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On some Grüss’ type inequalities for the complex integral

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

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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References

  1. Alomari, M.W.: A companion of Grüss type inequality for Riemann-Stieltjes integral and applications. Mat. Vesnik 66(2), 202–212 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Andrica, D., Badea, C.: Grüss’ inequality for positive linear functionals. Period. Math. Hungar. 19(2), 155–167 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baleanu, D., Purohit, S.D., Uçar, F.: On Grüss type integral inequality involving the Saigo’s fractional integral operators. J. Comput. Anal. Appl. 19(3), 480–489 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Beck M., Marchesi G., Pixton, D., Sabalka, L.: A First Course in Complex Analysis. Orthogonal Publishing L3C, Michigan (2002). http://math.sfsu.edu/beck/complex.html

  5. Chebyshev, P.L.: Sur les expressions approximatives des intè grals dèfinis par les outres prises entre les même limites. Proc. Math. Soc. Charkov 2, 93–98 (1882)

    Google Scholar 

  6. Cerone, P.: On a Čebyšev-type functional and Grüss-like bounds. Math. Inequal. Appl. 9(1), 87–102 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Cerone, P., Dragomir, S.S.: A refinement of the Grüss inequality and applications, Tamkang J. Math., 38(1), 37–49 (2007). Preprint RGMIA Res. Rep. Coll., 5(2) (2002), Article 14. http://rgmia.org/papers/v5n2/RGIApp.pdf

  8. Cerone, P., Dragomir, S.S.: Some new Ostrowski-type bounds for the Čebyšev functional and applications. J. Math. Inequal. 8(1), 159–170 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cerone, P., Dragomir, S.S., Roumeliotis, J.: Grüss inequality in terms of \(\Delta \)-seminorms and applications. Integral Transforms Spec. Funct. 14(3), 205–216 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cheng, X.L., Sun, J.: A note on the perturbed trapezoid inequality. J. Inequal. Pure Appl. Math. 3(2), 29 (2002)

    MathSciNet  MATH  Google Scholar 

  11. Dragomir, S.S.: A generalization of Grüss’s inequality in inner product spaces and applications. J. Math. Anal. Appl. 237(1), 74–82 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dragomir, S.S.: A Grüss’ type integral inequality for mappings of \(r\)-Hölder’s type and applications for trapezoid formula. Tamkang J. Math. 31(1), 43–47 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Dragomir, S.S.: Some integral inequalities of Grüss type. Indian J. Pure Appl. Math. 31(4), 397–415 (2000)

    MathSciNet  MATH  Google Scholar 

  14. Dragomir, S.S.: Integral Grüss inequality for mappings with values in Hilbert spaces and applications. J. Korean Math. Soc. 38(6), 1261–1273 (2001)

    MathSciNet  MATH  Google Scholar 

  15. Dragomir, S.S., Fedotov, I.A.: An inequality of Grüss’ type for Riemann-Stieltjes integral and applications for special means. Tamkang J. Math. 29(4), 287–292 (1998)

    MathSciNet  MATH  Google Scholar 

  16. Dragomir, S.S., Gomm, I.: Some integral and discrete versions of the Grüss inequality for real and complex functions and sequences. Tamsui Oxf. J. Math. Sci. 19(1), 67–77 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Fink, A.M.: A Treatise on Grüss’ Inequality Analytic and Geometric Inequalities and Applications, Math Appl, vol. 478, pp. 93–113. Kluwer Academic Publisher, Dordrecht (1999)

    Google Scholar 

  18. Grüss, G.: Über das Maximum des absoluten Betrages von \( \frac{1}{b-a}\int _{a}^{b}f(x)g(x)dx-\frac{1}{\left( b-a\right) ^{2}} \int _{a}^{b}f(x)dx\int _{a}^{b}g(x)dx\). Math. Z. 39, 215–226 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jankov Maširević, D., Pogány, T.K.: Bounds on Čebyšev functional for \(C_{\varphi }[0,1]\) function class. J. Anal. 22, 107–117 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Liu, Z.: Refinement of an inequality of Grüss type for Riemann-Stieltjes integral. Soochow J. Math. 30(4), 483–489 (2004)

    MathSciNet  MATH  Google Scholar 

  21. Liu, Z.: Notes on a Grüss type inequality and its application. Vietnam J. Math. 35(2), 121–127 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Lupaş, A.: The best constant in an integral inequality. Mathematica (Cluj, Romania) 15(38)(2), 219–222 (1973)

    MathSciNet  MATH  Google Scholar 

  23. Mercer, A.McD, Mercer, P.R.: New proofs of the Grüss inequality. Aust. J. Math. Anal. Appl 1(2), 6 (2004). Art. 12

    MATH  Google Scholar 

  24. Minculete, N., Ciurdariu, L.: A generalized form of Grüss type inequality and other integral inequalities. J. Inequal. Appl 2014(119), 18 (2014)

    MATH  Google Scholar 

  25. Pachpatte, B.G.: A note on some inequalities analogous to Grü ss inequality. Octogon Math. Mag. 5(2), 62–66 (1997)

    MathSciNet  Google Scholar 

  26. Pečarić, J., Ungar, Š.: On a inequality of Grüss type. Math. Commun. 11(2), 137–141 (2006)

    MathSciNet  MATH  Google Scholar 

  27. Sarikaya, M.Z., Budak, H.: An inequality of Grüss like via variant of Pompeiu’s mean value theorem. Konuralp J. Math. 3(1), 29–35 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Ujević, N.: A generalization of the pre-Grüss inequality and applications to some quadrature formulae. J. Inequal. Pure Appl. Math. 3(1), 9 (2002). Article 13

    MATH  Google Scholar 

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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

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