On approximately convex functions
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Abstract:
A real valued function $f$ defined on a real interval $I$ is called $(\varepsilon ,\delta )$-convex if it satisfies \[ f(tx+(1-t)y)\le tf(x)+(1-t)f(y) + \varepsilon t(1-t)|x-y| + \delta \quad \text {for}\ x,y\in I, t\in [0,1]. \] The main results of the paper offer various characterizations for $(\varepsilon ,\delta )$-convexity. One of the main results states that $f$ is $(\varepsilon ,\delta )$-convex for some positive $\varepsilon$ and $\delta$ if and only if $f$ can be decomposed into the sum of a convex function, a function with bounded supremum norm, and a function with bounded Lipschitz-modulus. In the special case $\varepsilon =0$, the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called $\delta$-convexity.References
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Additional Information
- Zsolt Páles
- Affiliation: Institute of Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary
- Email: pales@math.klte.hu
- Received by editor(s): April 2, 2001
- Received by editor(s) in revised form: September 4, 2001
- Published electronically: June 5, 2002
- Additional Notes: This research was supported by the Hungarian Scientific Research Fund (OTKA) Grant T-038072 and by the Higher Education, Research and Development Fund (FKFP) Grant 0215/2001.
- Communicated by: Jonathan M. Borwein
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 243-252
- MSC (2000): Primary 26A51, 26B25
- DOI: https://doi.org/10.1090/S0002-9939-02-06552-8
- MathSciNet review: 1929044