A functional generalization of Ostrowski inequality via Montgomery identity
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Abstract
We show in this paper amongst other that, if $f : [a; b] \to R$ isabsolutely continuous on [a; b] and $\Phi : R \to R$ is convex (concave) on R then$$\Phi (f (x)-\frac{1}{b-a}\int_a^b f(t) dt \leq (\geq )\frac{1}{b-a}[\int_a^b \Phi[(t-a)f'(t)]dt-\int_a^b \Phi[(t-b)f'(t)]dt]$$for any $x \in[a; b]$.Natural applications for power and exponential functions are provided as well. Bounds for the Lebesgue p-norms of the deviation of a function from itsintegral mean are also given.
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Dragomir, S.
(2015).
A functional generalization of Ostrowski inequality via Montgomery identity.
Acta Mathematica Universitatis Comenianae, 84(1), 63-78.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/54/127
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References
[1] G. A. Anastassiou, Univariate Ostrowski inequalities, revisited. Monatsh. Math., 135 (2002), no. 3, 175189.
[2] P. Cerone and S. S. Dragomir, Midpoint-type rules from an inequalities point of view, Ed. G. A. Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics,
CRC Press, New York. 135-200.
[3] P. Cerone and S. S. Dragomir, New bounds for the three-point rule involving the Riemann-Stieltjes integrals, in Advances in Statistics Combinatorics and Related Areas, C. Gulati, et al. (Eds.), World Science Publishing, 2002, 53-62.
[4] P. Cerone, S. S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time di¤erentiable mappings and applications, Demonstratio Mathematica, 32(2) (1999), 697
712.
[5] S. S. Dragomir, Ostrowskis inequality for monotonous mappings and applications, J. KSIAM, 3(1) (1999), 127-135.
[6] S. S. Dragomir, The Ostrowskis integral inequality for Lipschitzian mappings and applications, Comp. Math. Appl., 38 (1999), 33-37.
[7] S. S. Dragomir, On the Ostrowskis inequality for Riemann-Stieltjes integral, Korean J. Appl. Math., 7 (2000), 477-485.
[8] S. S. Dragomir, On the Ostrowskis inequality for mappings of bounded variation and applications, Math. Ineq. & Appl., 4(1) (2001), 33-40.
[9] S. S. Dragomir, On the Ostrowski inequality for Riemann-Stieltjes integral Rb a f (t) du (t) where f is of Hölder type and u is of bounded variation and applications, J. KSIAM, 5(1)
(2001), 35-45.
[10] S. S. Dragomir, Ostrowski type inequalities for isotonic linear functionals, J. Inequal. Pure & Appl. Math., 3(5) (2002), Art. 68.
[11] S. S. Dragomir, An inequality improving the rst Hermite-Hadamard inequality for convex functions de ned on linear spaces and applications for semi-inner products. J. Inequal. Pure Appl. Math. 3 (2002), no. 2, Article 31, 8 pp.
[12] S. S. Dragomir, An Ostrowski like inequality for convex functions and applications, Revista Math. Complutense, 16(2) (2003), 373-382.
[13] S. S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics. Springer, New York, 2012. x+112 pp. ISBN: 978-1-4614-1778-1
[14] S. S. Dragomir, P. Cerone, J. Roumeliotis and S. Wang, A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis, Bull. Math.
Soc. Sci. Math. Romanie, 42(90) (4) (1999), 301-314.
[15] S. S. Dragomir and Th. M. Rassias (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publisher, 2002.
[16] S. S. Dragomir and S. Wang, A new inequality of Ostrowskis type in L1
[17] S. S. Dragomir and S. Wang, Applications of Ostrowskis inequality to the estimation of error bounds for some special means and some numerical quadrature rules, Appl. Math. Lett., 11 (1998), 105-109.
[18] S. S. Dragomir and S. Wang, A new inequality of Ostrowskis type in Lp-norm and applications to some special means and to some numerical quadrature rules, Indian J. of Math., 40(3) (1998), 245-304.
[19] A. M. Fink, Bounds on the deviation of a function from its averages, Czechoslovak Math. J., 42(117) (1992), No. 2, 298-310.
[20] A. Ostrowski, Über die Absolutabweichung einer di¤erentienbaren Funktionen von ihren Integralmittelwert, Comment. Math. Hel, 10 (1938), 226-227.
[2] P. Cerone and S. S. Dragomir, Midpoint-type rules from an inequalities point of view, Ed. G. A. Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics,
CRC Press, New York. 135-200.
[3] P. Cerone and S. S. Dragomir, New bounds for the three-point rule involving the Riemann-Stieltjes integrals, in Advances in Statistics Combinatorics and Related Areas, C. Gulati, et al. (Eds.), World Science Publishing, 2002, 53-62.
[4] P. Cerone, S. S. Dragomir and J. Roumeliotis, Some Ostrowski type inequalities for n-time di¤erentiable mappings and applications, Demonstratio Mathematica, 32(2) (1999), 697
712.
[5] S. S. Dragomir, Ostrowskis inequality for monotonous mappings and applications, J. KSIAM, 3(1) (1999), 127-135.
[6] S. S. Dragomir, The Ostrowskis integral inequality for Lipschitzian mappings and applications, Comp. Math. Appl., 38 (1999), 33-37.
[7] S. S. Dragomir, On the Ostrowskis inequality for Riemann-Stieltjes integral, Korean J. Appl. Math., 7 (2000), 477-485.
[8] S. S. Dragomir, On the Ostrowskis inequality for mappings of bounded variation and applications, Math. Ineq. & Appl., 4(1) (2001), 33-40.
[9] S. S. Dragomir, On the Ostrowski inequality for Riemann-Stieltjes integral Rb a f (t) du (t) where f is of Hölder type and u is of bounded variation and applications, J. KSIAM, 5(1)
(2001), 35-45.
[10] S. S. Dragomir, Ostrowski type inequalities for isotonic linear functionals, J. Inequal. Pure & Appl. Math., 3(5) (2002), Art. 68.
[11] S. S. Dragomir, An inequality improving the rst Hermite-Hadamard inequality for convex functions de ned on linear spaces and applications for semi-inner products. J. Inequal. Pure Appl. Math. 3 (2002), no. 2, Article 31, 8 pp.
[12] S. S. Dragomir, An Ostrowski like inequality for convex functions and applications, Revista Math. Complutense, 16(2) (2003), 373-382.
[13] S. S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics. Springer, New York, 2012. x+112 pp. ISBN: 978-1-4614-1778-1
[14] S. S. Dragomir, P. Cerone, J. Roumeliotis and S. Wang, A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis, Bull. Math.
Soc. Sci. Math. Romanie, 42(90) (4) (1999), 301-314.
[15] S. S. Dragomir and Th. M. Rassias (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publisher, 2002.
[16] S. S. Dragomir and S. Wang, A new inequality of Ostrowskis type in L1
[17] S. S. Dragomir and S. Wang, Applications of Ostrowskis inequality to the estimation of error bounds for some special means and some numerical quadrature rules, Appl. Math. Lett., 11 (1998), 105-109.
[18] S. S. Dragomir and S. Wang, A new inequality of Ostrowskis type in Lp-norm and applications to some special means and to some numerical quadrature rules, Indian J. of Math., 40(3) (1998), 245-304.
[19] A. M. Fink, Bounds on the deviation of a function from its averages, Czechoslovak Math. J., 42(117) (1992), No. 2, 298-310.
[20] A. Ostrowski, Über die Absolutabweichung einer di¤erentienbaren Funktionen von ihren Integralmittelwert, Comment. Math. Hel, 10 (1938), 226-227.