On some Grüss’ type inequalities for the complex integral

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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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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Keywords

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

Mathematics Subject Classification

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