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Hardy’s, Carleman’s and Related Inequalities

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Inequalities Involving Functions and Their Integrals and Derivatives

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

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Abstract

Some preliminaries. Theorem 1. If p > 1, a n ≥ 0, and A n = a 1 + a 2 +... + a n, then

$${\sum\limits_1^\infty {\left( {\frac{{{A_n}}}{n}} \right)} ^p} < {\left( {\frac{p}{{p - 1}}} \right)^p}\sum\limits_1^\infty {a_n^p} ,$$
(1.1)

unless all the a are zero. The constant \({\left( {\frac{p}{{p - 1}}} \right)^p}\) is the best possible.

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References

  1. Hardy, G. H., Note on a theorem of Hilbert, Math. Zeitschr. 6 (1920), 314–317.

    Article  MATH  Google Scholar 

  2. Landau, E., A note on a theorem concerning series of positive terms, J. Lond. Math. Soc. 1 (1926), 38–39.

    Article  MATH  Google Scholar 

  3. Broadbent, T. A. A., A proof of Hardy’s convergence theorem, J. Lond. Math. Soc. 3 (1928), 93–96.

    Article  Google Scholar 

  4. Elliott, E. B., A simple exposition of some recently proved facts as a convergency, J. Lond. Math. Soc. 1 (1926), 93–96.

    Article  MATH  Google Scholar 

  5. Grandjot, K., On some identities relating to Hardy’s convergence theorem, J. Lond. Math. Soc. 3 (1928), 114–117.

    Article  MathSciNet  MATH  Google Scholar 

  6. Hardy, G. H., Notes on some points in the integral calculus (LX), Messenger of Math. 54 (1925), 150–156.

    Google Scholar 

  7. Kaluza, Th. and G. Szegö, Über Reihen mit lauter positiven Gliedern, J. Lond. Math. Soc. 2 (1927), 266–272.

    Article  MATH  Google Scholar 

  8. Knopp, K., Über Reihen mit positiven Gliedern, J. Lond. Math. Soc. 3 (1928), 205–211.

    Article  MathSciNet  MATH  Google Scholar 

  9. Carleman, T., Sur les fonctions quasi-analytiques, in “Conférences faites au cinquième congrès des mathématiciens Scandinaves,” Helsingfores, 1923, pp. 181–196.

    Google Scholar 

  10. PÓlya, G., Proof of an inequality, Proc. London Math. Soc. 242 (1926), Records of Proc. 1vii.

    Google Scholar 

  11. Valiron, G., “Lectures on the General Theory of Integral Functions,” Toulouse, 1923.

    Google Scholar 

  12. Ostrowski, A. M., Über quasi-analytische Funktionen und Bestimmtheit asymptotischer Entwicklungen, Acta Math. 53 (1929), 181–266.

    Article  MathSciNet  MATH  Google Scholar 

  13. Hardy, G. H., J. E. Littlewood and G. Pólya, “Inequalities,” Cambridge, 1934.

    Google Scholar 

  14. Knopp, K., Über Reihen mit positiven Gliedern (2te Mitteilung), J. Lond. Math. Soc. 5 (1930), 13–21.

    Article  MathSciNet  MATH  Google Scholar 

  15. Hardy, G. H., Notes on some points in the integral calculus (LXIV), Messenger of Math. 57 (1928), 12–16.

    Google Scholar 

  16. Copson, E. T., Note on series of positive terms, J. Lond. Math. Soc. 2 (1927), 9–12.

    Article  MathSciNet  MATH  Google Scholar 

  17. Copson, E. T. —, Note on series of positive terms, J. Lond. Math. Soc. 3 (1928), 49–51.

    Article  MathSciNet  MATH  Google Scholar 

  18. Hardy, G. H., Remarks on three recent notes in the Journal, J. Lond. Math. Soc. 3 (1928), 166–169.

    Article  MATH  Google Scholar 

  19. Hardy, G. H. —, The constants of certain inequalities, J. Lond. Math. Soc. 8 (1933), 114–119.

    Article  Google Scholar 

  20. Hardy, G. H. and J. E. Littlewood, Notes on the theory of seris (XII): On certain inequalities connected with the calculus of variations, J. Lond. Math. Soc. 5 (1930), 283–290.

