A refinement of Grüss inequality for the complex integral

Dragomir, Silvestru Sever

doi:10.2478/gm-2020-0006

Abstract

Assume that f and g are continuous on γ, γ ⊂ 𝔺 is a piecewise smooth path parametrized by z (t), t ∈ [a, b] from z (a) = u to z (b) = w with w ≠ u and the complex Čebyšev functional is defined by

$𝒟_{γ} (f, g) : = \frac{1}{w - u} \int_{γ} f (z) g (z) d z - \frac{1}{w - u} \int_{γ} f (z) d z \frac{1}{w - u} \int_{γ} g (z) d z .$ {{\cal D}_\gamma}\left({f,g} \right): = {1 \over {w - u}}\int_\gamma {f\left(z \right)} g\left(z \right)dz - {1 \over {w - u}}\int_\gamma {f\left(z \right)} dz{1 \over {w - u}}\int_\gamma {g\left(z \right)} dz.

In this paper we establish some Grüss type inequalities for 𝒟 (f, g) under some complex boundedness conditions for the functions f and g.

References

[1] M. W. Alomari, A companion of Grüss type inequality for Riemann-Stieltjes integral and applications, Mat. Vesnik, vol. 66, no. 2, 2014, 202–212.
Search in Google Scholar Back to article
[2] D. Andrica, C. Badea, Grüss’ inequality for positive linear functionals, Period. Math. Hungar., vol. 19, no. 2, 1988, 155–167.10.1007/BF01848061
Search in Google Scholar Back to article
[3] D. Baleanu, S. D. Purohit, F. Uçar, On Grüss type integral inequality involving the Saigo’s fractional integral operators, J. Comput. Anal. Appl., vol. 19, no. 3, 2015, 480–489.
Search in Google Scholar Back to article
[4] P. L. Chebyshev, Sur les expressions approximatives des intègrals dèfinis par les outres prises entre les même limites, Proc. Math. Soc. Charkov, vol. 2, 1882, 93–98.
Search in Google Scholar Back to article
[5] P. Cerone, On a Čebyšev-type functional and Grüss-like bounds, Math. Inequal. Appl., vol. 9, no. 1, 2006, 87–102.10.7153/mia-09-09
Search in Google Scholar Back to article
[6] P. Cerone, S. S. Dragomir, A refinement of the Grüss inequality and applications, Tamkang J. Math., vol. 38, no. 1, 2007, 37–49. Preprint RGMIA Res. Rep. Coll., vol. 5, no. 2, 2002, Article 14.10.5556/j.tkjm.38.2007.92
Search in Google Scholar Back to article
[7] P. Cerone, S. S. Dragomir, Some new Ostrowski-type bounds for the Čebyšev functional and applications, J. Math. Inequal., vol. 8, no. 1, 2014, 159–170.10.7153/jmi-08-10
Search in Google Scholar Back to article
[8] P. Cerone, S. S. Dragomir, J. Roumeliotis, Grüss inequality in terms of -seminorms and applications, Integral Transforms Spec. Funct., vol. 14, no. 3, 2003, 205–216.10.1080/1065246031000074353
Search in Google Scholar Back to article
[9] X. L. Cheng, J. Sun, A note on the perturbed trapezoid inequality, J. Ineq. Pure and Appl. Math., vol. 3, no. 2, art. 29, 2002.
Search in Google Scholar Back to article
[10] S. S. Dragomir, A generalization of Grüss’s inequality in inner product spaces and applications, J. Math. Anal. Appl., vol. 237, no. 1, 1999, 74–82.10.1006/jmaa.1999.6452
Search in Google Scholar Back to article
[11] S. S. Dragomir, A Grüss’ type integral inequality for mappings of r-Hölder’s type and applications for trapezoid formula, Tamkang J. Math., vol. 31, no. 1, 2000, 43–47.10.5556/j.tkjm.31.2000.413
Search in Google Scholar Back to article
[12] S. S. Dragomir, Some integral inequalities of Grüss type, Indian J. Pure Appl. Math., vol. 31, no. 4, 2000, 397–415.
Search in Google Scholar Back to article
