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Norm Inequalities

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

Abstract

In this Chapter we look at inequalities for norms which are related to the triangle inequality. Several of these are attached to the names, e.g. Clarkson’s, Dunkl-Williams’ and Hlawka’s.

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References

  1. DUNKL, C. F. and K. S. WILLIAMS, A simple inequality, Amer. Math. Monthly 71 (1964), 53–54.

    Article  MathSciNet  MATH  Google Scholar 

  2. MASSERA, J. L. and J. J. SCHAFFER, Linear differential equations and functional analysis, I, Ann. of Math., (2) 67 (1958), 517–573.

    Article  MathSciNet  MATH  Google Scholar 

  3. KIRK, W. A. and M. F. SMILEY, Another characterization of inner product spaces, Amer. Math. Monthly 71 (1964), 890–891.

    Article  MathSciNet  Google Scholar 

  4. DRAGOMIR, S. S. and I. SANDOR, Some inequalities in prehilbertian spaces, Studia Univ. Babe§-Bolyai, Mathematica 32 (1987), 71–78.

    MATH  Google Scholar 

  5. GURARII, V. I., Strengtheining the Dunkl-Williams inequality on the norms of elements of Banach spaces, Dopovidi Akad. Nauk. Ukrain. RSR, 1966, 35–38.

    Google Scholar 

  6. BOURBAKI, N., “Integration”, Paris 1965, Chap. 4, §6, Exer. 10, p. 257.

    Google Scholar 

  7. HILE, G. N., Entire solutions of linear elliptic equations with Laplacian principal part, Pacific J. Math. 62 (1976), 127–140.

    MathSciNet  MATH  Google Scholar 

  8. GLOWINSKI, R. and A. MARROCCO, Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet nonlinéaires,Rapport de Recherche #115, I.R.I.A. Paris, 1975.

    Google Scholar 

  9. FIX, G. J. and B. NETA, Finite element approximation of a nonlinear diffusion problem, Comput. Math. Applics 3 (1977), 287–298.

    MathSciNet  MATH  Google Scholar 

  10. NETA, B., On three inequalities,Comput. Math. Applics 6 (1980), 301304.

    Google Scholar 

  11. GOLDSTEIN, A. A., J. V. RYFF and L. E. CLARKE, Problem 5473, Amer. Math. Monthly 75 (1968), 309–310.

    Article  MathSciNet  Google Scholar 

  12. ZAIDMAN, S., Some elementary inequalities, Boll. Un. Mat. Ital. IV. Series 3 (1970), 213–216.

    MathSciNet  MATH  Google Scholar 

  13. BLUMENTHAL, L. M., Note on normed linear spces, Revista Acad. Ci. Madrid 62 (1968), 307–310.

    MathSciNet  MATH  Google Scholar 

  14. JANOUS, W., Private communications.

    Google Scholar 

  15. FISHER, D. and M. MARTELLI, Problem E3192, Amer. Math. Monthly, 94 (1987), 181.

    Google Scholar 

  16. KADEV, M. L., Bezuslovno shodjaséesja rjady v ravnomerno vypuklom prostranstve, Uspehi Mat. Nauk 11, Vyp 5, 1956, 185–190.

    Google Scholar 

  17. ABILOV, V. A., 0 rabotah I. V. Cenova po konstruktivnoi teorii funkcii Serdica (Sofia) vol. 10, fasc. 3 (1984), 317–336.

    MathSciNet  Google Scholar 

  18. SKARDA, V., 2 1 -bound for inner products, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 634–677 (1979), 143–147.

    MathSciNet  Google Scholar 

  19. GROSS, W., Bedingt konvergente Reihen, Monatsch. Math. Phys. 28 (1917), 221–237.

    MATH  Google Scholar 

  20. BERGSTROM, V., Zwei Sätze ûber ebene Vektor-polygone, Abh. Math. Seminar, Univ. Hamburg 8 (1930), 206–214.

    Google Scholar 

  21. DAMSTEEG, I. and I. HALPERIN, The Steinitz-Gross theorem on sums of vectors, Trans. Roy. Soc. Canada Sect. III (3) 44 (1950), 31–35.