    Article  Google Scholar 

  21. Bliss, G. A., An integral inequality, J. Lond. Math. Soc. 5 (1930), 40–46.

    Article  MathSciNet  MATH  Google Scholar 

  22. Sunouchi, G. and N. Takagi, A generalization of the Carleman inequality theorem, Proceedings Phys.-Math. Soc. Japan 16 (1934), 164–166.

    Google Scholar 

  23. Knopp, K., Neuere Sätze: Reihen mit positiven Gliedern, Math. Zeitschr. 30 (1929), 387–413.

    Article  MathSciNet  Google Scholar 

  24. Mulholland, H. P., On the generalisation of Hardy’s inequality, J. Lond. Math. Soc. 7 (1934), 208–214.

    Article  MathSciNet  Google Scholar 

  25. Mulholland, H. P. —, Concerning the generalization of the Young-Hausdorff theorem and the Hardy-Littlewood theorems on Fourier constants, Proc. Lond. Math. Soc. 35 (1934), 257–293.

    Article  MathSciNet  Google Scholar 

  26. Mulholland, H. P. —, The generlization of certain inequality theorems involving powers, Proc. Lond. Math. Soc. 332 (1932), 481–516.

    Article  MathSciNet  Google Scholar 

  27. Mulholland, H. P. —, Some theorems on Dirichlet series with positive coefficients and related integrals, Proc. Lond. Math. Soc. 29 (1927), 141–158.

    Google Scholar 

  28. Hardy, G. H. and J. E. Littlewood, Elementary theorems concerning power series with positive coefficients and moment constants of positive functions, J. f. Math. 157 (1927), 141–158.

    MATH  Google Scholar 

  29. Levin, V., Two remarks on van der Corput’s generalisations of Knopp’s inequality, Proc. Akad. Wet. Amsterdam 40 (1937), 429–431.

    Google Scholar 

  30. Van Der Corput, J. G., Generalizations of an inequality of Knopp, Proc. Akad. Wet. Amsterdam 39 (1936), 1053–1055.

    Google Scholar 

  31. Hardy, G. H. and J. E. Littlewood, Some new properties of Fourier constants, Math. Annalen 97 (1927), 159–209 (199).

    Article  MathSciNet  Google Scholar 

  32. Levin, V., O neravenstvah. III: neravenstva, vypolnjamye geometriceskim srednim neotricatel’noi funkcii, Matem. Sbornik 446 (1938), N.2, 325–331.

    Google Scholar 

  33. Van Der Corput, J. G., Generalisations of Carleman’s inequality, Proc. Acad. Wet. Amsterdam 39 (1936), 906–911.

    Google Scholar 

  34. Chow, Y. C., A note on Hardy’s inequality and similar inequalities, J. London Math. Soc. 14 (1939), 88–93.

    Article  Google Scholar 

  35. Flett, T. M., A note on some inequalities, Proc. Glasgow Math. Assoc. 4 (1959), 7–15.

    Article  MathSciNet  Google Scholar 

  36. Levinson, N., Generalizations of an inequality of Hardy, Duke Math. J. 31 (1964), 389–394.

    Article  MathSciNet  MATH  Google Scholar 

  37. Boas, R. P., Inequalities for monotonic series, J. Math. Anal. Appl. 1 (1960), 121–126.

    Article  MathSciNet  MATH  Google Scholar 

  38. Konyuskov, A. A., Beste approximationer durch trigonometrische Polynome und Fourierkoeffizienten, Mat. Sb. (N.S.) 4486 (1958), 53–84.

    MathSciNet  Google Scholar 

  39. Levin, V. I. and S. B. Steckin, Inequalities, Amer. Math. Soc. Translations, Series 214 (1960), 1–29.

    MathSciNet  Google Scholar 

  40. Beesack, P. R., Hardy’s inequality and its extensions, Pacif. J. Math. 11 (1961), 39–61.

    MathSciNet  MATH  Google Scholar 

  41. De Bruijn, N. G., Carleman’s inequality for finite series, Proc. Akad. Wet. Amsterdam Ser. A. 44 (1963), 505–514.

    Google Scholar 

  42. Wilf, H. S., Finite sections of the classical inequalities, Nederl. Akad. Wet. Amsterdam. Proc. Ser. A. 65 (1962), 340–343, (= Indagationes Mathematicae 24).

    MathSciNet  MATH  Google Scholar 

  43. Sysoeva, F. A., Verallgemeinerung einer Ungleichung von Hardy, Izvestija Vysš. Učebn. Zaved., Mat. 1965 496 (1965), 140–141.

