M athematical I nequalities & A pplications Volume 4, Number 2 (2001), 203–207 EQUIVALENCE OF THE H ¨ OLDER–ROGERS AND MINKOWSKI INEQUALITIES LECH MALIGRANDA Abstract. It is well-known that the H¨ older-Rogers inequality implies the Minkowski inequality. Infantozzi [6] observed implicitely and Royden [15] proved explicitely that the reverse implication is also true. In this note we discuss and give a new proof of this perhaps surprising fact. Mathematics subject classification (2000): 26D15. Key words and phrases: H¨ older-Rogers inequality, Minkowski inequality, Bernoulli inequality, power means. REFERENCES [1] P. S. BULLEN, D. S. MITRINOVI ´ C AND P. M. V ASI ´ C, Means and Their Inequalities, D. Reidel Publishing Company, Dordrecht, 1988. [2] S. B. CHAE, Lebesgue Integration, Springer Verlag, New York, 1995. [3] U. DUDLEY, Real Analysis and Probability, Wadsworth, 1989. [4] G. B. FOLLAND, Real Analysis, Modern Techniques and Their Applications, Wiley, New York, 1984. [5] G. H. HARDY , J. E. LITLEWOOD AND G. P ´ OLYA, Inequalities, Cambridge Univ. Press, 1934. [6] C. A. INFANTOZZI, An introduction to relations among inequalities, Amer. Math. Soc. Meeting 700 Cleavlend, Ohio 1972; Notices Amer. Math. Soc. 14 (1972), A819–A820. [7] J. LINDENSTRAUSS AND L. TZAFRIRI, Classical Banach Spaces II. Function Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1979. [8] L. MALIGRANDA, Why H¨ older’s inequality should be called Rogers’ inequality, Math. Inequalities and Appl. 1 (1998), 69–83. [9] L. MALIGRANDA AND L. E. PERSSON, Generalized duality of some Banach function spaces, Indagationes Math. 51 (1989), 323–338. [10] A. W. MARSHALL AND I. OLKIN, Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979. [11] D. S. MITRINOVI ´ C, Analytic Inequalities, Springer-Verlag, Berlin-Heidelberg-New York, 1970. [12] D. S. MITRINOVI ´ C, J. E. PE ˇ CARI ´ C AND A. M. FINK, Classical and New Inequalities in Analysis, Kluwer Academic Publ., Dordrecht, 1993. [13] F. RIESZ, Untersuchungen ¨ uber Systeme integrierbarer Funktionen, Math. Ann. 69 (1910), 449–497. [14] F. RIESZ, Les Syst` emes D’´ equations Lin´ eaires ` a Une Infinit´ e D’inconnues, Gauthier-Villars, Paris, 1913. [15] H. L. ROYDEN, Real Analysis, Third Edition, Macmillan Publishing Company, New York, 1988. c  , Zagreb Paper MIA-04-18 Mathematical Inequalities & Applications www.ele-math.com mia@ele-math.com