Have a personal or library account? Click to login
An Elementary Proof for the Decomposition Theorem of Wright Convex Functions Cover

An Elementary Proof for the Decomposition Theorem of Wright Convex Functions

By: Zsolt Páles  
Open Access
|Jul 2020

References

  1. [1] M. Adamek, Almost λ -convex and almost Wright-convex functions, Math. Slovaca 53 (2003), no. 1, 67–73.
  2. [2] A. Bahyrycz and J. Olko, Stability of the equation of (p, q)-Wright functions, Acta Math. Hungar. 146 (2015), no. 1, 71–85.
  3. [3] N.G. de Bruijn, Functions whose differences belong to a given class, Nieuw Arch. Wisk. (2) 23 (1951), 194–218.
  4. [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.
  5. [5] A. Gilányi and Zs. Páles, On convex functions of higher order, Math. Inequal. Appl. 11 (2008), no. 2, 271–282.
  6. [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.
  7. [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.
  8. [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.
  9. [9] M. Lewicki, A remark on quasiaffine functions, Demonstratio Math. 39 (2006), no. 4, 743–750.
  10. [10] M. Lewicki, Wright-convexity with respect to arbitrary means, J. Math. Inequal. 1 (2007), no. 3, 419–424.
  11. [11] M. Lewicki, Baire measurability of (M, N)-Wright convex functions, Comment. Math. Prace Mat. 48 (2008), no. 1, 75–83.
  12. [12] M. Lewicki, Measurability of (M, N)-Wright convex functions, Aequationes Math. 78 (2009), no. 1-2, 9–22.
  13. [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.
  14. [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.2009.05.047
  15. [15] J. Matkowski, On a-Wright convexity and the converse of Minkowski’s inequality, Aequationes Math. 43 (1992), no. 1, 106–112.
  16. [16] J. Matkowski and M. Wróbel, A generalized a-Wright convexity and related functional equation, Ann. Math. Sil. 10 (1996), 7–12.
  17. [17] J. Mrowiec, On the stability of Wright-convex functions, Aequationes Math. 65 (2003), no. 1-2, 158–164.
  18. [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_35
  19. [19] K. Nikodem, On some class of midconvex functions, Ann. Polon. Math. 50 (1989), no. 2, 145–151.
  20. [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.
  21. [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.
  22. [22] A. Olbryś, On the measurability and the Baire property of t-Wright-convex functions, Aequationes Math. 68 (2004), no. 1-2, 28–37.
  23. [23] A. Olbryś, Some conditions implying the continuity of t-Wright convex functions, Publ. Math. Debrecen 68 (2006), no. 3-4, 401–418.
  24. [24] A. Olbryś, A characterization of (t1, . . ., tn)-Wright affine functions, Comment. Math. Prace Mat. 47 (2007), no. 1, 47–56.
  25. [25] A. Olbryś, A support theorem for t-Wright-convex functions, Math. Inequal. Appl. 14 (2011), no. 2, 399–412.
  26. [26] A. Olbryś, Representation theorems for t-Wright convexity, J. Math. Anal. Appl. 384 (2011), no. 2, 273–283.
  27. [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.
  28. [28] A. Olbryś, On some inequalities equivalent to the Wright-convexity, J. Math. Inequal. 9 (2015), no. 2, 449–461.
  29. [29] A. Olbryś, On support, separation and decomposition theorems for t-Wright-concave functions, Math. Slovaca 67 (2017), no. 3, 719–730.
  30. [30] Zs. Páles, On Wright- but not Jensen-convex functions of higher order, Ann. Univ. Sci. Budapest. Sect. Comput. 41 (2013), 227–234.
  31. [31] J.E. Pečarić and I. Raşa, Inequalities for Wright-convex functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 50 (1996), 185–190.
  32. [32] T. Rajba, A generalization of multiple Wright-convex functions via randomization, J. Math. Anal. Appl. 388 (2012), no. 1, 548–565.
  33. [33] A.W. Roberts and D.E. Varberg, Convex Functions, Pure and Applied Mathematics, vol. 57, Academic Press, New York–London, 1973.
  34. [34] G. Rodé, Eine abstrakte Version des Satzes von Hahn–Banach, Arch. Math. (Basel) 31 (1978), 474–481.10.1007/BF01226477
  35. [35] E.M. Wright, An inequality for convex functions, Amer. Math. Monthly 61 (1954), 620–622.
DOI: https://doi.org/10.2478/amsil-2020-0010 | Journal eISSN: 2391-4238 | Journal ISSN: 0860-2107
Language: English
Page range: 142 - 150
Submitted on: Dec 31, 2019
Accepted on: Jun 3, 2020
Published on: Jul 9, 2020
Published by: University of Silesia in Katowice, Institute of Mathematics
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year

© 2020 Zsolt Páles, published by University of Silesia in Katowice, Institute of Mathematics
This work is licensed under the Creative Commons Attribution 4.0 License.