    MathSciNet  Google Scholar 

  44. Kadlec, J. and A. Kufner, Characterization of functions with zero traces by integrals with weight functions, I, Cas. Pešt. Mat 91 (1966), 463–470; II ibid, 92 (1967), 16-28.

    MathSciNet  MATH  Google Scholar 

  45. Godunova, E. K., Inequalities with convex functions, Izv. Vyss. Ucebn. Zaved. Math. 4 (1965), 45–63.

    MathSciNet  Google Scholar 

  46. Akerberg, B., A variant of the proofs of some inequalities, Proc. Cambridge Philos. Soc. 571 (1961), 184–186.

    Article  MathSciNet  Google Scholar 

  47. Godunova, E. K., Nekotorye neravenstva tipa neravenstv Karlemans, Ucen. Zap. Gos. Ped. Inst, im Lenina, Moskva 277 (1970), 176–191.

    Google Scholar 

  48. Boyd, D. W., On the exponent of an osculatory packing, Can. J. Math. 23 (1971), 355–363.

    Article  MATH  Google Scholar 

  49. Godunova, E. K., Integral’nye neravenstva s vypuklymi funkcijami, Mat. šestoĭ mez. fiz.-mat. nauč. Konf. Dal’nego Vostoka, Diff. int. ur. 3 (1967), 65–69.

    Google Scholar 

  50. Boas, R. P., Some integral inequalities related to Hardy’s inequality, Journal d’Analyse Math. 23 (1970), 53–63.

    Article  MathSciNet  MATH  Google Scholar 

  51. Bennett, C and K. Rudnick, On Lorentz-Zygmund spaces, (98 page preprint).

    Google Scholar 

  52. Talenti, G., Una diseguaglianza integrale, Boll. Un. mat. Ital 213 (1966), 25–34.

    MathSciNet  Google Scholar 

  53. Talenti, G. —, Sopra una diseguaglianza integrale, Ann. Scuola Norm. Sup. Pisa 313 (1967), 167–188.

    MathSciNet  Google Scholar 

  54. Tomaselli, G., A class of inequalities, Bol. Un. mat. Ital. IV Ser. 2 (1969), 622–631.

    MathSciNet  MATH  Google Scholar 

  55. Muckenhoupt, B., Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38.

    MathSciNet  MATH  Google Scholar 

  56. Izumi, M. and S. I. Izumi, On some inequalities for Fourier series, Journal d’Analyse Math. 21 (1968), 277–291.

    Article  MathSciNet  MATH  Google Scholar 

  57. Hardy, G. H., “Divergent Series,” Cambridge, 1948.

    Google Scholar 

  58. Petersen, G. M., An inequality of Hardy’s, Quart. J. Math. Oxford Ser. 132 (1962), 237–240.

    Article  Google Scholar 

  59. Davies, G. S. and G. M. Petersen, On an inequality of Hardy’s (II), Quart. J. Math. Oxford Ser. 152 (1964), 35–40.

    Article  MathSciNet  Google Scholar 

  60. Izumi, M., S. I. Izumi and G M. Petersen, On Hardy’s inequality and its generalization, Tôhoku Math. J. 21 (1969), 601–613.

    Article  MathSciNet  MATH  Google Scholar 

  61. Leindler, L., Generalization of inequalities of Hardy and Littlewood, Acta Sci. Math. (Szeged) 31 (1970), 279–285.

    MathSciNet  MATH  Google Scholar 

  62. Németh, J., Generalizations of the Hardy-Littlewood inequality, Acta Sci. Math. (Szeged) 32 (1971), 295–299.

    MathSciNet  MATH  Google Scholar 

  63. Németh, J. —, Generalizations of the Hardy-Littlewood inequality, II, Acta Sci. Math. (Szeged) 35, 127–134.

    Google Scholar 

  64. Leindler, L., Inequalities of Hardy-Littlewood type, Analysis Math. 2 (1976), 117–123.

    Article  MathSciNet  MATH  Google Scholar 

  65. Shum, D. T., On integral inequalities related to Hardy’s, Can. Math. Bull. 14 (1971), 225–230.

    Article  MathSciNet  MATH  Google Scholar 

  66. Florkiewicz, B. and A. Rybarski, A certain modification of Hardy’s inequality, Colloquium Math. 26, (1972), 345–348.

    MathSciNet  MATH  Google Scholar 

  67. Florkiewicz, B. —, On an integral inequality connected with Hardy’s inequality (III), Colloquium Math. 27 (1973), 293–296.

