. Math. Monthly 61 (1954), 463–466.

    Article  MathSciNet  MATH  Google Scholar 

  25. KovaČec, A., A note on Popoviciu’s inequality for bilinear forms, Anal. Numer. Th. Approx. 122 (1983), 151–155.

    MATH  Google Scholar 

  26. KovaČec, A. —, On positive definite differences of multilinear forms and their representation, Proc. Roy. Soc. Edinburg 96 (1984), 97–108.

    Article  MATH  Google Scholar 

  27. Daróczy, Z., Rearrangement inequalities, Publ. Math., Debrecen 20 (1973), 273–279.

    MathSciNet  MATH  Google Scholar 

  28. Metcalf, F. T., A theorem concerning rearrangements, Amer. Math. Monthly 71 (1964), 172–174.

    Article  MathSciNet  MATH  Google Scholar 

  29. Gabriel, R. M., The rearrangement of positive Fourier coefficients, Proc. Lond. Math. Soc. 332 (1932), 32–51.

    Article  MathSciNet  Google Scholar 

  30. Hardy, G. H., J. E. Littlewood and G. Pólya, The maximum of a certain bilinear form, Proc. Lond. Math. Soc. 252 (1926), 265–282.

    Article  MATH  Google Scholar 

  31. Lehman, A. L., A result on rearrangements, Israel J. Math. 1 (1963), 22–28.

    Article  MathSciNet  MATH  Google Scholar 

  32. Duffin, R. J. and A. G. Schaeffer, A redefinement of an inequality of the brothers Markoff, Trans. Amer. Math. Soc. 50 (1941), 517–528.

    Article  MathSciNet  Google Scholar 

  33. Burkill, H., A note on rearrangements of functions, Amer. Math. Monthly 71 (1964), 887–888.

    Article  MathSciNet  MATH  Google Scholar 

  34. Luttinger, J. M. and R. Friedberg, A new rearrangement inequality for multiple integrals, Arch. Rat. Mech. Anal. 61 (1976), 45–64.

    MathSciNet  MATH  Google Scholar 

  35. Duff, G. F. D., Differences, derivatives and decreasing rearrangements, Canad. J. Math. 19 (1967), 1153–1178.

    Article  MathSciNet  MATH  Google Scholar 

  36. Ryff, J. V., Measure preserving transformations and rearrangements, J. Math. Anal. Appl. 31 (1970), 449–458.

    Article  MathSciNet  MATH  Google Scholar 

  37. Duff, G. F. D., Integral inequalities for equimeasurable rearrangements, Can. J. Math. 22 (1970), 408–430.

    Article  MathSciNet  MATH  Google Scholar 

  38. MitrinoviĆ, D. S. and J. E. PeČariĆ, A general integral inequality for the derivative of an equimeasurable rearrangement, C. R. Math. Rep. Acad. Sci. Canada 11 (1989), 201–205.

    MathSciNet  MATH  Google Scholar 

  39. Duff, G. F. D., A general integral inequality for the derivative of an equimeasurable rearrangement, Can. J. Math. 28 (1976), 793–804.

    Article  MathSciNet  MATH  Google Scholar 

  40. Duff, G. F. D. —, Navier Stokes derivative estimates in three space dimensions with boundary values and body forces, C. R. Math. Rep. Acad. Sci. Canada 11 (1989), 195–204.

    MathSciNet  MATH  Google Scholar 

  41. Burkill, J. C., Rearrangements of functions, J. London Math. Soc. 27 (1952), 393–401.

    Article  MathSciNet  MATH  Google Scholar 

  42. Chen, Y. M., Theorems of asymptotoic approximation, Math. Annalen 140 (1960), 360–407.

    Article  MATH  Google Scholar 

  43. Babenko, K. I., On conjugate functions (Russian), Doklady Akad. Nauk SSSR, N. S. 62 (1948), 157–160.

    MathSciNet  MATH  Google Scholar 

  44. Fefferman, C. and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115.

    Article  MathSciNet  MATH  Google Scholar 

  45. Alonso, I. P., A general result on the equivalence between derivation of integrals and weak inequalities for the Hardy-Littlewood maximal operator, Studia Math. 60 (1977), 91–96.

    MathSciNet  MATH  Google Scholar 

  46. Kurtz, D. S., Weighted norm inequalities for the Hardy-Littlewood maximal function for one parameter rectangles, Studia Math. 53 (1975), 39–54.

    MathSciNet  MATH  Google Scholar 

  47. Caffarelli, L. A. and C. P. Calderón, Weak type estimates for the Hardy-Littlewood maximal functions, Studia Math. 49 (1974), 217–223.

    MATH  Google Scholar 

  48. Heinig, H., A weighted norm inequality for a maximal function, Notices AMS 26 (1979), A–206.

    Google Scholar 

  49. Imoru, C. O., Variations of integral means and Hardy-Littlewood maximal functions, Ann. Soc. Sci. Bruxelles 96 (1983), 165–173.

    MathSciNet  Google Scholar 

  50. Muckenhoupt, B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.

    Article  MathSciNet  MATH  Google Scholar 

  51. Chen,Y. M., On conjugate functions, Can. J.Math. 15 (1963), 486–494.

    Article  MATH  Google Scholar 

  52. Muckenhoupt, B. and R. L. Wheeden, Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform, Studia Math. 55 (1976), 279–294.

