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On approximately convex functions. (English) Zbl 0823.26006

Let \(D\) be a convex subset of a real vector space \(X\) and \(\varepsilon\) a nonnegative constant. A real valued function \(f\) on \(D\) is \(\varepsilon\)- convex if \(f(tx+ (1- t)y)- tf(x)- (1- t) f(y)\leq \varepsilon\) and \(\varepsilon\)-Wright-convex if \(f(tx+ (1- t)y)+ f((1- t)x+ ty)- f(x)- f(y)\leq 2\varepsilon\) \((x, y\in [0, 1])\). A typical result states that \(\varepsilon\)-Wright-convex functions, bounded from below on a neighbourhood of an interior point of \(D\), are \(2\varepsilon\)-convex.

MSC:

26A51 Convexity of real functions in one variable, generalizations
26B25 Convexity of real functions of several variables, generalizations
39B72 Systems of functional equations and inequalities
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[1] E. F. Bechenbach, Convex functions, Bull. Amer. Math. Soc. 54 (1948), 439 – 460. · Zbl 0041.38003
[2] D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952), 821 – 828. · Zbl 0047.29505
[3] Z. Kominek, On additive and convex functionals, Rad. Mat. 3 (1987), no. 2, 267 – 279 (English, with Serbo-Croatian summary). · Zbl 0643.39006
[4] Zygfryd Kominek, Convex functions in linear spaces, Prace Naukowe Uniwersytetu Śląskiego w Katowicach [Scientific Publications of the University of Silesia], vol. 1087, Uniwersytet Śląski, Katowice, 1989. With Polish and Russian summaries. · Zbl 0699.46005
[5] Z. Kominek and M. Kuczma, Theorems of Bernstein-Doetsch, Piccard and Mehdi and semilinear topology, Arch. Math. (Basel) 52 (1989), no. 6, 595 – 602. · Zbl 0683.46006
[6] 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. · Zbl 0555.39004
[7] Marek Kuczma, An example of semilinear topologies, Stochastica 12 (1988), no. 2-3, 197 – 205. · Zbl 0708.46011
[8] C. T. Ng, On midconvex functions with midconcave bounds, Proc. Amer. Math. Soc. 102 (1988), no. 3, 538 – 540. · Zbl 0659.39004
[9] C. T. Ng, Functions generating Schur-convex sums, General inequalities, 5 (Oberwolfach, 1986) Internat. Schriftenreihe Numer. Math., vol. 80, Birkhäuser, Basel, 1987, pp. 433 – 438. · Zbl 0634.39014
[10] 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. · Zbl 0271.26009
[11] Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. · Zbl 0129.37203
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