Download PDF - Weighted inequalities for monotone and concave functionsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Weighted inequalities for monotone and concave functions

Authors 1, 2

Affiliations

  1. Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
  2. Department of Mathematics, Luleå University, S-971 87 Luleå, Sweden

Abstract

Characterizations of weight functions are given for which integral inequalities of monotone and concave functions are satisfied. The constants in these inequalities are sharp and in the case of concave functions, constitute weighted forms of Favard-Berwald inequalities on finite and infinite intervals. Related inequalities, some of Hardy type, are also given.

Keywords

ENG
weighted integral inequalities, weighted Hardy inequalities, weighted Hardy inequalities for monotone functions, weighted Favard-Berwald inequality, reverse Hölder inequality, concave functions

Bibliography

  1. K. F. Andersen, Weighted generalized Hardy inequalities for nonincreasing functions, Canad. J. Math. 43 (1991), 1121-1135.
  2. M. Ariño and B. Muckenhoupt, Maximal functions on classical spaces and Hardy's inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (1990), 727-735.
  3. R. W. Barnard and J. Wells, Weighted inverse Hölder inequalities, J. Math. Anal. Appl. 147 (1990), 198-213.
  4. D. C. Barnes, Some complements of Hölder's inequality, ibid. 26 (1969), 82-87.
  5. E. F. Beckenbach and R. Bellman, Inequalities, Springer, New York, 1983.
  6. R. Bellman, Converses of Schwarz's inequality, Duke Math. J. 23 (1956), 429-434.
  7. J. Bergh, V. Burenkov and L. E. Persson, Best constants in reversed Hardy's inequalities for quasimonotone functions, Acta Sci. Math. (Szeged) 59 (1994), 221-239.
  8. L. Berwald, Verallgemeinerung eines Mittelwertsatzes von J. Favard, für positive konkave Funktionen, Acta Math. 79 (1947), 17-37.
  9. C. Borell, Inverse Hölder inequalities in one and several dimensions, J. Math. Anal. Appl. 41 (1973), 300-312.
  10. J. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405-408.
  11. H. Bückner, Untere Schranken für skalare Produkte von Vektoren und für analoge Integralausdrücke, Ann. Mat. Pura Appl. 28 (1949), 237-261.
  12. V. I. Burenkov, On the best constant in Hardy's inequality for 0 < p < 1, Trudy Mat. Inst. Steklov. 194 (1992), 58-62 (in Russian).
  13. R. J. Bushell and W. Okrasiński, Nonlinear Volterra equations with convolution kernel, J. London Math. Soc. 41 (1990), 503-510.
  14. A. P. Calderón and R. Scott, Sobolev type inequalities for p> 0, Studia Math. 62 (1978), 75-92.
  15. M. J. Carro and J. Soria, Weighted Lorentz spaces and the Hardy operator, J. Funct. Anal. 112 (1993), 480-494.
  16. M. J. Carro and J. Soria, Boundedness of some linear operators, Canad. J. Math. 45 (1993), 1155-1166.
  17. A. Clausing, Disconjugacy and integral inequalities, Trans. Amer. Math. Soc. 260 (1980), 293-307.
  18. J. Favard, Sur les valeurs moyennes, Bull. Sci. Math. 57 (1933), 54-64.
  19. P. Frank und G. Pick, Distanzschätzungen im Funktionenraum. I, Math. Ann. 76 (1915), 354-375.
  20. A. García del Amo, On reverse Hardy's inequality, Collect. Math. 44 (1993), 115-123.
  21. G. Grűss, Über das Maximum des absoluten Betrages von (1b-a)ʃabf(x)g(x)dx-(1(b-a)2)ʃabf(x)dxʃabg(x)dx, Math. Z. 39 (1935), 215-226.
  22. G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, 1952.
  23. H. Heinig and L. Maligranda, Chebyshev inequality in function spaces, Real Anal. Exchange 17 (1991-92), 211-247.
  24. H. Heinig and V. D. Stepanov, Weighted Hardy inequalities for increasing functions, Canad. J. Math. 45 (1993), 104-116.
  25. R. A. Hunt, On L(p, q) spaces, Enseign. Math. 12 (1966), 249-276.
  26. S. Karlin and W. J. Studden, Tchebycheff Systems: with Applications in Analysis and Statistics, Pure Appl. Math. 15, Wiley, New York, 1966.
  27. S. Lai, Weighted norm inequalities for general operators on monotone functions, Trans. Amer. Math. Soc. 340 (1993), 811-836.
  28. G. G. Lorentz, Some new functional spaces, Ann. of Math. 51 (1950), 37-55.
  29. L. Maligranda, J. Pečarić and L. E. Persson, Weighted Favard and Berwald inequalities, J. Math. Anal. Appl. 190 (1995), 248-262.
  30. V. M. Manakov, On the best constant in weighted inequalities for Riemann-Liouville integrals, Bull. London Math. Soc. 24 (1991), 442-448.
  31. W. G. Mazja [V. G. Maz'ya], Einbettungssätze für Sobolewsche Räume, Teil 1, Teubner, Leipzig, 1979.
  32. B. Muckenhoupt, Hardy's inequality with weights, Studia Math. 44 (1972), 31-38.
  33. E. Myasnikov, L. E. Persson and V. Stepanov, On the best constants in certain integral inequalities for monotone functions, Acta Sci. Math. (Szeged) 59 (1994), 613-624.
  34. C. J. Neugebauer, Weighted norm inequalities for averaging operators of monotone functions, Publ. Mat. 35 (1991), 429-447.
  35. R. O'Neil, Convolution operators and L(p,q) spaces, Duke Math. J. 30 (1963), 129-142.
  36. B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes in Math. Ser. 219, Longman, 1990.
  37. M. Petschke, Extremalstrahlen konvexer Kegel und komplementäre Ungleichungen, Ph.D. Dissertation, Darmstadt, 1989, 106 pp.
  38. E. Sawyer, Weighted Lebesgue and Lorentz norm inequalities, Trans. Amer. Math. Soc. 281 (1984), 329-337.
  39. E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158.
  40. G. Sinnamon, Weighted Hardy and Opial-type inequalities, J. Math. Anal. Appl. 160 (1991), 434-445.
  41. E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971.
  42. V. D. Stepanov, On boundedness of linear integral operators on the class of monotone functions, Sibirsk. Mat. Zh. 32 (1991), 222-224 (in Russian).
  43. V. D. Stepanov, Integral operators on the cone of monotone functions, J. London Math. Soc. 48 (1994), 465-487.
  44. V. D. Stepanov, Weighted Hardy's inequality for nonincreasing functions, Trans. Amer. Math. Soc. 338 (1993), 173-186.
  45. W. Walter and V. Weckesser, An integral inequality of convolution type, Aequationes Math. 46 (1993), 212-219.
  46. C.-L. Wang, An extension of a Bellman inequality, Utilitas Math. 8 (1975), 251-256.
  47. H.-T. Wang and S.-Y. Chen, Inverse Hölder inequalities with weight tα, J. Math. Anal. Appl. 176 (1993), 92-107.
Studia Mathematica
1995/116/2
Export to PBN, Opens in new tab
Pages:
133-165
Main language of publication
English
Received
1994-09-27
Accepted
1995-03-20
Published
1995
Exact and natural sciences