Conditionally δ-midconvex functions

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• 作者：Jacek Chudziak ; Jacek Tabor ; Józef Tabor
• 关键词：26A51 ; 26B25 ; 39B62 ; Midconvex function ; approximately midconvex function ; extremal point
• 刊名：Aequationes Mathematicae
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：89
• 期：4
• 页码：981-990
• 全文大小：454 KB
• 参考文献：1.Hyers, D.H., Ulam, S.: Approximately convex functions. Proc. Am. Math. Soc. 3, 821-28 (1952)MATH MathSciNet View Article
  2.Najdecki, A., Tabor, J., Tabor, J.: On conditionally δ-convex functions. Acta Math. Hungar. 128, 103-08 (2010)MathSciNet View Article
  3.Ng, T., Nikodem, K.: On approximately convex functions. Proc. Am. Math. Soc. 118, 103-08 (1993)MATH MathSciNet View Article
  4.Niculescu, C.P., Persson, L.-E.: Convex functions and their applications. A contemporary approach. CMS Books in Mathematics, vol. 23, Springer, New York (2006)
  5.Peters, H.J.M., Wakker, P.P.: Convex functions on non-convex domains. Econ. Lett. 22, 251-55 (1987)MathSciNet View Article
  6.Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)MATH
  
• 作者单位：Jacek Chudziak (1)
  Jacek Tabor (2)
  Józef Tabor (1)
  
  1. Faculty of Mathematics and Sciences, University of Rzeszów, ul.Prof. St. Pigonia 1, 35-310, Rzeszów, Poland
  2. Department of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Combinatorics
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8903

文摘

Let X be a real linear space, V be a nonempty subset of X and δ be a nonnegative real number. A function \({f : V \to \mathbb{R}}\) is said to be conditionally δ-midconvex provided \({f(\frac{x+y}{2}) \leq \frac{f(x) + f(y)}{2} + \delta}\) for every \({x, y \in V}\) such that \({\frac{x + y}{2} \in V}\) . We show that if V satisfies some reasonable assumptions, then for every bounded from above conditionally δ-midconvex function \({f : V \to \mathbb{R}}\) the following estimation holds: \({\sup f(V) \leq \sup f(ext \, V) + k (V)\delta}\) , where ext ? V denotes the set of all extremal points of V and k(V) is a respective constant depending on V.