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 $|(v-u)f\left(u\right)+(w-v)f\left(w\right)-{\int }_{\gamma }f\left(z\right)dz|$  under the assumptions that $γ$ is a smooth path parametrized by $z\left(t\right),t\in [a,b],u=z\left(a\right),v=z\left(x\right)$ with $x\in (a,b)$ and $w=z\left(b\right)$ while $f$ is holomorphic in $G$, an open domain and $\gamma \in G$. 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.