    MathSciNet  MATH  Google Scholar 

  22. BEHREND, F. A., The Steinitz-Gross theorem on sums of vectors, Canada J. Math. 6 (1954), 108–124.

    MathSciNet  MATH  Google Scholar 

  23. STEINITZ, E., Bedingt konvergente Reihen und konvexe Systeme, J. Reine Angew. Math. 143 (1913), 128–175; 144 (1914), 1–40.

    Google Scholar 

  24. BERGSTROM, V., Ein neuer Beweis eines Satzes von E. Steinitz, Abh. Math. Seminar, Univ. Hamburg 8 (1930), 148–152.

    Article  Google Scholar 

  25. HARTMAN, P., On the limits of Riemann approximating sums, Quarterly J. Math. 18 (1947), 124–127.

    Article  MathSciNet  MATH  Google Scholar 

  26. HALPERIN, I. and N. MILLER, An inequality of Steinitz and the limits of Riemann sums, Trans. Roy. Soc. Canada Sect. III (3) 48 (1954), 27–29.

    MathSciNet  MATH  Google Scholar 

  27. HORNICH, H., Eine Ungleichung fzïr Vektorlängen, Math. Z. 48 (1942), 268–274.

    Article  MathSciNet  Google Scholar 

  28. LUCIC, R., Sur une inégalité de Hornich, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 101–106 (1963), 5–6.

    MathSciNet  Google Scholar 

  29. DJOKOVIC, D. Z., Hornich inequality and some generalizations, Bull. Soc. Math. Phys. Serbie 15 (1963), 33–36.

    MathSciNet  MATH  Google Scholar 

  30. ADAMOVIC, D. D., Généralisation d’une identité de Hlawka et de l’inégalité correspondante, Mat. Vesnik 1 (16) (1964), 39–43.

    MathSciNet  Google Scholar 

  31. VASIC, P. M., Les inégalités pour les fonctions convexes d’ordre n, Mat. Vesnik 5 (20) (1968), 327–331.

    MathSciNet  Google Scholar 

  32. DJOKOVIC, D. Z., Generalizations of Hlawka’s inequality, Glasnik Mat.Fiz. Astromon. Ser. II. Drustvo Mat. Fiz. Hrvatske 18 (1963), 169–175.

    MATH  Google Scholar 

  33. SMILEY, D. M. and M. F. SMILEY, The polynomial inequalities, Amer. Math. Monthly 71 (1964), 755–760.

    Article  MathSciNet  MATH  Google Scholar 

  34. ADAMOVIC, D. D., Quelques remarques relatives aux généralisations des inégalités de Hlawka et de Hornich, Mat. Vesnik 1 (16) (1964), 241–242.

    MathSciNet  Google Scholar 

  35. SUDBERY, A., The quadrilateral inequality in two dimensions, A.er. Math. Monthly 82 (1975), 629–632.

    Article  MathSciNet  MATH  Google Scholar 

  36. KELLY, L. M., D. M. SMILEY and M. F. SMILEY, Two dimensional spaces are quadrilateral spaces,Amer. Math. Monthly 72 (1965), 753754.

    Google Scholar 

  37. WILLIAMS, L. R. and J. H. WELLS, LP inequalities, J. Math. Anal. Appl. 64 (1978), 518–529.

    Article  MathSciNet  MATH  Google Scholar 

  38. FREUDENTHAL, H., Problem 141, Wisk. Opgaven 21 (1963), 137–139.

    Google Scholar 

  39. LEVI, F. M., Ein Reduktionsverfahren für lineare Vektorungleichungen, Arch. Math. 2 (1949), 24–26.

    Google Scholar 

  40. MARJANOVIC, M., An elementary inequality, Mat. Vesnik 1 (16) (1964), 153–156.

    MathSciNet  Google Scholar 

  41. WITSENHAUSEN, H. S., Metric inequalities and the zonoid problem, Proc. Amer. Math. Soc. 40 (1973), 517–520.

    Article  MathSciNet  MATH  Google Scholar 

  42. KELLY, J. B., Metric inequalities and symmetric differences, “Inequalities 2” (Proc. Sec. Symp. U.S. Air Force Acad. Colo., 1967), New York, 1970, pp. 193–212.

