Opuscula Math. 35, no. 3 (2015), 397-410

Opuscula Mathematica

Generalized Levinson's inequality and exponential convexity

Josip Pečarić
Marjan Praljak
Alfred Witkowski

Abstract. We give a probabilistic version of Levinson's inequality under Mercer's assumption of equal variances for the family of 3-convex functions at a point. We also show that this is the largest family of continuous functions for which the inequality holds. New families of exponentially convex functions and related results are derived from the obtained inequality.

Keywords: Levinson's inequality, exponential convexity.

Mathematics Subject Classification: 26D15.

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  • Josip Pečarić
  • Faculty Of Textile Technology, University Of Zagreb, Prilaz baruna Filipovića 28a, 10000 Zagreb, Croatia
  • Marjan Praljak
  • Faculty of Food Technology and Biotechnology, University Of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia
  • Alfred Witkowski
  • Institute of Mathematics and Physics, UTP University of Science and Technology, al. prof. Kaliskiego 7, 85-796 Bydgoszcz, Poland
  • Communicated by Zbigniew Szkutnik.
  • Received: 2013-11-16.
  • Revised: 2014-09-16.
  • Accepted: 2014-09-23.
  • Published online: 2014-12-15.
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Cite this article as:
Josip Pečarić, Marjan Praljak, Alfred Witkowski, Generalized Levinson's inequality and exponential convexity, Opuscula Math. 35, no. 3 (2015), 397-410, http://dx.doi.org/10.7494/OpMath.2015.35.3.397

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