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Midconvex functions in locally compact groups

Authors: A. Chademan and F. Mirzapour
Journal: Proc. Amer. Math. Soc. 127 (1999), 2961-2968
MSC (1991): Primary 26A51; Secondary 22A10
Published electronically: June 17, 1999
MathSciNet review: 1610936
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Abstract: The theorems of Bernstein-Doetsch, and Ostrowski, concerning the continuity of midconvex functions are extended to open subsets of locally compact and root-approximable topological groups.

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  • [1] F. Bernstein and G. Doetsch, Zur Theorie der Konvexen Funktionen, Math. Ann. 76 (1915), 514-526.
  • [2] H. Blumberg, On convex functions, Trans. Amer. Math. Soc. 20 (1919), 40-44.
  • [3] N. Bourbaki, Éléments de mathématique. Première partie. (Fascicule III.) Livre III; Topologie générale. Chap. 3: Groupes topologiques. Chap. 4: Nombres réels, Troisième édition revue et augmentée, Actualités Sci. Indust., No. 1143. Hermann, Paris, 1960 (French). MR 0140603
  • [4] N. Bourbaki, Espaces vectoriels topologiques, chap. 1 et 2, 2-ième éd. (Actualités Scientifiques et Industrielles no. 1198), Hermann, Paris, (1966).
  • [5] A. Chademan, Sur les notions élémentaires de la théorie spectrale, Thèse de Doctorat 3-ième cycle, Univ. Paris VI, 1970.
  • [6] A. Chademan and F. Mirzapour, Boundedness properties of midconvex functions in locally comact groups, (in Proc. of the 26-th AIMC, March 1995, published by the Iranian Math. Society and University of Kerman, Kerman, Iran), 1995, 59-63. CMP 98:07
  • [7] R. Ger, Some remarks on convex functions, Fund. Math. 66 (1969/1970), 255–262. MR 0254195
  • [8] Roman Ger, 𝑛-convex functions in linear spaces, Aequationes Math. 11 (1974), 172–176. MR 0358119,
  • [9] Roman Ger and Marek Kuczma, On the boundedness and continuity of convex functions and additive functions, Aequationes Math. 4 (1970), 157–162. MR 0264006,
  • [10] I.I. Hirschman and D.V. Widder, The convolution transform, Princeton University Press, Princeton, New Jersey, (1955). MR 17:479c
  • [11] J. L. W. V. Jensen, Sur les fonctions convexes et les inégalités entre les valeurs moyennes, Acta Math. 30(1906), 175-193.
  • [12] Z. Kominek, On additive and convex functionals, Rad. Mat. 3 (1987), no. 2, 267–279 (English, with Serbo-Croatian summary). MR 931983
  • [13] Z. Kominek and M. Kuczma, Theorems of Bernstein-Doetsch, Piccard and Mehdi and semilinear topology, Arch. Math. (Basel) 52 (1989), no. 6, 595–602. MR 1007635,
  • [14] Marek Kuczma, An introduction to the theory of functional equations and inequalities, Prace Naukowe Uniwersytetu Śląskiego w Katowicach [Scientific Publications of the University of Silesia], vol. 489, Uniwersytet Śląski, Katowice; Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1985. Cauchy’s equation and Jensen’s inequality; With a Polish summary. MR 788497
  • [15] M. R. Mehdi, On convex functions, J. London Math. Soc. 39 (1964), 321–326. MR 0161949,
  • [16] C. T. Ng and K. Nikodem, On approximately convex functions, Proc. Amer. Math. Soc. 118 (1993), no. 1, 103–108. MR 1159176,
  • [17] A. Ostrowski, Uber die Funktionalgleichung der Exponentialfunktionen und verwande Funktionalgleichungen, Jahresber. Deut. Math. Ver., 38(1929), 54-62.
  • [18] A. Wayne Roberts and Dale E. Varberg, Convex functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Pure and Applied Mathematics, Vol. 57. MR 0442824
  • [19] W. Sierpinski, Sur les fonctions convexes mesurables; Fund. Math. 1 (1920), 125-129.
  • [20] A. Weil, L' intégration dans les groupes topologiques et ses applications, 2-ième éd., (Actualités Scientifiques et Industrielles no. 1145), Hermann, Paris, (1965).

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Additional Information

A. Chademan
Affiliation: Department of Mathematics and Computer Science, Faculty of Science, University of Tehran, Tehran, Iran

F. Mirzapour
Affiliation: Department of Mathematics, University of Zanjan, Zanjan, Iran

Keywords: Midconvex functions, locally compact groups, Bernstein-Doetsch theorem, Jensen's theorem, Ostrowski's theorem
Received by editor(s): December 12, 1996
Received by editor(s) in revised form: January 1, 1998
Published electronically: June 17, 1999
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1999 American Mathematical Society