    Google Scholar 

  43. CHIRITA, M. and R. CONSTANTINESCU, Asupra unei inegalitati care caracterizeazu functiile convexe,Gaz. Mat. (Bucharest) 89 (1984), 241242.

    Google Scholar 

  44. BENCZE, M., About Hlawka’s inequality (Romanian), Gaz. Matematicâ 6 (1985), 48–50.

    MATH  Google Scholar 

  45. POPOVICIU, T., Sur certaines inégalités qui caractérisent les fonctions convexes,An. Sti. Univ. “Al. I. Cuza” Iasi Sect. la Mat. 11 (1965), 155–164.

    Google Scholar 

  46. VASIC, P. M. and D. D. ADAMOVIC, Sur un système infini d’inégalités fonctionnelles, Publ. Inst. Math. (Beograd) 9 (23) (1969), 107–114.

    MathSciNet  Google Scholar 

  47. KECKIC, J. D., On an infinite system of functional inequalities, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 274–301 (1969), 79–81.

    MathSciNet  Google Scholar 

  48. PECARIC, J. E., Modified version of a general result of Vasié- Adamovié Keckie and some remarks concerning inequalities for convex functions, Glasnik Matematicki 21 (41) (1986), 331–341.

    Google Scholar 

  49. BURKILL, J. C., The concavity of discrepancies in inequalities of means and of Hölder, J. London Math. Soc. 7 (2) (1974), 617–626.

    Article  MathSciNet  MATH  Google Scholar 

  50. BASTON, V. J., On some Hlawka-type inequalities of Burkill, J. London Math. Soc. 12 (2) (1976), 402–404.

    Article  MathSciNet  MATH  Google Scholar 

  51. VASIC, P. M. and Lj. R. STANKOVIC, Some inequalities for convex functions, Math. Balkanica 6 (1976), 281–288.

    MathSciNet  MATH  Google Scholar 

  52. VASIC, P. M. and Z. MIJALKOVIC, On an index set function connected with Jensen inequality, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 544–576 (1976), 110–112.

    MathSciNet  Google Scholar 

  53. HARDY, G. H., J. E. LITTLEWOOD and G. POLYA, “Inequalities”, Cambridge, 1934, 1952.

    Google Scholar 

  54. POPOVICIU, T., On an inequality, (Romanian), Gaz. Mat. (Bucharest) 51 (1946), 81–85.

    MathSciNet  MATH  Google Scholar 

  55. VASIC, P. M., Les inégalités pour les fonctions convexes d’orde n, Mat. Vesnik 5 (20) (1968), 327–331.

    MathSciNet  Google Scholar 

  56. PECARIC, J. E., Notes on some inequalities of P. M. Vasié, Publ. Inst. Math. (Beograd) N. S. 27 (41) (1980), 189–194.

    Google Scholar 

  57. KECKIC, J. D., Some inequalities for convex functions of order n, Mat. Vesnik 7 (22) (1970), 61–67.

    MathSciNet  Google Scholar 

  58. LACKOVIC, I. B., Neki novi resultati za konveksne funkcije i za nejednakosti koje su u vezi sa njima, Doktorska disertacija, Elektronski Fakultet, Nis, 1975. ( Not published. )

    Google Scholar 

  59. CLARKSON, J. A., Uniformly convex spaces, Trans. A.er. Math. Soc. 40 (1936), 396–413.

    Article  MathSciNet  Google Scholar 

  60. SOBOLEV, S. L., On an inequality (Russian), Uspehi Mat. Nauk (N.S.) 1, No. 3–4 (13–14) (1946), 197.

    Google Scholar 

  61. MITRINOVIC, D. S. and P. M. VASIC, Addenda to the monograph ‘Analytic Inequalities’ I, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 577-598 (1977), 3–10.