    MathSciNet  MATH  Google Scholar 

  68. Bennett, C., Intermediate spaces and the class L log+ L, Arkiv for Math. 11 (1973), 215–228.

    Article  MATH  Google Scholar 

  69. Gluk, L., On an extension of Hardy’s inequality and some weighted norm inequalities for the indefinite integral of conjugagte functions, Notices Amer. Math. Soc. 11 (1974), A-163.

    Google Scholar 

  70. Flett, T. M., Some elementary inequalities for integrals with applications to Fourier Transforms, Proc. London Math. Soc. 293 (1974), 538–556.

    Article  MathSciNet  Google Scholar 

  71. Andersen, K. F., On Hardy’s inequality and Laplace transforms in weighted rearrangement invariant space, Proc. Amer. Math. Soc. 39 (1973), 295–299.

    MathSciNet  MATH  Google Scholar 

  72. Copson, E. T., Some integral inequalities, Proc. Roy. Soc. Edinburgh A 75 13 (1975-76), 157–164.

    MathSciNet  Google Scholar 

  73. Heinig, H., Some extensions of Hardy’s inequality, SIAM J. Math. Anal. 6 (1975), 698–713.

    Article  MathSciNet  MATH  Google Scholar 

  74. Imoru, C. O., On some integral inequalities related to Hardy’s, Canad. Math. Bull. 20 (1977), 307–312.

    Article  MathSciNet  MATH  Google Scholar 

  75. Musielak, H., An inequality for double fractional integrals, Funct. Approx., Comment. Math. 2 (1976), 183–195.

    MathSciNet  MATH  Google Scholar 

  76. Peaslee, D. C. and W. A. Coppel, On an inequality of Bombieri, J. Australian Math. Soc. 9 (1969), 339–404.

    Article  MathSciNet  Google Scholar 

  77. Bombieri, E., Sulla seconds variazione della fuzione di Koebe, Boll. Un. Mat. Ital. 22 (1967), 25–32.

    MathSciNet  MATH  Google Scholar 

  78. Shum, D. T., A Generalization of Bombieri’s Inequality, manuscript.

    Google Scholar 

  79. Bradley, J. S., Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405–408.

    Article  MathSciNet  MATH  Google Scholar 

  80. Heinig, H., Variations of Hardy’s inequality, Real Analysis Exchange 5 (1979-80), 61–81.

    MathSciNet  Google Scholar 

  81. Beesack, P. R. and H. Heinig, Hardy’s inequalities with indices less than 1, Proc. Amer. Math. Soc. 83 (1981), 532–536.

    MathSciNet  MATH  Google Scholar 

  82. Kufner, A. and H. Triebel, Generalizations of Hardy’s inequality, Conf. del Sem. di Mat. dell’ Università di Bari 156 (1978), 1–21.

    MathSciNet  Google Scholar 

  83. Chan, L. Y., Some further extensions of Hardy’s inequality, Canad. Math. Bull. 24 (1981), 393–400.

    Article  MathSciNet  MATH  Google Scholar 

  84. Chan, L. Y. —, Some extensions of Hardy’s inequalities, Canad. Math. Bull. 22 (1979), 165–169.

    Article  MathSciNet  MATH  Google Scholar 

  85. Beesack, P. R., On some integral inequalities of E. T. Copson, in “General Inequalities 2,” ed. E. F. Beckenbach, Basel, 1980, pp. 151–159.

    Google Scholar 

  86. Feher, F., A generalized Schur-Hardy inequality on normed Käthe spaces, in “General Inequalties 2,” ed. E. F. Beckenbach, Basel, 1980, pp. 277–286.

    Google Scholar 

  87. Mohapatra, R. N., Remarks on a generalization of the Schur-Hardy inequality, in “General Inequalities 2,” ed. E. F. Beckenbach, Basel, 1980, pp. 459–460.

    Google Scholar 

  88. Butzer, P. L. and F. Fehér, Generlized Hardy and Hardy-Littlewood inequalities in rearrangement-invariant spaces, Comment. Math. Prace Mat. Tomus Specialis in Honorem Ladislai Orlicz I (1978), 41–64.

