Research and Innovation Articles
Published 2019-07-29
Keywords
- Complex integral,
- Continuous functions,
- Holomorphic functions,
- Chebyshev inequality
How to Cite
Dragomir, S. S. (2019). On some Chebyshev type inequalities for the complex integral. Revista Integración, Temas De matemáticas, 37(2), 307–317. https://doi.org/10.18273/revint.v37n2-2019006
Abstract
Assume that f and g are continuous on γ, γ ⊂ C is a piecewise
smooth path parametrized by z (t) , t ∈ [a, b] from z (a) = u to z (b) = w with
w 6= u, and the complex Chebyshev functional is defined by
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References
[1] Alomari M.W., “A companion of Grüss type inequality for Riemann-Stieltjes integral and applications”, Mat. Vesnik 66 (2014), No. 2, 202–212.
[2] Andrica D. and Badea C., “Grüss’ inequality for positive linear functionals”, Period. Math. Hungar. 19 (1988), No. 2, 155–167.
[3] Baleanu D., Purohit S.D. and Uçar F., “On Grüss type integral inequality involving the Saigo’s fractional integral operators”, J. Comput. Anal. Appl. 19 (2015), No. 3, 480–489.
[4] Cerone P., “On a Čebyšev-type functional and Grüss-like bounds”, Math. Inequal. Appl. 9 (2006), No. 1, 87–102.
[5] Cerone P. and Dragomir S.S., “A refinement of the Grüss inequality and applications”, Tamkang J. Math., 38 (2007), No. 1, 37–49.
[6] Cerone P. and Dragomir S.S., “Some new Ostrowski-type bounds for the Čebyšev functional and applications”, J. Math. Inequal. 8 (2014), No. 1, 159–170.
[7] Cerone P., Dragomir S.S. and Roumeliotis J., “Grüss inequality in terms of -seminorms and applications”, Integral Transforms Spec. Funct. 14 (2003), No. 3, 205–216.
[8] Chebyshev P.L., “Sur les expressions approximatives des intègrals dèfinis par les outres prises entre les même limites”, Proc. Math. Soc. Charkov, 2 (1882), 93-98.
[9] Cheng X.L. and Sun J., “A note on the perturbed trapezoid inequality”, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), No. 2, Art. 29, 7 pp.
[10] Dragomir S.S., “A generalization of Grüss’s inequality in inner product spaces and applications”, J. Math. Anal. Appl. 237 (1999), No. 1, 74–82.
[11] Dragomir S.S., “A Grüss’ type integral inequality for mappings of r-Hölder’s type and applications for trapezoid formula”, Tamkang J. Math. 31 (2000), No. 1, 43–47.
[12] Dragomir S.S., “Some integral inequalities of Grüss type”, Indian J. Pure Appl. Math. 31 (2000), No. 4, 397–415.
[13] Dragomir S.S., “Integral Grüss inequality for mappings with values in Hilbert spaces and applications”, J. Korean Math. Soc. 38 (2001), No. 6, 1261–1273.
[14] Dragomir S.S. and Fedotov I.A., “An inequality of Grüss’ type for Riemann-Stieltjes integral and applications for special means”, Tamkang J. Math. 29 (1998), No. 4, 287–292.
[15] Dragomir S.S. and Gomm I., “Some integral and discrete versions of the Grüss inequality for real and complex functions and sequences”, Tamsui Oxf. J. Math. Sci. 19 (2003), No. 1, 67–77.
[16] Fink A.M., “A treatise on Grüss’ inequality”, in Analytic and Geometric Inequalities and Applications (eds. RassiasHari T. and Srivastava M.), Math. Appl. 478, Kluwer Acad. Publ. (1999), 93–113.
17] Grüss G., “Über das Maximum des absoluten Betrages von 1/b−a Integral de a-b de f(x)g(x)dx −1/(b−a)2 Integral de a-b de f(x)dx por integral de a-b de g(x)dx", Math. Z. 39 (1935), No. 1, 215–226.
[18] Jankov Maširević D. and Pogány T.K., “Bounds on Čebyšev functional for C'[0, 1] function class”, J. Anal. 22 (2014), 107–117.
[19] Liu Z., “Refinement of an inequality of Grüss type for Riemann-Stieltjes integral”, Soochow J. Math. 30 (2004), No. 4, 483–489.
[20] Liu Z., “Notes on a Grüss type inequality and its application”, Vietnam J. Math. 35 (2007), No. 2, 121–127.
[21] Lupas A., “The best constant in an integral inequality”, Mathematica (Cluj, Romania), 15 (38) (1973), 219–222.
[22] Mercer A.Mc.D. and Mercer P.R., “New proofs of the Grüss inequality”,Aust. J. Math. Anal. Appl. 1 (2004), No. 2, Art. 12, 6 pp.
[23] Minculete N. and Ciurdariu L., “A generalized form of Grüss type inequality and other integral inequalities”, J. Inequal. Appl. 2014, 2014:119, 18 pp.
[24] Ostrowski A.M., “On an integral inequality”, Aequationes Math. 4 (1970), No. 3, 358-373.
[25] Pachpatte B.G., “A note on some inequalities analogous to Grüss inequality”, Octogon Math. Mag. 5 (1997), No. 2, 62–66.
[26] Pečarić J. and Ungar Š., “On a inequality of Grüss type”, Math. Commun. 11 (2006), No. 2, 137–141.