[13] S. S. Dragomir, Integral Grüss inequality for mappings with values in Hilbert spaces and applications, J. Korean Math. Soc., vol. 38, no. 6, 2001, 1261–1273.
Search in Google Scholar Back to article
[14] S. S. Dragomir, I. A. Fedotov, An inequality of Grüss’ type for Riemann-Stieltjes integral and applications for special means, Tamkang J. Math., vol. 29, no. 4, 1998, 287–292.10.5556/j.tkjm.29.1998.4257
Search in Google Scholar Back to article
[15] S. S. Dragomir, I. Gomm, Some integral and discrete versions of the Grüss inequality for real and complex functions and sequences, Tamsui Oxf. J. Math. Sci., vol. 19, no. 1, 2003, 67–77.
Search in Google Scholar Back to article
[16] A. M. Fink, A treatise on Grüss’ inequality, Analytic and Geometric Inequalities and Applications, 93–113, Math. Appl., 478, Kluwer Acad. Publ., Dordrecht, 1999.10.1007/978-94-011-4577-0_7
Search in Google Scholar Back to article
[17] G. Grüss, Über das Maximum des absoluten Betrages von1b-a∫abf(x)g(x)dx-1(b-a)2∫abf(x)dx∫abg(x)dx{1 \over {b - a}}\int_a^b {f\left(x \right)} g\left(x \right)dx - {1 \over {{{\left({b - a} \right)}^2}}}\int_a^b {f\left(x \right)} dx\int_a^b {g\left(x \right)} dx, Math. Z., vol. 39, 1935, 215–226.10.1007/BF01201355
Search in Google Scholar Back to article
[18] D. Jankov Maširević, T. K. Pogány, Bounds on Čebyšev functional for C’[0, 1] function class, J. Anal., vol. 22, 2014, 107–117.
Search in Google Scholar Back to article
[19] Z. Liu, Refinement of an inequality of Grüss type for Riemann-Stieltjes integral, Soochow J. Math., vol. 30, no. 4, 2004, 483–489.
Search in Google Scholar Back to article
[20] Z. Liu, Notes on a Grüss type inequality and its application, Vietnam J. Math., vol. 35, no. 2, 2007, 121–127.
Search in Google Scholar Back to article
[21] A. Lupaş, The best constant in an integral inequality, Mathematica (Cluj, Romania), 15, vol. 38, no. 2, 1973, 219–222.10.1137/1015010
Search in Google Scholar Back to article
[22] A. Mc. D. Mercer, P. R. Mercer, New proofs of the Grüss inequality, Aust. J. Math. Anal. Appl., vol. 1, no. 2, 2004, art. 12.
Search in Google Scholar Back to article
[23] N. Minculete, L. Ciurdariu, A generalized form of Grüss type inequality and other integral inequalities, J. Inequal. Appl., 2014, 2014:119, 18 pp.10.1186/1029-242X-2014-119
Search in Google Scholar Back to article
[24] A. M. Ostrowski, On an integral inequality, Aequat. Math., vol. 4, 1970, 358–373.10.1007/BF01844168
Search in Google Scholar Back to article
[25] B. G. Pachpatte, A note on some inequalities analogous to Grüss inequality, Octogon Math. Mag., vol. 5, no. 2, 1997, 62–66.
Search in Google Scholar Back to article
[26] J. Pečarić, Š. Ungar, On a inequality of Grüss type, Math. Commun., vol. 11, no. 2, 2006, 137–141.
Search in Google Scholar Back to article
[27] M. Z. Sarikaya, H. Budak, An inequality of Grüss like via variant of Pompeiu’s mean value theorem, Konuralp J. Math., vol. 3, no. 1, 2015, 29–35.
Search in Google Scholar Back to article
[28] N. Ujević, A generalization of the pre-Grüss inequality and applications to some quadrature formulae, J. Inequal. Pure Appl. Math., vol. 3, no. 1, 2002, art. 13, 9 pp.
Search in Google Scholar Back to article

A refinement of Grüss inequality for the complex integral

Abstract

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