    Google Scholar 

  89. Fehér, F., A note on a paper of E. R. Love, Bull. Austral. Math. Soc. 19 (1978), 67–75.

    Article  MathSciNet  MATH  Google Scholar 

  90. Love, E. R. Generlizations of a classical inequality, Appl. Anal. 8 (1978), 47–59.

    Article  MathSciNet  MATH  Google Scholar 

  91. Love, E. R. —, Links between some generalizations of Hardy’s integral inequality, in “General Inequalities 4,” ed. W. Walter, Basel, 1984, pp. 47–57.

    Google Scholar 

  92. Stolarsky, K. B., Another Proof of Carleman’s Inequality, manuscript.

    Google Scholar 

  93. KokiaŠvili, V. N., O. neravenstvah Hardi v vesovyh prostranstvah, Soob. Akad. Nauk Gruz. SSR, Matematika 962 (1979), 37–40.

    Google Scholar 

  94. Sysoeva, F. A., Odno neravensivo tipa Hardi— Litlvuda, Sb. Tr. Mosk. Gos. Ped. Inst. im. Lenia 6 (1978), 136–141.

    Google Scholar 

  95. Kufner, A. and A. Triebel, Ob odnom obobščenii neravenstva Hardi, Akad. Nauk SSSR, Sib. Otd. Inst. Mat., Teorija kubaturnyh formul i priloženija funkcional’nogo analiza k zadacam matematiceskoĭ fiziki, Tr. Sem. S. L. Soboleva 1 (1978), 61–68.

    MathSciNet  Google Scholar 

  96. Ding, Y., On a generalization of Hardy’s inequality (Chinese), J. Math. (Wuhan) 11 (1981), 31–39.

    Google Scholar 

  97. VasiĆ, P. M. and J. E. PeČariĆ, Notes on some inequalities for convex functions, Mat. Vesnik 634 (1982), 185–193.

    MathSciNet  Google Scholar 

  98. Chandra, P. and R. N. Mohapatra, Inequalities which yield inclusions among sequence spaces containing l p, in “General Inequalities 3,” ed. E. F. Beckenbach, Basel, 1983, 455–470.

    Google Scholar 

  99. Mohapatra, R. N. and D. C. Russell, Integral inequalities related to Hardy’s inequality, Aequationes Math. 28, (1985), 199–207.

    Article  MathSciNet  MATH  Google Scholar 

  100. Imoru, C. O., On some extensions of Hardy’s inequality, Internat. J. Math. Sci. 81 (1985), 165–171; 9 1 (1986), 207.

    Article  MathSciNet  MATH  Google Scholar 

  101. Imoru, C. O. —, An extension of a type of generalized Hardy’s integral inequality, Simon Stevin, A Quart. J. Pure Appl. Math. 59 (1985), 109–112.

    MathSciNet  MATH  Google Scholar 

  102. Imoru, C. O. —, Generalizations of Hardy’s integral inequality, J.Math. Anal. Appl. 122 (1987), 7–15.

    Article  MathSciNet  MATH  Google Scholar 

  103. Pachpatte, B. G., On Hardy type integral inequalities, Tamkang J. Math. 182 (1987), 27–41.

    MathSciNet  MATH  Google Scholar 

  104. Love, E. Ft. Inequalities related to Carleman’s inequality, unpublished lecture to Inequalities Conference at Birmingham, July 1987.

    Google Scholar 

  105. Love, E. Ft. —, Generlizations of Hardy’s and Copson’s inequalities, J. London Math. Soc. 302 (1984), 431–440.

    Article  MathSciNet  MATH  Google Scholar 

  106. Love, E. Ft. —, Generalizations of Hardy’s integral inequality, Proc. Roy. Soc. Edinburgh 100A (1985), 237–262.

    Article  MathSciNet  Google Scholar 

  107. Love, E. Ft. —, Hardy’s inequality in Orlicz-type sequence spaces for operators related to generalized Hausdorff matrices, Math. Z. 193 (1986), 481–490.

    Article  MathSciNet  MATH  Google Scholar 

  108. Love, E. Ft. —, Inequalities related to those of Hardy and of Cochran and Lee, Math. Proc. Comb. Phil. Soc. 99 (1986), 395–408.

    Article  MathSciNet  MATH  Google Scholar 

  109. Love, E. Ft. —, Some inequalities for geometric means, in “General Inequalities 5,” ed. W. Walter, Basel/Boston, 1987, pp. 87–93.

