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:
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 [\(a,b\)].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Anastassiou, G.A.: Univariate Ostrowski inequalities, revisited. Monatsh. Math. 135(3), 175–189 (2002)
Cerone, P., Dragomir, S.S.: Midpoint-type rules from an inequalities point of view. In: Anastassiou, G. (ed.) Handbook of Analytic-Computational Methods in Applied Mathematics, pp. 135–200. CRC Press, New York (2000)
Cerone, P., Dragomir, S.S.: Trapezoidal-type rules from an inequalities point of view. In: Anastassiou, G. (ed.) Handbook of Analytic-Computational Methods in Applied Mathematics, pp. 65–134. CRC Press, New York (2000)
Cerone, P., Dragomir, S.S.: New bounds for the three-point rule involving the Riemann–Stieltjes integrals. In: Gulati, C., et al. (eds.) Advances in Statistics Combinatorics and Related Areas, pp. 53–62. World Science Publishing, Singapore (2002)
Cerone, P., Dragomir, S.S., Roumeliotis, J.: Some Ostrowski type inequalities for n-time differentiable mappings and applications. Demonstr. Math. 32(2), 697–712 (1999)
Cerone, P., Dragomir, S.S., Pearce, C.E.M.: A generalised trapezoid inequality for functions of bounded variation. Turk. J. Math. 24(2), 147–163 (2000)
Dragomir, S.S.: The Ostrowski’s integral inequality for mappings of bounded variation. Bull. Aust. Math. Soc. 60, 495–508 (1999)
Dragomir, S.S.: The Ostrowski’s integral inequality for Lipschitzian mappings and applications. Comp. Math. Appl. 38, 33–37 (1999)
Dragomir, S.S.: Ostrowski’s inequality for monotonous mappings and applications. J. KSIAM 3(1), 127–135 (1999)
Dragomir, S.S.: On the Ostrowski’s inequality for Riemann–Stieltjes integral. Korean J. Appl. Math. 7, 477–485 (2000)
Dragomir, S.S.: On the Ostrowski’s inequality for mappings of bounded variation and applications. Math. Inequal. Appl. 4(1), 33–40 (2001)
Dragomir, S.S.: On the Ostrowski inequality for Riemann-Stieltjes integral \(\int_{a}^{b}f(t) du(t)\) where f is of Hölder type and u is of bounded variation and applications. J. KSIAM 5(1), 35–45 (2001)
Dragomir, S.S.: An inequality improving the first Hermite–Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. J. Inequal. Pure Appl. Math. 3(2), Article 31 (2002)
Dragomir, S.S.: An inequality improving the second Hermite–Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products. J. Inequal. Pure Appl. Math. 3(3), Article 35 (2002)
Dragomir, S.S.: Ostrowski type inequalities for isotonic linear functionals. J. Inequal. Pure Appl. Math. 3(3), Article 68 (2002)
Dragomir, S.S.: An Ostrowski like inequality for convex functions and applications. Rev. Math. Complut. 16(2), 373–382 (2003)
Dragomir, S.S.: Operator Inequalities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics. Springer, New York (2012)
Dragomir, S.S., Rassias, Th.M. (eds.): Ostrowski Type Inequalities and Applications in Numerical Integration. Kluwer Academic, Dordrecht (2002)
Dragomir, S.S., Wang, S.: A new inequality of Ostrowski’s type in \(L1-\)norm and applications to some special means and to some numerical quadrature rules. Tamkang J. Math. 28, 239–244 (1997)
Dragomir, S.S., Wang, S.: Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and some numerical quadrature rules. Appl. Math. Lett. 11, 105–109 (1998)
Dragomir, S.S., Wang, S.: A new inequality of Ostrowski’s type in \(Lp-\)norm and applications to some special means and to some numerical quadrature rules. Indian J. Math. 40(3), 245–304 (1998)
Dragomir, S.S., Cerone, P., Roumeliotis, J., Wang, S.: A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis. Bull. Math. Soc. Sci. Math. Rom. 42(4), 301–314 (1999)
Fink, A.M.: Bounds on the deviation of a function from its averages. Czechoslov. Math. J. 42(2), 298–310 (1992)
Kechriniotis, A.I., Assimakis, N.D.: Generalizations of the trapezoid inequalities based on a new mean value theorem for the remainder in Taylor’s formula. J. Inequal. Pure Appl. Math. 7(3), Article 90 (2006)
Liu, Z.: Some inequalities of perturbed trapezoid type. J. Inequal. Pure Appl. Math. 7(2), Article 47 (2006)
Mercer, A.McD.: On perturbed trapezoid inequalities. J. Inequal. Pure Appl. Math. 7(4), Article 118 (2006)
Ostrowski, A.: Über die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert. Comment. Math. Helv. 10, 226–227 (1938)
Ujević, N.: Error inequalities for a generalized trapezoid rule. Appl. Math. Lett. 19(1), 32–37 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-1-4939-1246-9_4
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-1245-2
Online ISBN: 978-1-4939-1246-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)