Theorem of Bernstein-Doetsh in Baire spaces
Abstract
Theorem of Bernstein and Doetsch is one of the most important in the theory of convex (or convex in the sense of Jensen) functions. In this paper it is shown that the original proof of Bernstein and Doetsch of this theorem can be adapted in a more general situation. Also some parts of the proof may be of interest for themselves.
References
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2. F. Bernstein, G. Doetsch, Zur Theorie der konvexen Funktionen, Math. Ann. 76 (1915), 514-526.
3. Z. Kominek, On additive and convex functionals, Radovi Mat. 3 (1987), 267-279.
4. Z. Kominek, M. Kuczma, On the lower hull of convex function, Aequationes Math. 38 (1989), 192-210.
5. Z. Kominek M. Kuczma, Theorems of Bernstein-Doetsch, Piccard and Mehdi, and semilinear topology, Arch. Math. (Basel) 52 (1989), 595-602.
6. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Canchy's equation and Jensen's inequality, PWN (Polish Scientific Publishers) i Uniwersytet Śląski, Warszawa-Kraków-Katowice, 1985.
7. N. Kuhn, A note on t-convex functions, International Ser. Numer. Math. 71. Birkhäuser Verlag, Basel-Stuttgart, 1984. General Inequalities 4, 269-276. [Proceedings of the Fourth International Conference on General Inequalities held in the Mathematical Institute at Obervolfach, Black Forest, May 8-14, 1983]. Edited by W. Walter.
8. K. Kuratowski, Topologie, vol. I. Monografie Mat. 20. PWN (Polish Scientific Publishers), Warszawa, 1958.
9. M.R. Mehdi, On convex functions, J. London Math. Soc. 39 (1964), 321-326.
10. A.W. Roberts, D.E. Varberg, Convex functions, Academic Press, New York and London, 1973.
11. F.A. Valentine, Convex sets, McGraw-Hill Book Company, New York-San Francisco-Toronto-London, 1964.
KominekZ., & KuczmaM. (1991). Theorem of Bernstein-Doetsh in Baire spaces. Annales Mathematicae Silesianae, 5, 10-17. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14276
Zygfryd Kominek
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Marek Kuczma
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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