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On some Grüss’ type inequalities for the complex integral

  • Silvestru Sever DragomirEmail author
Original Paper

Abstract

Assume that f and g are continuous on \(\gamma \), \(\gamma \subset {\mathbb {C}}\) is a piecewise smooth path parametrized by \(z\left( t\right) ,\)\(t\in \left[ a,b\right] \) from \(z\left( a\right) =u\) to \(z\left( b\right) =w\) with \(w\ne u\) and the complexČebyšev functional is defined by
$$\begin{aligned} {\mathcal {D}}_{\gamma }\left( f,g\right) :=\frac{1}{w-u}\int _{\gamma }f\left( z\right) g\left( z\right) dz-\frac{1}{w-u}\int _{\gamma }f\left( z\right) dz \frac{1}{w-u}\int _{\gamma }g\left( z\right) dz. \end{aligned}$$
In this paper we establish some bounds for the magnitude of the functional \( {\mathcal {D}}_{\gamma }\left( f,g\right) \) under various assumptions for the functions f and g and provide a complex version for the well known Grüss inequality.

Keywords

Complex integral Continuous functions Holomorphic functions Grüss inequality 

Mathematics Subject Classification

26D15 26D10 30A10 30A86 

Notes

Acknowledgements

The author would like to thank the anonymous referee for valuable suggestions that have been implemented in the final version of the paper.

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Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Mathematics, College of Engineering and ScienceVictoria UniversityMelbourneAustralia
  2. 2.DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science and Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

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