Abstract
We discuss the connections between boundedness and continuity of t-Wright convex functions, moreover, we generalize some results of P. Fischer and Z. Słodkowski [4] concerning the Christensen measurability of Jensen convex functions to the case of t-Wright convex functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Bernstein and G. Doetsch, Zur Theorie der konvexen Funktionen, Math. Ann., 76 (1915), 514–526.
J. Brzdęk, The Christensen measurable solutions of a generalization of the Gołab–Schinzel functional equation, Ann. Polon. Math., 64 (1996), 195–205.
J. P. R. Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math., 13 (1972), 255–260.
P. Fischer and Z. Słodkowski, Christensen zero sets and measurable convex functions, Proc. Amer. Math. Soc., 79 (1980), 449–453.
Z. Gajda, Christensen measurability of polynomial functions and convex functions of higher orders, Ann. Polon. Math., XLVII (1986), 25–40.
R. Ger, Convex functions of higher orders in Euclidean spaces, Ann. Polon. Math., 25 (1972), 293–302.
R. Ger, n-convex functions in linear spaces, Aequationes Math., 10 (1974), 172–176.
R. Ger and M. Kuczma, On the boundedness and continuity of convex functions and additive functions, Aequationes Math., 4 (1970), 157–162.
E. Jabłońska, Jensen convex functions bounded above on nonzero Christensen measurable sets, Annal. Math. Sil., 23 (2009), 53–55.
Z. Kominek, A continuity result on t-Wright convex functions, Publ. Math. Debrecen, 63 (2003), 213–219.
Z. Kominek, Convex Functions in Linear Spaces, Prace Naukowe Uniwersytetu Śląskiego w Katowicach 1087 (Katowice, 1989).
M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Publishers and Silesian University Press (Warszawa–Kraków–Katowice, 1985).
Gy. Maksa, K. Nikodem and Zs. Páles, Result on t-Wright convexity, C.R. Math. Rep. Acad. Sci. Canada, 13 (1991), 274–278.
A. Olbryś, On the measurability and the Baire property of t-Wright convex functions, Aequationes Math., 68 (2004), 28–37.
A. Olbryś, Some conditions implying the continuity of t-Wright convex functions, Publ. Math. Debrecen, 68 (2006), 401–418.
A. Olbryś, Representation theorems for t-Wright convexity, J. Math. Anal. Appl., 384 (2011), 273–283.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Olbryś, A. On the boundedness, Christensen measurability and continuity of t-Wright convex functions. Acta Math Hung 141, 68–77 (2013). https://doi.org/10.1007/s10474-013-0330-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-013-0330-z