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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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Correspondence to Eder Kikianty .

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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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