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(u)+(w−v)f(w)−∫γf(z)dz||(v−u)f(u)+(w−v)f(w)−∫γf(z)dz| under the assumptions that $γ$ is a smooth path parametrized by z(t),t∈[a,b],u=z(a),v=z(x)z(t),t∈[a,b],u=z(a),v=z(x) with x∈(a,b)x∈(a,b) and w=z(b)w=z(b) while ff is holomorphic in GG, an open domain and γ∈Gγ∈G. An application for circular paths is also given.