Complex Multivariate Montgomery Type Identity Leading to Complex Multivariate Ostrowski and Grüss Inequalities

Year 2019, Volume: 2 Issue: 2, 161 - 175, 27.06.2019

George Anastassıou

https://doi.org/10.33434/cams.556815

Cited By: 2

Abstract

We give a general complex multivariate Montgomery type identity which is a representation formula for a complex multivariate function. Using it we produce general tight complex multivariate high order Ostrowski and Grüss type inequalities. The estimates involve $L_{p}$ norms, any $1\leq p\leq \infty $. We include also applications.

Keywords

Multivariate complex integral , Multivariate complex continuous functions , Multivariate complex analytic functions , Multivariate complex montgomery identity , Multivariate complex Ostrowski and Gr\"{u}ss inequalities

References

• [1] A. Ostrowski, Über die Absolutabweichung einer differentiebaren Funktion von ihrem Integral mittelwert, Comment. Math. Helv. 10 (1938), 226-227.
• [2] G. Grüss, Über das Maximum des absoluten Betrages von $\frac{1}{b-a}\int_{a}^{b}f\left( x\right) g\left( x\right) dx-\frac{1}{\left( b-a\right) ^{2}}\int_{a}^{b}f\left( x\right)dx\int_{a}^{b}g\left( x\right) dx$, Math. Z., 39(1935), 215-226.
• [3] D. S. Mitrinovic, J.E. Pecaric, A.M. Fink, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1994.
• [4] S. S. Dragomir, An extension of Ostrowski’s inequality to the Complex integral, RGMIA Res. Rep. Call. 21 (2018), Art. 112, 17 pp, [http://rgmia.org/papers/v21/v21/v21a112.pdf].
• [5] S. S. Dragomir, On some Gr¨uss type inequalities for the complex integral, RGMIA Res. Rep. Call. 21 (2018), Art. 121, 12 pp, [http://rgmia.org/papers/v21/v21a121.pdf].