An extension of trapezoid inequality to the complex integral

Sever Dragomır

doi:10.31801/cfsuasmas.798863

Research Article

Year 2021, Volume: 70 Issue: 2, 1113 - 1130, 31.12.2021

Sever Dragomır

https://doi.org/10.31801/cfsuasmas.798863

Abstract

References

Cerone, P., Dragomir, S. S., Trapezoidal-Type Rules From an Inequalities Point of View, Handbook of Analytic-Computational Methods in Applied Mathematics, G. Anastassiou (Ed.), CRC Press, NY, 2000, 65-134.
Cerone, P., Dragomir, S. S., Pearce, C. E. M., A generalised trapezoid inequality for functions of bounded variation, Turkish J. Math., 24(2) (2000), 147-163.
Dragomir, S. S., An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Ineq. Pure & Appl. Math., 3(2) (2002), Art. 31. https://www.emis.de/journals/JIPAM/article183.html?sid=183
Dragomir, S. S., An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Ineq. Pure & Appl. Math., 3(3) (2002), Art. 35. https://www.emis.de/journals/JIPAM/article187.html?sid=187
Dragomir, S. S., The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc., 60 (1999), 495–508.
Dragomir, S. S., Rassias, Th. M. (Eds.), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht, 2002.
Kechriniotis, A. I., Assimakis, N. D., Generalizations of the trapezoid inequalities based on a new mean value theorem for the remainder in Taylor’s formula, J. Inequal. Pure Appl. Math., 7(3) (2006), Article 90, 13 pp.
Liu, Z., Some inequalities of perturbed trapezoid type, J. Inequal. Pure Appl. Math., 7(2) (2006), Article 47, 9 pp.
Mercer, McD. A., On perturbed trapezoid inequalities, J. Inequal. Pure Appl. Math., 7(4) (2006), Article 118, 7 pp. (electronic).
Ostrowski, A., Uber die absolutabweichung einer differentiierbaren funktion von ihrem integralmittelwert, Comment. Math. Helv., 10 (1938), 226–227.
Ujevic, N., Error inequalities for a generalized trapezoid rule, Appl. Math. Lett. 19(1) (2006), 32–37.

An extension of trapezoid inequality to the complex integral

Year 2021, Volume: 70 Issue: 2, 1113 - 1130, 31.12.2021

Sever Dragomır

https://doi.org/10.31801/cfsuasmas.798863

Abstract

In this paper we extend the trapezoid inequality to the complex integral by providing upper bounds for the quantity $| (v - u) f (u) + (w - v) f (w) - \int_{γ} f (z) d z |$ under the assumptions that $γ$ is a smooth path parametrized by $z (t), t \in [a, b], u = z (a), v = z (x)$ with $x \in (a, b)$ and $w = z (b)$ while $f$ is holomorphic in $G$ , an open domain and $γ \in G$ . An application for circular paths is also given.

Keywords

Complex integral, continuous functions, holomorphic functions, trapezoid inequality

References

Cerone, P., Dragomir, S. S., Trapezoidal-Type Rules From an Inequalities Point of View, Handbook of Analytic-Computational Methods in Applied Mathematics, G. Anastassiou (Ed.), CRC Press, NY, 2000, 65-134.
Cerone, P., Dragomir, S. S., Pearce, C. E. M., A generalised trapezoid inequality for functions of bounded variation, Turkish J. Math., 24(2) (2000), 147-163.
Dragomir, S. S., An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Ineq. Pure & Appl. Math., 3(2) (2002), Art. 31. https://www.emis.de/journals/JIPAM/article183.html?sid=183
Dragomir, S. S., An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Ineq. Pure & Appl. Math., 3(3) (2002), Art. 35. https://www.emis.de/journals/JIPAM/article187.html?sid=187
Dragomir, S. S., The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc., 60 (1999), 495–508.
Dragomir, S. S., Rassias, Th. M. (Eds.), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht, 2002.
Kechriniotis, A. I., Assimakis, N. D., Generalizations of the trapezoid inequalities based on a new mean value theorem for the remainder in Taylor’s formula, J. Inequal. Pure Appl. Math., 7(3) (2006), Article 90, 13 pp.
Liu, Z., Some inequalities of perturbed trapezoid type, J. Inequal. Pure Appl. Math., 7(2) (2006), Article 47, 9 pp.
Mercer, McD. A., On perturbed trapezoid inequalities, J. Inequal. Pure Appl. Math., 7(4) (2006), Article 118, 7 pp. (electronic).
Ostrowski, A., Uber die absolutabweichung einer differentiierbaren funktion von ihrem integralmittelwert, Comment. Math. Helv., 10 (1938), 226–227.
Ujevic, N., Error inequalities for a generalized trapezoid rule, Appl. Math. Lett. 19(1) (2006), 32–37.

There are 11 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Research Articles
Authors	Sever Dragomır 0000-0003-2902-6805
Publication Date	December 31, 2021
Submission Date	September 23, 2020
Acceptance Date	April 17, 2021
Published in Issue	Year 2021 Volume: 70 Issue: 2

Cite

APA	Dragomır, S. (2021). An extension of trapezoid inequality to the complex integral. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(2), 1113-1130. https://doi.org/10.31801/cfsuasmas.798863
AMA	Dragomır S. An extension of trapezoid inequality to the complex integral. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. December 2021;70(2):1113-1130. doi:10.31801/cfsuasmas.798863
Chicago	Dragomır, Sever. “An Extension of Trapezoid Inequality to the Complex Integral”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70, no. 2 (December 2021): 1113-30. https://doi.org/10.31801/cfsuasmas.798863.
EndNote	Dragomır S (December 1, 2021) An extension of trapezoid inequality to the complex integral. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 2 1113–1130.
IEEE	S. Dragomır, “An extension of trapezoid inequality to the complex integral”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 70, no. 2, pp. 1113–1130, 2021, doi: 10.31801/cfsuasmas.798863.
ISNAD	Dragomır, Sever. “An Extension of Trapezoid Inequality to the Complex Integral”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70/2 (December 2021), 1113-1130. https://doi.org/10.31801/cfsuasmas.798863.
JAMA	Dragomır S. An extension of trapezoid inequality to the complex integral. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70:1113–1130.
MLA	Dragomır, Sever. “An Extension of Trapezoid Inequality to the Complex Integral”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 70, no. 2, 2021, pp. 1113-30, doi:10.31801/cfsuasmas.798863.
Vancouver	Dragomır S. An extension of trapezoid inequality to the complex integral. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70(2):1113-30.