## 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 *complex* *Čebyšev functional* is defined by

In this paper we establish some bounds for the magnitude of the functional \(\mathcal {D}_{\gamma }\left( f,g\right)\) and a related version of this under various assumptions for the functions *f* and *g* and provide some examples for circular paths.

This is a preview of subscription content, access via your institution.

## References

Cerone, P., and S.S. Dragomir. 2000. Three point identities and inequalities for n-time differentiable functions.

*SUT Journal of Mathematics*36 (2): 351–383.Cerone, P., and S.S. Dragomir. 2003. Three-point inequalities from Riemann–Stieltjes integrals.

*Inequality Theory and Applications*3: 57–83. (Nova Sci. Publ., Hauppauge, NY).Dragomir, S.S. 2016. Trapezoid type inequalities for complex functions defined on the unit circle with applications for unitary operators in Hilbert spaces.

*Georgian Mathematical Journal*23 (2): 199–210.Dragomir, S.S. 2015. Generalised trapezoid-type inequalities for complex functions defined on unit circle with applications for unitary operators in Hilbert spaces.

*Mediterranean Journal of Mathematics*12 (3): 573–591.Dragomir, S.S. 2015. Ostrowski’s type inequalities for complex functions defined on unit circle with applications for unitary operators in Hilbert spaces.

*Archivum Mathematicum (Brno)*51 (4): 233–254.Dragomir, S.S. 2015. Grüss type inequalities for complex functions defined on unit circle with applications for unitary operators in Hilbert spaces.

*Revista Colombiana de Matematicas*49 (1): 77–94.Dragomir, S.S. 2016. Quasi Grüss type inequalities for complex functions defined on unit circle with applications for unitary operators in Hilbert spaces.

*Extracta Mathematicae*31 (1): 47–67.Dragomir, S.S. 2018. An extension of Ostrowski’s inequality to the complex integral. Preprint RGMIA Res. Rep. Coll., vol. 21, Art. 112, 17 pp. [Online]. https://rgmia.org/papers/v21/v21a112.pdf.

Hanna, G., P. Cerone, and J. Roumeliotis. 2000. An Ostrowski type inequality in two dimensions using the three point rule. Proceedings of the 1999 International Conference on Computational Techniques and Applications (Canberra).

*ANZIAM Journal*42 (C): C671–C689.Klaričić Bakula, M., J. Pečarić, and Ribičić Penava, M., Vukelić, A. 2015. Some Grüss type inequalities and corrected three-point quadrature formulae of Euler type.

*Journal of Inequalities and Applications*2015: 76.Liu, Z. 2007. A note on perturbed three point inequalities.

*SUT Journal of Mathematics*43 (1): 23–34.Liu, W. 2017. A unified generalization of perturbed mid-point and trapezoid inequalities and asymptotic expressions for its error term.

*Analele Stiintifice ale Universitatii Al I Cuza din Iasi Matematica*63 (1): 65–78.Liu, W., and J. Park. 2017. Some perturbed versions of the generalized trapezoid inequality for functions of bounded variation.

*Journal of Computational Analysis and Applications*22 (1): 11–18.Pečarić, Josip, and M. Ribičić Penava. 2012. Sharp integral inequalities based on general three-point formula via a generalization of Montgomery identity.

*Analele Universitatii din Craiova Seria Matematica Informatica*39 (2): 132–147.Tseng, K.L., and S.R. Hwang. 2016. Some extended trapezoid-type inequalities and applications.

*Hacettepe Journal of Mathematics and Statistics*45 (3): 827–850.

## Author information

### Authors and Affiliations

### Corresponding author

## Ethics declarations

### Conflict of interest

Author declares that he has no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants performed by the author.

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

## About this article

### Cite this article

Dragomir, S.S. Several Grüss’ type inequalities for the complex integral.
*J Anal* **29**, 337–351 (2021). https://doi.org/10.1007/s41478-020-00268-4

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s41478-020-00268-4