An elementary proof for the decomposition theorem of Wright convex functions
Abstract
The main goal of this paper is to give a completely elementary proof for the decomposition theorem of Wright convex functions which was discovered by C. T. Ng in 1987. In the proof, we do not use transfinite tools, i.e., variants of Rodé’s theorem, or de Bruijn’s theorem related to functions with continuous differences.
Keywords
Wright convexity; Jensen convexity; decomposition theorem
References
1. M. Adamek, Almost λ-convex and almost Wright-convex functions, Math. Slovaca 53 (2003), no. 1, 67–73.
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. N.G. de Bruijn, Functions whose differences belong to a given class, Nieuw Arch. Wisk. (2) 23 (1951), 194–218.
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. A. Gilányi and Zs. Páles, On convex functions of higher order, Math. Inequal. Appl. 11 (2008), no. 2, 271–282.
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9. M. Lewicki, A remark on quasiaffine functions, Demonstratio Math. 39 (2006), no. 4, 743–750.
10. M. Lewicki, Wright-convexity with respect to arbitrary means, J. Math. Inequal. 1 (2007), no. 3, 419–424.
11. M. Lewicki, Baire measurability of (M,N)-Wright convex functions, Comment. Math. Prace Mat. 48 (2008), no. 1, 75–83.
12. M. Lewicki, Measurability of (M,N)-Wright convex functions, Aequationes Math. 78 (2009), no. 1-2, 9–22.
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. Gy. Maksa and Zs. Páles, Decomposition of higher-order Wright-convex functions, J. Math. Anal. Appl. 359 (2009), 439–443.
15. J. Matkowski, On a-Wright convexity and the converse of Minkowski’s inequality, Aequationes Math. 43 (1992), no. 1, 106–112.
16. J. Matkowski and M. Wróbel, A generalized a-Wright convexity and related functional equation, Ann. Math. Sil. 10 (1996), 7–12.
17. J. Mrowiec, On the stability of Wright-convex functions, Aequationes Math. 65 (2003), no. 1-2, 158–164.
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.
19. K. Nikodem, On some class of midconvex functions, Ann. Polon. Math. 50 (1989), no. 2, 145–151.
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. K. Nikodem, T. Rajba, and Sz. Wąsowicz, On the classes of higher-order Jensenconvex functions and Wright-convex functions, J. Math. Anal. Appl. 396 (2012), no. 1, 261–269.
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. A. Olbryś, Some conditions implying the continuity of t-Wright convex functions, Publ. Math. Debrecen 68 (2006), no. 3-4, 401–418.
24. A. Olbryś, A characterization of (t_1,…,t_n)-Wright affine functions, Comment. Math. Prace Mat. 47 (2007), no. 1, 47–56.
25. A. Olbryś, A support theorem for t-Wright-convex functions, Math. Inequal. Appl. 14 (2011), no. 2, 399–412.
26. A. Olbryś, Representation theorems for t-Wright convexity, J. Math. Anal. Appl. 384 (2011), no. 2, 273–283.
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. A. Olbryś, On some inequalities equivalent to the Wright-convexity, J. Math. Inequal. 9 (2015), no. 2, 449–461.
29. A. Olbryś, On support, separation and decomposition theorems for t-Wright-concave functions, Math. Slovaca 67 (2017), no. 3, 719–730.
30. Zs. Páles, On Wright- but not Jensen-convex functions of higher order, Ann. Univ. Sci. Budapest. Sect. Comput. 41 (2013), 227–234.
31. J.E. Pečarić and I. Raşa, Inequalities for Wright-convex functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 50 (1996), 185–190.
32. T. Rajba, A generalization of multiple Wright-convex functions via randomization, J. Math. Anal. Appl. 388 (2012), no. 1, 548–565.
33. A.W. Roberts and D.E. Varberg, Convex Functions, Pure and Applied Mathematics, vol. 57, Academic Press, New York-London, 1973.
34. G. Rodé, Eine abstrakte Version des Satzes von Hahn-Banach, Arch. Math. (Basel) 31 (1978), 474–481.
35. E.M. Wright, An inequality for convex functions, Amer. Math. Monthly 61 (1954), 620–622.
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. N.G. de Bruijn, Functions whose differences belong to a given class, Nieuw Arch. Wisk. (2) 23 (1951), 194–218.