    Google Scholar 

  62. FRIEDRICHS, K. O., On Clarkson’s inequality, Comm. Pure Appl. Math. 23 (1970), 603–607.

    MathSciNet  MATH  Google Scholar 

  63. HEWITT, E. and K. STROMBERG, “Real and Abstract Analysis”, Berlin-Heidelberg-New York, 1965, see in particular pp. 223–226.

    Google Scholar 

  64. LAFIÈVRE, G., Problem 5844,Revue Math. Spéc. 79 (1968/69), 369370 and 597–602.

    Google Scholar 

  65. BOAS, R. P., Some uniformly convex spaces, Bull. Amer. Math. Soc. 26 (1940), 304–311.

    Article  MathSciNet  Google Scholar 

  66. HANNER, O., On the uniform convexity of LP and QP, Arkiv for Matematik 3 (1955), 239–244.

    Article  MathSciNet  Google Scholar 

  67. MORAWETZ, C. S., Two LP inequalities, Bull. Amer. Math. Soc. 75 (1969), 1299–1302.

    Article  MathSciNet  MATH  Google Scholar 

  68. McSHANE, E. J., Linear functionals on certain Banach spaces, Proc. Amer. Math. Soc. 1 (1950), 402–408.

    Article  MathSciNet  MATH  Google Scholar 

  69. NIKOL’SKIÎ, S. M., “Approximations of Functions of Several Variables and Imbedding Theorems”, Berlin-Heidelberg-New York, 1975, pp. 2223.

    Google Scholar 

  70. SHAPIRO, H. S., Regularity properties of the element of closest approximation, Trans. Amer. Math. Soc. 181 (1973), 127–142.

    Article  MATH  Google Scholar 

  71. RAMASWAMY, S., A simple proof of Clarkson’s inequality, Proc. Amer. Math. Soc. 68 (1978), 249–250.

    MathSciNet  MATH  Google Scholar 

  72. CLARKSON, J. A., The von Neumann-Jordan constant for the Lebesgue spaces, Ann. of Math. 38 (1937), 114–115.

    Article  MathSciNet  Google Scholar 

  73. von NEUMANN, J. and R. JORDAN, On inner products in linear metric spaces, Ann. of Math. 36 (1935), 719–724.

    Google Scholar 

  74. BYNUM, W. L. and J. H. DREW, A weak parallelogram law for £ P, Amer. Math. Monthly 79 (1972), 1012–1015.

    Article  MathSciNet  MATH  Google Scholar 

  75. BYNUM, W. L., Weak parallelogram laws for Banach spaces,Notices Amer. Math. Soc. 22 (1975), A-175.

    Google Scholar 

  76. SOBOLEV, S. L., “Introduction to the Theory of Cubature Formulas” ( Russian ), Moscow, 1974.

    Google Scholar 

  77. KOSKELA, M., Some generalizations of Clarkson’s inequalities, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. M.t. Fiz. No. 634–677 (1979), 89–93.

    MathSciNet  Google Scholar 

  78. BENEDEK, A. and R. PANZONE, The spaces LP, with mixed norm, Duke Math. J. 28 (1961), 301–324.

    MathSciNet  MATH  Google Scholar 

  79. WILLIAMS, L. R. and J. H. WELLS, LP inequalities, J. Math. Anal. Appl. 64 (1978), 518–529.

    Article  MathSciNet  MATH  Google Scholar 

  80. PERSSON, L. E., Some elementary inequalities in connection with X-spaces, Constructive Theory of Functions, Sofija, 1988, 367–376.

    Google Scholar 

  81. COBOS, F., Clarkson’s inequalities for Sobolev spaces, Math. Japonica 31, No. 1 (1986), 17–22.

    MathSciNet  MATH  Google Scholar 

  82. MILMAN, M., Complex interpolation and geometry of Banach spaces, Ann. M.t. Pura Appl. 136 (1984), 317–328.

    Article  MathSciNet  MATH  Google Scholar 

  83. TUNGE, A., Random Clarkson inequalities and L P version of Grothendieck’s Inequality, Math. Nadir. 131 (1987), 335–343.

