Some characterizations of functions generating K-Schur concave sums and of K-concave set-valued functions
Abstract
In this note we establish some characterizations of (single valued) unctions, that assume values in a Banach space, generating K-Schur concave sums. These results improve some theorems obtained in [13] and [11]. Moreover we prove that a set-valued function is K-concave if and only of it is K-t-concave and K-quasi concave (where t is a fixed number in (0,1)). This result improves the theorems obtained in [11], [2], [5] and extends the theorem of [3].
References
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2. F.A. Behringer, Convexity is equivalent to midpoint convexity combined with strict quasiconvexity, Optimization (ed. K.H. Elstev, Ilmenan, Germany), (to appear).
3. T. Cardinali, K. Nikodem, F. Papalini, Some results on stability and on characterization of K-convexity of set-valued functions, Annales Polonici Mathematici, LVIII.2 (1993), 185-192.
4. Z. Daróczy, Z. Páles, Convexity with given infinity weight sequences, Stochastica 11 (1987), 76-86.
5. Z. Kominek, A characterization of convex functions in linear spaces, Zeszyty Naukowe Akademii Górniczo-Hutniczej Im. Stanisława Staszica, No 1277 Opuscula Math. 5 Kraków, 1989, 71-75.
6. N. Kuhn, A note on t-convex functions, General Inequalities 5 (Proc. of the 5th International Conference on General Inequalities, Oberwolfach, 1968), 269-276.
7. A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979.
8. C.T. Ng, K. Nikodem, On approximately convex functions, Proc. Amer. Math. Soc. 118 (1993), 103-108.
9. K. Nikodem, Continuity of K-convex set-valued functions, Bull. Polish Acad. Sci. Math. 35 (1986), 392-399.
10. K. Nikodem, Approximately quasiconvex functions, C. R. Math. Rep. Acad. Sci. Canada - Vol. X, No 6 (1988), 291-295.
11. K. Nikodem, On some class of midconvex functions, Annales Polonici Mathematici L (1989), 151-155.
12. K. Nikodem, K-convex and K-concave set-valued functions, Zeszyty Nauk. Politech. Łódz. 559 (Rozprawy Mat. 115), Łódź 1989.
13. C.T. Ng, Functions generating Schur-convex sums, General Inequalities 5 (Proc. Oberwolfach, 1986), 533-538.
14. F. Papalini, Decomposition of a K-midconvex (K-midconcave) function in a Banach space, Riv. Mat. Univ. Parma (5) 2 (1993).
15. H. Rådström, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165-169.
CardinaliT. (1995). Some characterizations of functions generating K-Schur concave sums and of K-concave set-valued functions. Annales Mathematicae Silesianae, 9, 17-28. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14201
Tiziana Cardinali
Department of Mathematics, Perugia University, Italy Italy
Department of Mathematics, Perugia University, Italy Italy
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