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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On approximately convex Takagi type functions
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by Judit Makó and Zsolt Páles PDF
Proc. Amer. Math. Soc. 141 (2013), 2069-2080 Request permission

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
  • 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
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 3034432