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On approximately convex Takagi type functions


Authors: Judit Makó and Zsolt Páles
Journal: Proc. Amer. Math. Soc. 141 (2013), 2069-2080
MSC (2010): Primary 39B62, 26B25
DOI: https://doi.org/10.1090/S0002-9939-2013-11486-3
Published electronically: January 23, 2013
MathSciNet review: 3034432
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Abstract: Given a nonnegative function $\phi :[0,\frac 12]\to \mathbb {R}_+$, we define the Takagi type function $S_\phi :\mathbb {R}\to \mathbb {R}$ by \begin{equation*} S_\phi (x):=\sum _{n=0}^{\infty }2\phi \big (\tfrac {1}{2^{n+1}}\big )d_Z(2^nx), \end{equation*} where $d_{\mathbb {Z}}(x):=\operatorname {dist}(x,\mathbb Z) :=\inf \{|x-k|: k\in \mathbb Z\}.$ The main result of the paper states that if $\phi (0)=0$ and the mapping $x\mapsto \phi (x)/x$ is concave, then the Takagi type function $S_\phi$ is approximately Jensen convex in the following sense: \begin{equation*} S_\phi \Big (\frac {x+y}{2}\Big ) \leq \frac {S_\phi (x)+S_\phi (y)}{2}+\phi \circ d_\mathbb Z\Big (\frac {x-y}{2}\Big ) \qquad (x,y\in \mathbb {R}). \end{equation*} Applications to the theory of approximately convex functions are also given.


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Additional Information

Judit Makó
Affiliation: Institute of Mathematics, University of Debrecen, Pf. 12, H-4010 Debrecen, Hungary
Email: makoj@science.unideb.hu

Zsolt Páles
Affiliation: Institute of Mathematics, University of Debrecen, Pf. 12, H-4010 Debrecen, Hungary
Email: pales@science.unideb.hu

Keywords: Approximate Jensen convexity, $\phi$-Jensen convexity, Takagi type function
Received by editor(s): March 23, 2010
Received by editor(s) in revised form: October 4, 2011
Published electronically: January 23, 2013
Additional Notes: This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grant NK81402 and by the TÁMOP 4.2.1/B-09/1/KONV-2010-0007 and 4.2.2/B-10/1-2010-0024 projects implemented through the New Hungary Development Plan co-financed by the European Social Fund and the European Regional Development Fund.
Communicated by: Sergei K. Suslov
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.