    Article  Google Scholar 

  84. KATO, M., Generalized Clarkson’s inequalities and the norms of the Littlewood matrices, Math. Nachr. 119 (1983), 163–170.

    Article  Google Scholar 

  85. COBOS, F., Clarkson’s inequalities for Sobolov spaces, Math. Japonica 31 (1986), 17–22.

    MathSciNet  MATH  Google Scholar 

  86. COBOS, F. and D. E. EDMONDS, Clarkson’s inequalities, Besov spaces and Triebel-Sobolev spaces, Z. Anal. Anwend. 7 (1988), 229–232.

    MathSciNet  MATH  Google Scholar 

  87. MALIGRANDA, L., and L. E. PERSSON, On Clarkson’s inequalities and

    Google Scholar 

  88. BETH, E. W. and J. G. van der CORPUT, Problem 172, Wisk Opgaven 16 (1937), 421–428.

    Google Scholar 

  89. DRAGOMIR, S. S. and I. SANDOR, Some inequalities in prehilbertian spaces, Studia Univ. Babes-Bolyai, Math. 32 (1987), 71–78.

    MATH  Google Scholar 

  90. RASSIAS, Th. M., New characterizations of inner product spaces, Bull. Sci. Math. (2) 108 (1984), 95–99.

    MathSciNet  MATH  Google Scholar 

  91. KLAMKIN, M. S., A vector norm inequality, Amer. Math. Monthly 82 (1975), 829–830.

    Article  Google Scholar 

  92. SHORE, T. R., On an inequality of van der Corput and Beth, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 678–715 (1980), 56–57.

    MathSciNet  Google Scholar 

  93. PECARIC, J. E. and R. R. JANIC, Note on a vector norm inequality, Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 735–762 (1982), 35–38.

    Google Scholar 

  94. DAY, M. M., On criteria of Kasahara and Blumenthal for inner-product spaces, Proc. Amer. M.th. Soc. 10 (1959), 92–100.

    MATH  Google Scholar 

  95. SCHOENBERG, I. J., A remark on M. M. Day’s characterization of inner-product spaces and a conjecture of L. M. Blumenthal, Proc. Amer. Math. Soc. 3 (1952), 961–964.

    MathSciNet  MATH  Google Scholar 

  96. HARRIS, L. A., Problem 5919, A.er. Math. Monthly 80 (1973), 697.

    Article  Google Scholar 

  97. RASSIAS, Th. M., On the theory of inner product spaces, Eleftheria 2 (1979), 481–484.

    Google Scholar 

  98. KATO, M., Generalized Clarkson’s inequality and the norms of Littlewood matrices, Math. Nachr. 114 (1983), 163–170.

    Article  MathSciNet  MATH  Google Scholar 

  99. KAMALY, A., Some new inequalities suggested by a conjecture of F. Holland,Trita-Mat. 1991:0047, Dept. of Math. Royal Inst. of Tech. Stockholm.

    Google Scholar 

  100. KAMALY, A., , Application of Fourier analysis in showing some new inequalities,Trita-Mat. 1991:0049, Dept. of Math. Royal Inst. of Tech. Stockholm.

    Google Scholar 

  101. KAMALY, A., , Some new results of Shapiro’s inequality,preprint.

    Google Scholar 

  102. TUDOR, M., Quelques inégalités intégales, Incr. Semin. Mat.,si, Fiz., Inst. Politehn. Timisoara (1988) No. Maj. C., 46–50.

    Google Scholar 

  103. XIAXI, D. and P. LUO, On some inequalities with weights, IEEE Trans. Ind. Electron. 36 (1989), 427–436.

    Google Scholar 

  104. ORLICZ, W., Ober unbedingte Konvergenz in Funktionenraûomen, Stu-dia Math. 4 (1933), 41–47.

    MATH  Google Scholar 

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Mitrinović, D.S., Pečarić, J.E., Fink, A.M. (1993). Norm 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_18

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