    Google Scholar 

  110. Love, E. Ft. —, Inequalities related to Knopp’s inequality, J. Math. Anal. Appl. 137 (1987), 173–180.

    Article  MathSciNet  Google Scholar 

  111. Jakimovski, A., B. E. Rhoades and J. Tzimbalario, Hausdorff matrices as bounded operators over l p, Math. Z. 138 (1974), 173–181.

    Article  MathSciNet  MATH  Google Scholar 

  112. Borwein, D., Generlized Hausdorff matrices as bounded operators on l p, Math. Z. 183 (1983), 483–487.

    Article  MathSciNet  MATH  Google Scholar 

  113. Cochran, J. A. and C.-S. Lee, Inequalities related to Hardy’s and Heinigs’s, Math. Proc. Cambridge Philos. Soc. 96 (1984), 1–7.

    Article  MathSciNet  MATH  Google Scholar 

  114. Moore, R. A. and Z. Nehari, Nonoscillation theorems for a class of nonlienar differential equations, Trans. Amer. Math. Soc. 93 (1959), 30–52.

    Article  MathSciNet  MATH  Google Scholar 

  115. Kim, W. J., On the zeros of solutions of y (n) + py = 0, J. Math. Anal. Appl.25 (1969), 189–

    Article  MATH  Google Scholar 

  116. Hinton, D. B. and R. T. Lewis, Singular differential operators with spectra discrete and bounded below, Proc. Roy. Soc. Edinburgh 84A (1979), 117–134.

    Article  MathSciNet  Google Scholar 

  117. Krzywicki, A. and A. Rybarski, On a linearization of an equation of an elastic rod (II), Zastosow. Mat. 7 (1964), 383–390.

    MathSciNet  MATH  Google Scholar 

  118. Rybarski, A., Angenäherte Schwingungsfrequenzformeln für konservative Systeme (2), Zastosow. Mat. 7 (1964), 255–269.

    MathSciNet  Google Scholar 

  119. Everitt, W. N., M. Giertz and J. Weidmann, Some remarks on a separation and limit-point criterion of second order, ordinary differential expressions, Math. Ann. 200 (1973), 335–346.

    Article  MathSciNet  MATH  Google Scholar 

  120. Everitt, W. N., An integral inequality with an application to ordinary differential operators, Proc. Roy. Soc. Edinburgh 80A (1978), 35–44.

    Article  MathSciNet  Google Scholar 

  121. Amos, R. J. and W. N. Everitt, On integral inqualities and compact embeddings associated with ordinary differntial expressions, Arch. Rational Mech. Anal. 71 (1979), 15–50.

    Article  MathSciNet  MATH  Google Scholar 

  122. Bradley, J. S., D. B. Hinton and R. M. Kauffman, On the minimization of singular quadratic functionals, Proc. Roy. Soc. Edinburgh 87A (1981), 193–208.

    Article  MathSciNet  Google Scholar 

  123. Putnam, C. R., An application of spectral theory to a singular calculus of variations problem, Amer. J. Math. 70 (1948), 780–803.

    Article  MathSciNet  MATH  Google Scholar 

  124. Everitt, W. N., Integral inequalities and spectral theory, Spectral theory and differential equations, in “Lecture Notes in Math.” 448, Berlin, 1976, pp. 73–105.

    Google Scholar 

  125. Beesack, P. R., Integral inequalities of the Wirtinger type, Duke Math. J. 25 (1958), 477–798.

    Article  MathSciNet  MATH  Google Scholar 

  126. Beesack, P. R. —, Minimum properties of eigenvalues — elementary proofs, in “General Inequalities 2,” E. F. Beckenbach, ed., Basel, 1980, pp. 109–120.

    Google Scholar 

  127. Redheffer, R., Inequalities with three functions, J. Math. Anal. Appl. 16 (1966), 219–242.

    Article  MathSciNet  MATH  Google Scholar 

  128. Benson, D. C., Inequalities involving integrals of functions and their derivatives, J. Math. Anal. Appl. 17 (1967), 292–308.

