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On some Grüss’ type inequalities for the complex integral

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas volume 113, pages 3531–3543 (2019)Cite this article

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.

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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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Authors and Affiliations

  1. Mathematics, College of Engineering and Science, Victoria University, PO Box 14428, Melbourne, MC, 8001, Australia

    Silvestru Sever Dragomir

  2. DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg, 2050, South Africa

    Silvestru Sever Dragomir

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  1. Silvestru Sever Dragomir
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Correspondence to Silvestru Sever Dragomir.

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Dragomir, S.S. On some Grüss’ type inequalities for the complex integral. RACSAM 113, 3531–3543 (2019). https://doi.org/10.1007/s13398-019-00712-6

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  • DOI: https://doi.org/10.1007/s13398-019-00712-6

Keywords

  • Complex integral
  • Continuous functions
  • Holomorphic functions
  • Grüss inequality

Mathematics Subject Classification

  • 26D15
  • 26D10
  • 30A10
  • 30A86