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Multiplicative Ostrowski and Trapezoid Inequalities

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Handbook of Functional Equations

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 95))

Abstract

We introduce the multiplicative Ostrowski and trapezoid inequalities, that is, providing bounds for the comparison of a function f and its integral mean in the following sense:

$$\begin{aligned} f(x) {\rm exp}\! \left[\!-\frac{1}{b-a}\int_a^b\!\! \log f(t)\, dt\!\right]\! \text{and}\!\ f(b)^{\frac{b-x}{b-a}} f(a)^{\frac{x-a}{b-a}} {\rm exp}\! \left[\!-\frac{1}{b-a}\int_a^b\!\! \log f(t)\, dt\!\right].\end{aligned}$$

We consider the cases of absolutely continuous and logarithmic convex functions. We apply these inequalities to provide approximations for the integral of f; and the first moment of f around zero, that is, \(\int_{a}^{b}xf(x)dx\); for an absolutely continuous function f on [\(a,b\)].

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Author information

Authors and Affiliations

  1. Department of Pure and Applied Mathematics, University of Johannesburg, PO Box 524, Auckland Park, 2006, South Africa

    Eder Kikianty

  2. Department of Mathematics and Statistics, La Trobe University, Bundoora, 3086, Australia

    Pietro Cerone

  3. School of Engineering and Science, Victoria University, Melbourne, 8001, Australia

    Sever S. Dragomir

  4. School of Computational and Applied Mathematics, University of the Witwatersrand, Witwatersrand, South Africa

    Sever S. Dragomir

Authors
  1. Pietro Cerone
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  2. Sever S. Dragomir
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  3. Eder Kikianty
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Corresponding author

Correspondence to Eder Kikianty .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Cerone, P., Dragomir, S., Kikianty, E. (2014). Multiplicative Ostrowski and Trapezoid Inequalities. In: Rassias, T. (eds) Handbook of Functional Equations. Springer Optimization and Its Applications, vol 95. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1246-9_4

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