An extension of trapezoid inequality to the complex integral

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

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