    Article  MathSciNet  MATH  Google Scholar 

  129. Portnov, V. R., Two embedding theorems for spaes and their applications (Russian), Dokl. Akad. Nauk SSSR155 (1964), 761–

    MathSciNet  Google Scholar 

  130. Kufner, A., Imbedding theorems for general Sobolev weight spaces, Ann. Scuola Norm. Sup. Pisa 23 (1969), 373–386.

    MathSciNet  MATH  Google Scholar 

  131. Kufner, A. —, “Weighted Sobolev Spaces,” Leipzig, 1980.

    Google Scholar 

  132. Maz’ya, V. G., “Spaces of S. L. Sobolev” (Russian), Leningard, 1985.

    Google Scholar 

  133. Hunt, R. A., On L(p,q) spaces, L’Enseignment Math. 12 (1966), 249–276.

    MATH  Google Scholar 

  134. Hunt, R. A. and G. Weiss, The Marcinkiewicz interpolation theorem, Proc. Amer. Math. Soc. 15 (1964), 996–998.

    MathSciNet  MATH  Google Scholar 

  135. Hunt, R. A., An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces, Bull. Amer. Math. Soc. 70 (1964), 803–807.

    Article  MathSciNet  MATH  Google Scholar 

  136. Nehari, Z., The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551.

    Article  MathSciNet  MATH  Google Scholar 

  137. Flanders, H., Remarks on Almgren’s interior regularity theorems, Illinois J. Math. 13 (1969), 707–716.

    MathSciNet  MATH  Google Scholar 

  138. Janet, M., Sur la méthode de Legendre-Jacobi-Clebsch et quelquesunes de ses applications, Bull. Sci. Math. 53 (1929), 144–160.

    Google Scholar 

  139. Troesch, B. A., An isoperimetric sloshing problem, Comm. Pure Appl. Math. 18 (1965), 319–338.

    Article  MathSciNet  MATH  Google Scholar 

  140. Beesack, P. R., Integral inequalities involving a function and its derivative, Amer. Math. Monthly 78 (1971), 705–741.

    Article  MathSciNet  MATH  Google Scholar 

  141. Feinberg, J. M., Some Wirtinger-like inequalities, SIAM J. Math. Anal. 10 (1979), 1258–1271.

    Article  MathSciNet  MATH  Google Scholar 

  142. Florkiewicz, B., Integral inequalities of Hardy type, Dissertation.

    Google Scholar 

  143. Beesack, P. R., D. S. Mitrinovic and P. M. VasiĆ, Integral inequalities, (The manuscript not achieved and not published — we used this manuscript in writing sections 15, 16, 21, 23-25, 28-32).

    Google Scholar 

  144. Pachpatte, B. G., On some extensions of Levinson’s generalizations of Hardy’s inequality, Soochow J. Math. 132 (1987), 203–210.

    MathSciNet  MATH  Google Scholar 

  145. Ostrogorski, T., Analogues of Hardy’s inequality in R n, Studia Math. 883 (1988), 209–219.

    MathSciNet  MATH  Google Scholar 

  146. Sinnamon, G., A note on the Stieltjes transformation, Proc. Roy. Soc. Edinburgh Sect. A 1101–2 (1988), 73–78.

    Article  MathSciNet  MATH  Google Scholar 

  147. Bennett, G., Some elementary inequalities II, Quart J. Math. Oxford Ser. (2) 39156 (1988), 385–400.

    Article  MathSciNet  MATH  Google Scholar 

  148. Pachpatte, B. G. and E. R. Love, On some new inequalities related to Hardy’s integral inequality, J. Math. Anal. Appl. 149 (1990), 17–25.

    Article  MathSciNet  MATH  Google Scholar 

  149. Mohapatra, R. N. and D. C. Russell, Integral inequalities related to Hardy’s inequality, Aeq. Math. 28 (1985), 199–207.

    Article  MathSciNet  MATH  Google Scholar 

  150. Mohapatra, R. N. and K. Vajravelu, Integral inequalities resembling Copson’s inequality, J. Aust. Math. Soc. Ser. A. 48 (1990), 124–132.

    Article  MathSciNet  MATH  Google Scholar 

  151. Martin-Reyes, F. J. and E. Sawyer, Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater, Proc. Amer. Math. Soc. 106 (1989), 727–733.