    MathSciNet  MATH  Google Scholar 

  53. Duff, G. F. D. The variation of maximal function, Bull. Malaysian Math. Soc. 22 (1979), 75–88.

    MathSciNet  MATH  Google Scholar 

  54. Pichorides, S. K., On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov, Studia Math. 44 (1972), 154–179.

    MathSciNet  Google Scholar 

  55. Davis, B., On the weak-type (1,1) inequality for conjugate functions, Proc. Amer. Math. Soc. 44 (1974), 307–311.

    MathSciNet  MATH  Google Scholar 

  56. O’neil, R. and G. Weiss, The Hilbert transform and rearrangement of functions, Studia Math. 23 (1963), 189–198.

    MathSciNet  Google Scholar 

  57. Bennett, C., A best constant for Zygmund’s conjugate function inequality, Proc. Amer. Math. Soc. 56 (1976), 256–260.

    MathSciNet  MATH  Google Scholar 

  58. Andersen, K. F., Weighted norm inequalities for Hilbert transforms and conjugate functions of even and odd functions, Proc. Amer. Math. Soc. 56 (1976), 99–107.

    Article  MathSciNet  MATH  Google Scholar 

  59. Stein, E. M., “Singular Integrals and Differentiability Properties of Functions,” Princeton, 1970.

    Google Scholar 

  60. Hardy, G. H. and J. E. Littlewood, Some properties of fractional integrals, Math. Z. 27 (1928), 565–606.

    Article  MathSciNet  MATH  Google Scholar 

  61. Sobolev, S. L., On a theorem of functional analysis, Mat. Sb. 446 (1938), 471–497. English transi., Amer. Math. Soc. Transi. (2) 34 (1963), 39-68.

    MATH  Google Scholar 

  62. Hedberg, L. I., On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510.

    Article  MathSciNet  Google Scholar 

  63. Luxemburg, W. A. J., Rearrangement invariant Banach function spaces, Queen’s Papers in Pure and Applied Math. 10 (1967), 83–144.

    Google Scholar 

  64. Lorentz, G. G. and T. Shimogaki, Interpolation theorems for operators in function spaces, J. Punc. Anal. 2 (1968), 31–51.

    Article  MathSciNet  MATH  Google Scholar 

  65. Chong, K. M. and N. M. Rice, Equimeasurable rearrangements of functions, Queen’s Papers in Pure and Applied Mathematics 28 (1971).

    Google Scholar 

  66. Hardy, G. H. and J. E. Littlewood, Notes on the theory of series (XIII): Some new properties of Fourier constants, J. London Math. Soc. 6 (1931), 3–9.

    Article  MathSciNet  Google Scholar 

  67. Hardy, G. H. —, A new proof of a theorem on rearrangements, J. London Math. Soc. 23 (1949), 163–168.

    Article  MATH  Google Scholar 

  68. Littlewood, J. E., On a theorem of Paley, J. Lon. Math. Soc. 29 (1954), 387–395.

    Article  MathSciNet  MATH  Google Scholar 

  69. Littlewood, J. E. —, On ineuqlities between f and f*, J. London Math. Soc. 35 (1960), 352–365.

    Article  MathSciNet  MATH  Google Scholar 

  70. Paley, R. E. A. C., Some theorems on orthogonal functions, Studia Math. 3 (1931), 336-238.

    Google Scholar 

  71. Keogh, F. R., On a problem of Hardy and Littlewood concerning rearrangements, J. London Math. Soc. 36 (1961), 353–361.

    Article  MathSciNet  MATH  Google Scholar 

  72. Keogh, F. R. —, On some inequalities of Littlewood and a problem on rearrangements, J. London Math. Soc. 36 (1961), 362-276.

    Google Scholar 

  73. Bachelis, G. A., On the upper and lower majorant properties of L p, Quart. J. Math. (Oxford) 242 (1973), 119–128.

    Article  MathSciNet  MATH  Google Scholar 

  74. Montgomery, H. L., Topics in multiplicative number theory, in “Lecture Notes in Mathematics,” Vol. 227, 1971, 187 pp.

    Article  Google Scholar 

  75. Montgomery, H. L. —, A note on rearrangements of Fourier coefficients, Ann. Inst. Fourier, Grenoble 262 (1976), 29–34.

    Article  MathSciNet  MATH  Google Scholar 

  76. Lehmer, D. H., On a problem of Hardy and Littlewood, J. London Math. Soc. 34 (1959), 395–396.

    Article  MathSciNet  MATH  Google Scholar 

  77. Kahane, J. P., Sur un problème de Littlewood, Proc. Kon. Ned. Ak. v. Wet., Amsterdam 60 (1957), 268–271.

    MathSciNet  MATH  Google Scholar 

  78. O’neil, R., Convolution operators and L(p,q) spaces, Duke Math. J. 30 (1963), 129–142.

    Article  MathSciNet  Google Scholar 

  79. Komaroff, N., Rearrangement and matrix product inequalities, Linear Alg. and Appl. 140 (1990), 155–161.

    Article  MathSciNet  MATH  Google Scholar 

  80. Saithoh, S., A fundamental inequality in the convolution of L 2 functions on the half line, Proc. Amer. Math. Soc. 91 (1984), 285–286.

    MathSciNet  Google Scholar 

  81. Saithoh, S. —, On the convolution of L 2 functions, Kodai Math. J. 9 (1986), 50–57.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Belgrade, Yugoslavia

    D. S. Mitrinović

  2. University of Zagreb, Yugoslavia

    J. E. Pečarić

  3. Iowa State University, Ames, 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

© 1991 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Mitrinović, D.S., Pečarić, J.E., Fink, A.M. (1991). Convolution, Rearrangement and Related Inequalities. In: Inequalities Involving Functions and Their Integrals and Derivatives. Mathematics and Its Applications, vol 53. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3562-7_10

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-3562-7_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5578-9

  • Online ISBN: 978-94-011-3562-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation