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ON APPROXIMATELY MIDCONVEX FUNCTIONS

  • ATTILA HÁZY (a1) and ZSOLT PÁLES (a2)
Abstract

A real-valued function $f$ defined on an open, convex set $D$ of a real normed space is called $(\varepsilon,\delta)$-midconvex if it satisfies $$f\left(\frac{x+y}{2}\right) \le \frac{f(x)+f(y)}{2} + \varepsilon|x-y| + \delta, \quad\hbox{for } x,y\in D.$$ The main result of the paper states that if $f$ is locally bounded from above at a point of $D$ and is $(\varepsilon,\delta)$-midconvex, then it satisfies the convexity-type inequality $$f(\lambda x+(1-\lambda) y) \leq \lambda f(x)+(1-\lambda)f(y)+2\delta +2\varepsilon \varphi(\lambda)|x-y| \quad\hbox{for } x,y\in D, \, \lambda\in[0,1],$$ where $\varphi:[0,1]\to{\mathbb R}$ is a continuous function satisfying $$\max(-\lambda\log_2\lambda,\,-(1-\lambda)\log_2(1-\lambda)) \le\varphi(\lambda)\le 1.4\max(-\lambda\log_2\lambda,\,-(1-\lambda)\log_2(1-\lambda))$$. The particular case $\varepsilon=0$ of this result is due to Ng and Nikodem (Proc. Amer. Math. Soc. 118 (1993) 103–108), while the specialization $\varepsilon=\delta=0$ yields the theorem of Bernstein and Doetsch (Math. Ann. 76 (1915) 514–526).

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This research has been supported by the Hungarian Scientific Research Fund (OTKA) Grants T-038072 and T-043080, and by the Higher Education, Research and Development Fund (FKFP) Grant 0215/2001.
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Bulletin of the London Mathematical Society
  • ISSN: 0024-6093
  • EISSN: 1469-2120
  • URL: /core/journals/bulletin-of-the-london-mathematical-society
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