    Article  MathSciNet  MATH  Google Scholar 

  152. Pittenger, A. P., Note on a square function inequality, Ann. Prob. 7 (1979), 907–908.

    Article  MathSciNet  MATH  Google Scholar 

  153. Wilf, H. S., “Finite sections of some classical inequalities,” Berlin, Heidelberg, New York, 1970.

    Google Scholar 

  154. Batuev, E. N. and V. D. Stepanov, Weighted inequalities of Hardy type (Russian), Sibirsk. Mat. Ž. 30 (1989), 13–22.

    MathSciNet  Google Scholar 

  155. Sinnamon, G., A weighted gradient inequality, Proc. Roy. Soc. Edinburgh Sect. A. 111 (1989), 329–335.

    Article  MathSciNet  MATH  Google Scholar 

  156. Stepanov, V. D., A weighted inequality of Hardy type for higher-order derivatives (Russian) Trudy Mat. Inst. Steklov. 187 (1989), 178–190.

    Google Scholar 

  157. Wannebo, A., Hardy inequalities, Proc. Amer. Math. Soc. 109 (1990), 85–95.

    Article  MathSciNet  MATH  Google Scholar 

  158. Brown, R. and D. Hinton, Integral inequalities with power weights for functions of one variable, preprint.

    Google Scholar 

  159. Opic, B. and A. Kufner, “Hardy-type inequalities,” New York, 1990.

    Google Scholar 

  160. Kufner, A., “Zur Hardyschen Ungleichung,” Sem. anal. 1982/83, 1–18, Akad. Wiss. DDR, Berlin 1983.

    Google Scholar 

  161. Lizorkin, P. I., Operatoren, die mit der gebrochenen differentiation zusammenhängen und Massen differenzierbarer functionen, Trudy Mat. Inst. AN SSSR 117 (1972), 212–243.

    MathSciNet  MATH  Google Scholar 

  162. Johnson, P. D. and R. N. Mohapatra, A Hardy-Davies-Petersen inequality for a class of matrices, Canad. J. Math. 30 (1978), 458–465.

    Article  MathSciNet  MATH  Google Scholar 

  163. Johnson, P. D. —, Inequalities involving infinite matrices with nonnegative entries, in “General Inequalities 2,” ed. E. F. Beckenbach, Basel, 1980, pp. 55–80.

    Google Scholar 

  164. Johnson, P. D. —, Inequalities involving lower-triangular matrices, Proc. London Math. Soc. 41 (1980), 83–132.

    Article  MathSciNet  Google Scholar 

  165. Johnson, P. D. —, Best possible results in a class of inequalities, Pacific J. Math. 103 (1982), 432–436.

    Google Scholar 

  166. Johnson, P. D. —, On inequalities related to sequences spaces as [p,q], in “General Inequalities 4,” ed. W. Walter, Basel, 1989, pp. 191–201.

    Google Scholar 

  167. Pachpatte, B. G., A note on Copson’s inequality involving series of positive terms, Tamkang J. Math. 21 (1990), 13–19.

    MathSciNet  MATH  Google Scholar 

  168. MitrinoviĆ, D. S. and J. E. PecariĆ A note on Copson’s inequality of B. G. Pachpatte, (to appear).

    Google Scholar 

  169. Heinig, H. P., Weighted inequalities in Fourier analysis, Proc. Summer School Roudnise, Czechoslovakie, May 1990 (Teubner).

    Google Scholar 

  170. Stepanov, V. D., Weighted inequalities of Hardy type for derivatives of higher order and their applications (Russian), Dokl. AN SSSR (1988), No. 5, 1059–1062.

    Google Scholar 

  171. Ding, Y., Weighted Hardy inequalty (Chinese), J. Hangzhou Univ. Nat. Sci. Ed. 11 (1984), 168–174.

    Google Scholar 

  172. Lee, Kin-Chun and Gou-Sheng Yang, On generalization of Hardy’s inequality, Tamkang J. Math. 17 (1986), 109–119.

    MathSciNet  MATH  Google Scholar 

  173. Carlson, L., A proof of an inequality of Carleman, Proc. Amer. Math. Soc. 5 (1954), 932–933.

    Article  MathSciNet  Google Scholar 

  174. Ahiezer, H. I., O vzvešennom približenii neperyonyh funkciĭ mnogočlenami na vseĭ cislovoĭ osi, Uspehi Mat. Nauk t. 11, byp 4 (70), 1956.

    Google Scholar 

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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

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  1. D. S. Mitrinović
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  2. J. E. Pečarić
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  3. A. M. Fink
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Mitrinović, D.S., Pečarić, J.E., Fink, A.M. (1991). Hardy’s, Carleman’s 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_4

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