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Hermite and convexity. (English) Zbl 0572.26004
The note deals with some historical informations concerning the well- known inequality \[ (*)\quad (b-a)f[(a+b)/2]<\int^{b}_{a}f(x)dx<(b- a)[(f(a)+f(b))/2] \] satisfied by any continuous strictly convex real function defined on a closed interval [a,b]. In the mathematical literature this inequality is usually connected to Hadamard’s name, who proved it in 1893. Only in 1974 D. S. Mitrinović found a modest note which was published in the journal Mathesis in 1883. It was an extract from a letter written by Ch. Hermite in 1881 announcing inequality (*). Therefore it seems that it was Ch. Hermite who obtained it for the first time. However it is interesting that Hermite’s note remained unknown for so long for experts in the history and theory of convex functions. Even the authors were not aware of it. The authors quote also some other inequalities and results (due, for example, to Fejér or Hardy, Littlewood, Pólya) which are consequences or improvements of Hermite’s inequality (*). Close connections with subharmonic functions are also stressed.
Reviewer: W.Jarczyk

26A51 Convexity of real functions in one variable, generalizations
26D15 Inequalities for sums, series and integrals
26-03 History of real functions
01A55 History of mathematics in the 19th century
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
Full Text: DOI EuDML
[1] Picard, É. (ed.),Oeuvres de Charles Hermite, 1–4. Paris, 1905–1917. · JFM 46.0034.11
[2] Jordan, C. andMansion, P.,Charles Hermite (1822–1901). Paris, 1901.
[3] Beckenbach, E. F.,Convex functions. Bull. Amer. Math. Soc.54 (1948), 439–460. · Zbl 0041.38003 · doi:10.1090/S0002-9904-1948-08994-7
[4] Hadamard, J.,Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. J. Math. Pures Appl.58 (1893), 171–215. · JFM 25.0698.03
[5] Hardy, G. H., Littlewood, J. E., andPólya, G.,Inequalities. Cambridge University Press. 1934.
[6] Timan, A. F. andTrofimoff, V. N.,Introduction to theory of harmonic functions (Russian). Moscow, 1968, p. 182.
[7] Lacković, I. B.,On some inequalities for convex functions (Serbian). InMatematička Biblioteka 42, Belgrade, 1970, pp. 138–141.
[8] Hartman, P.,Convex functions and mean value inequalities. Duke Math. J.39 (1972), 351–360. · Zbl 0247.26009 · doi:10.1215/S0012-7094-72-03942-7
[9] Radó, T.,Subharmonic functions. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 5, Berlin, 1937. · JFM 63.0458.05
[10] Hayman, W. K. andKennedy, P. B.,Subharmonic functions. Vol. 1, Academic Press, London-New York-San Francisco, 1976.
[11] Fejér, L.,Über die Fourierreihen, II. (Hungarian). Math. Naturwiss, Anz. Ungar. Akad. Wiss.24 (1906), 369–390. Gesammelte Arbeiten I, Budapest, 1970, pp. 280–297. · JFM 37.0286.01
[12] Mitrinović, D. S.,Inequalities (Serbian). Beograd, 1965.
[13] Mitrinović, D. S.,Analytic inequalities. Springer, Berlin-Heidelberg-New York, 1970. · Zbl 0199.38101
[14] Mitrinović, D. S.,Elementary inequalities (Polish). Warszawa, 1972.
[15] Radó, T.,On convex functions. Trans. Amer. Math. Soc.37 (1935), 266–285. · JFM 61.0198.01
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