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An Elementary Proof for the Decomposition Theorem of Wright Convex Functions

   | Jul 09, 2020

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[1] M. Adamek, Almost λ -convex and almost Wright-convex functions, Math. Slovaca 53 (2003), no. 1, 67–73.Search in Google Scholar

[2] A. Bahyrycz and J. Olko, Stability of the equation of (p, q)-Wright functions, Acta Math. Hungar. 146 (2015), no. 1, 71–85.Search in Google Scholar

[3] N.G. de Bruijn, Functions whose differences belong to a given class, Nieuw Arch. Wisk. (2) 23 (1951), 194–218.Search in Google Scholar

[4] A. Gilányi, N. Merentes, K. Nikodem, and Zs. Páles, Characterizations and decomposition of strongly Wright-convex functions of higher order, Opuscula Math. 35 (2015), no. 1, 37–46.Search in Google Scholar

[5] A. Gilányi and Zs. Páles, On convex functions of higher order, Math. Inequal. Appl. 11 (2008), no. 2, 271–282.Search in Google Scholar

[6] Z. Kominek, Convex Functions in Linear Spaces, Prace Naukowe Uniwersytetu ‚Śląskiego w Katowicach [Scientific Publications of the University of Silesia], 1087, Uniwersytet ‚Śląski, Katowice, 1989.Search in Google Scholar

[7] Z. Kominek and J. Mrowiec, Nonstability results in the theory of convex functions, C. R. Math. Acad. Sci. Soc. R. Can. 28 (2006), no. 1, 17–23.Search in Google Scholar

[8] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Prace Naukowe Uniwersytetu ‚Śląskiego w Katowicach, vol. 489, Państwowe Wydawnictwo Naukowe — Uniwersytet ‚Śląski, Warszawa–Kraków–Katowice, 1985, 2nd edn. (ed. by A. Gilányi), Birkhäuser, Basel, 2009.Search in Google Scholar

[9] M. Lewicki, A remark on quasiaffine functions, Demonstratio Math. 39 (2006), no. 4, 743–750.Search in Google Scholar

[10] M. Lewicki, Wright-convexity with respect to arbitrary means, J. Math. Inequal. 1 (2007), no. 3, 419–424.Search in Google Scholar

[11] M. Lewicki, Baire measurability of (M, N)-Wright convex functions, Comment. Math. Prace Mat. 48 (2008), no. 1, 75–83.Search in Google Scholar

[12] M. Lewicki, Measurability of (M, N)-Wright convex functions, Aequationes Math. 78 (2009), no. 1-2, 9–22.Search in Google Scholar

[13] Gy. Maksa, K. Nikodem, and Zs. Páles, Results on t-Wright convexity, C. R. Math. Rep. Acad. Sci. Canada 13 (1991), no. 6, 274–278.Search in Google Scholar

[14] Gy. Maksa and Zs. Páles, Decomposition of higher-order Wright-convex functions, J. Math. Anal. Appl. 359 (2009), 439–443.10.1016/j.jmaa.2009.05.047Search in Google Scholar

[15] J. Matkowski, On a-Wright convexity and the converse of Minkowski’s inequality, Aequationes Math. 43 (1992), no. 1, 106–112.Search in Google Scholar

[16] J. Matkowski and M. Wróbel, A generalized a-Wright convexity and related functional equation, Ann. Math. Sil. 10 (1996), 7–12.Search in Google Scholar

[17] J. Mrowiec, On the stability of Wright-convex functions, Aequationes Math. 65 (2003), no. 1-2, 158–164.Search in Google Scholar

[18] C.T. Ng, Functions generating Schur-convex sums, in: W. Walter (ed.), General Inequalities, 5 (Oberwolfach, 1986), International Series of Numerical Mathematics, vol. 80, Birkhäuser, Basel–Boston, 1987, pp. 433–438.10.1007/978-3-0348-7192-1_35Search in Google Scholar

[19] K. Nikodem, On some class of midconvex functions, Ann. Polon. Math. 50 (1989), no. 2, 145–151.Search in Google Scholar

[20] K. Nikodem and Zs. Páles, On approximately Jensen-convex and Wright-convex functions, C. R. Math. Rep. Acad. Sci. Canada 23 (2001), no. 4, 141–147.Search in Google Scholar

[21] K. Nikodem, T. Rajba, and Sz. Wąsowicz, On the classes of higher-order Jensen-convex functions and Wright-convex functions, J. Math. Anal. Appl. 396 (2012), no. 1, 261–269.Search in Google Scholar

[22] A. Olbryś, On the measurability and the Baire property of t-Wright-convex functions, Aequationes Math. 68 (2004), no. 1-2, 28–37.Search in Google Scholar

[23] A. Olbryś, Some conditions implying the continuity of t-Wright convex functions, Publ. Math. Debrecen 68 (2006), no. 3-4, 401–418.Search in Google Scholar

[24] A. Olbryś, A characterization of (t1, . . ., tn)-Wright affine functions, Comment. Math. Prace Mat. 47 (2007), no. 1, 47–56.Search in Google Scholar

[25] A. Olbryś, A support theorem for t-Wright-convex functions, Math. Inequal. Appl. 14 (2011), no. 2, 399–412.Search in Google Scholar

[26] A. Olbryś, Representation theorems for t-Wright convexity, J. Math. Anal. Appl. 384 (2011), no. 2, 273–283.Search in Google Scholar

[27] A. Olbryś, On the boundedness, Christensen measurability and continuity of t-Wright convex functions, Acta Math. Hungar. 141 (2013), no. 1–2, 68–77.Search in Google Scholar

[28] A. Olbryś, On some inequalities equivalent to the Wright-convexity, J. Math. Inequal. 9 (2015), no. 2, 449–461.Search in Google Scholar

[29] A. Olbryś, On support, separation and decomposition theorems for t-Wright-concave functions, Math. Slovaca 67 (2017), no. 3, 719–730.Search in Google Scholar

[30] Zs. Páles, On Wright- but not Jensen-convex functions of higher order, Ann. Univ. Sci. Budapest. Sect. Comput. 41 (2013), 227–234.Search in Google Scholar

[31] J.E. Pečarić and I. Raşa, Inequalities for Wright-convex functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 50 (1996), 185–190.Search in Google Scholar

[32] T. Rajba, A generalization of multiple Wright-convex functions via randomization, J. Math. Anal. Appl. 388 (2012), no. 1, 548–565.Search in Google Scholar

[33] A.W. Roberts and D.E. Varberg, Convex Functions, Pure and Applied Mathematics, vol. 57, Academic Press, New York–London, 1973.Search in Google Scholar

[34] G. Rodé, Eine abstrakte Version des Satzes von Hahn–Banach, Arch. Math. (Basel) 31 (1978), 474–481.10.1007/BF01226477Search in Google Scholar

[35] E.M. Wright, An inequality for convex functions, Amer. Math. Monthly 61 (1954), 620–622.Search in Google Scholar

eISSN:
2391-4238
ISSN:
0860-2107
Language:
English
Publication timeframe:
2 times per year
Journal Subjects:
Mathematics, General Mathematics