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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: January 23, 2013
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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

$\displaystyle S_\phi (x):=\sum _{n=0}^{\infty }2\phi \big (\tfrac {1}{2^{n+1}}\big )d_Z(2^nx),$    

where $ d_{\mathbb{Z}}(x):=\operatorname {dist}(x,\mathbb{Z}) :=\inf \{\vert x-k\vert: 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:

$\displaystyle S_\phi \Big (\frac {x+y}{2}\Big ) \leq \frac {S_\phi (x)+S_\phi (... ...}+\phi \circ d_\mathbb{Z}\Big (\frac {x-y}{2}\Big ) \qquad (x,y\in \mathbb{R}).$    

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

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11486-3
PII: S 0002-9939(2013)11486-3
Keywords: Approximate Jensen convexity, $𝜑$-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.