[27] Sarikaya M. Z. and Budak H., “An inequality of Grüss like via variant of Pompeiu’s mean value theorem”, Konuralp J. Math. 3 (2015), No. 1, 29–35.
[28] Ujević N., “A generalization of the pre-Grüss inequality and applications to some quadrature formulae”, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), No. 1, Art. 13, 9 pp.
[2] Andrica D. and Badea C., “Grüss’ inequality for positive linear functionals”, Period. Math. Hungar. 19 (1988), No. 2, 155–167.
[3] Baleanu D., Purohit S.D. and Uçar F., “On Grüss type integral inequality involving the Saigo’s fractional integral operators”, J. Comput. Anal. Appl. 19 (2015), No. 3, 480–489.
[4] Cerone P., “On a Čebyšev-type functional and Grüss-like bounds”, Math. Inequal. Appl. 9 (2006), No. 1, 87–102.
[5] Cerone P. and Dragomir S.S., “A refinement of the Grüss inequality and applications”, Tamkang J. Math., 38 (2007), No. 1, 37–49.
[6] Cerone P. and Dragomir S.S., “Some new Ostrowski-type bounds for the Čebyšev functional and applications”, J. Math. Inequal. 8 (2014), No. 1, 159–170.
[7] Cerone P., Dragomir S.S. and Roumeliotis J., “Grüss inequality in terms of -seminorms and applications”, Integral Transforms Spec. Funct. 14 (2003), No. 3, 205–216.
[8] Chebyshev P.L., “Sur les expressions approximatives des intègrals dèfinis par les outres prises entre les même limites”, Proc. Math. Soc. Charkov, 2 (1882), 93-98.
[9] Cheng X.L. and Sun J., “A note on the perturbed trapezoid inequality”, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), No. 2, Art. 29, 7 pp.
[10] Dragomir S.S., “A generalization of Grüss’s inequality in inner product spaces and applications”, J. Math. Anal. Appl. 237 (1999), No. 1, 74–82.
[11] Dragomir S.S., “A Grüss’ type integral inequality for mappings of r-Hölder’s type and applications for trapezoid formula”, Tamkang J. Math. 31 (2000), No. 1, 43–47.
[12] Dragomir S.S., “Some integral inequalities of Grüss type”, Indian J. Pure Appl. Math. 31 (2000), No. 4, 397–415.
[13] Dragomir S.S., “Integral Grüss inequality for mappings with values in Hilbert spaces and applications”, J. Korean Math. Soc. 38 (2001), No. 6, 1261–1273.
[14] Dragomir S.S. and Fedotov I.A., “An inequality of Grüss’ type for Riemann-Stieltjes integral and applications for special means”, Tamkang J. Math. 29 (1998), No. 4, 287–292.
[15] Dragomir S.S. and Gomm I., “Some integral and discrete versions of the Grüss inequality for real and complex functions and sequences”, Tamsui Oxf. J. Math. Sci. 19 (2003), No. 1, 67–77.
[16] Fink A.M., “A treatise on Grüss’ inequality”, in Analytic and Geometric Inequalities and Applications (eds. RassiasHari T. and Srivastava M.), Math. Appl. 478, Kluwer Acad. Publ. (1999), 93–113.
17] Grüss G., “Über das Maximum des absoluten Betrages von 1/b−a Integral de a-b de f(x)g(x)dx −1/(b−a)2 Integral de a-b de f(x)dx por integral de a-b de g(x)dx", Math. Z. 39 (1935), No. 1, 215–226.
[18] Jankov Maširević D. and Pogány T.K., “Bounds on Čebyšev functional for C'[0, 1] function class”, J. Anal. 22 (2014), 107–117.
[19] Liu Z., “Refinement of an inequality of Grüss type for Riemann-Stieltjes integral”, Soochow J. Math. 30 (2004), No. 4, 483–489.
[20] Liu Z., “Notes on a Grüss type inequality and its application”, Vietnam J. Math. 35 (2007), No. 2, 121–127.
[21] Lupas A., “The best constant in an integral inequality”, Mathematica (Cluj, Romania), 15 (38) (1973), 219–222.
[22] Mercer A.Mc.D. and Mercer P.R., “New proofs of the Grüss inequality”,Aust. J. Math. Anal. Appl. 1 (2004), No. 2, Art. 12, 6 pp.
[23] Minculete N. and Ciurdariu L., “A generalized form of Grüss type inequality and other integral inequalities”, J. Inequal. Appl. 2014, 2014:119, 18 pp.
[24] Ostrowski A.M., “On an integral inequality”, Aequationes Math. 4 (1970), No. 3, 358-373.
[25] Pachpatte B.G., “A note on some inequalities analogous to Grüss inequality”, Octogon Math. Mag. 5 (1997), No. 2, 62–66.
[26] Pečarić J. and Ungar Š., “On a inequality of Grüss type”, Math. Commun. 11 (2006), No. 2, 137–141.
[27] Sarikaya M. Z. and Budak H., “An inequality of Grüss like via variant of Pompeiu’s mean value theorem”, Konuralp J. Math. 3 (2015), No. 1, 29–35.
[28] Ujević N., “A generalization of the pre-Grüss inequality and applications to some quadrature formulae”, JIPAM. J. Inequal. Pure Appl. Math. 3 (2002), No. 1, Art. 13, 9 pp.