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.
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.
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://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
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