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. A. Gilányi and Zs. Páles, On convex functions of higher order, Math. Inequal. Appl. 11 (2008), no. 2, 271–282.
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. 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. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Prace Naukowe Uniwersytetu Slaskiego w Katowicach, vol. 489, Państwowe Wydawnictwo Naukowe - Uniwersytet Śląski, Warszawa-Kraków-Katowice, 1985, 2nd edn. (ed. by A. Gilányi), Birkhäuser, Basel, 2009.
9. M. Lewicki, A remark on quasiaffine functions, Demonstratio Math. 39 (2006), no. 4, 743–750.
10. M. Lewicki, Wright-convexity with respect to arbitrary means, J. Math. Inequal. 1 (2007), no. 3, 419–424.
11. M. Lewicki, Baire measurability of (M,N)-Wright convex functions, Comment. Math. Prace Mat. 48 (2008), no. 1, 75–83.
12. M. Lewicki, Measurability of (M,N)-Wright convex functions, Aequationes Math. 78 (2009), no. 1-2, 9–22.
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. Gy. Maksa and Zs. Páles, Decomposition of higher-order Wright-convex functions, J. Math. Anal. Appl. 359 (2009), 439–443.
15. J. Matkowski, On a-Wright convexity and the converse of Minkowski’s inequality, Aequationes Math. 43 (1992), no. 1, 106–112.
16. J. Matkowski and M. Wróbel, A generalized a-Wright convexity and related functional equation, Ann. Math. Sil. 10 (1996), 7–12.
17. J. Mrowiec, On the stability of Wright-convex functions, Aequationes Math. 65 (2003), no. 1-2, 158–164.
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.
19. K. Nikodem, On some class of midconvex functions, Ann. Polon. Math. 50 (1989), no. 2, 145–151.
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. K. Nikodem, T. Rajba, and Sz. Wąsowicz, On the classes of higher-order Jensenconvex functions and Wright-convex functions, J. Math. Anal. Appl. 396 (2012), no. 1, 261–269.
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. A. Olbryś, Some conditions implying the continuity of t-Wright convex functions, Publ. Math. Debrecen 68 (2006), no. 3-4, 401–418.
24. A. Olbryś, A characterization of (t_1,…,t_n)-Wright affine functions, Comment. Math. Prace Mat. 47 (2007), no. 1, 47–56.
25. A. Olbryś, A support theorem for t-Wright-convex functions, Math. Inequal. Appl. 14 (2011), no. 2, 399–412.
26. A. Olbryś, Representation theorems for t-Wright convexity, J. Math. Anal. Appl. 384 (2011), no. 2, 273–283.
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. A. Olbryś, On some inequalities equivalent to the Wright-convexity, J. Math. Inequal. 9 (2015), no. 2, 449–461.
29. A. Olbryś, On support, separation and decomposition theorems for t-Wright-concave functions, Math. Slovaca 67 (2017), no. 3, 719–730.
30. Zs. Páles, On Wright- but not Jensen-convex functions of higher order, Ann. Univ. Sci. Budapest. Sect. Comput. 41 (2013), 227–234.
31. J.E. Pečarić and I. Raşa, Inequalities for Wright-convex functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 50 (1996), 185–190.
32. T. Rajba, A generalization of multiple Wright-convex functions via randomization, J. Math. Anal. Appl. 388 (2012), no. 1, 548–565.
33. A.W. Roberts and D.E. Varberg, Convex Functions, Pure and Applied Mathematics, vol. 57, Academic Press, New York-London, 1973.
34. G. Rodé, Eine abstrakte Version des Satzes von Hahn-Banach, Arch. Math. (Basel) 31 (1978), 474–481.
35. E.M. Wright, An inequality for convex functions, Amer. Math. Monthly 61 (1954), 620–622.
PálesZ. (2020). An elementary proof for the decomposition theorem of Wright convex functions. Annales Mathematicae Silesianae, 34(1), 142-150. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/13639
Zsolt Páles
pales@science.unideb.hu
Institute of Mathematics, University of Debrecen, Hungary Hungary
https://orcid.org/0000-0003-2382-6035
Institute of Mathematics, University of Debrecen, Hungary Hungary
https://orcid.org/0000-0003-2382-6035
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