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 ) − ∫ γ f ( z ) d z | |(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 f f is holomorphic in G G , an open domain and γ ∈ G γ∈G . An application for circular paths is also given.

Highlights

  • Inequalities providing upper bounds for the quantity b (t − a) f (a) + (b − t) f (b) − f (s) ds, t ∈ [a, b] (1)a are known in the literature as generalized trapezoid inequalities and it has been shown in [2] that b (t − a) f (a) + (b − t) f (b) − f (s) ds (2) a ≤ 1 + t − a+b 22 b−a b (b − a) (f )2020 Mathematics Subject Classification. 26D15, 26D10, 30A10, 30A86

  • 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) − f (z) dz γ under the assumptions that γ is a smooth path parametrized by z (t), t ∈ [a, b], u = z (a), v = z (x) with x ∈ (a, b) and w = z (b) while f is holomorphic in G, an open domain and γ ⊂ G

  • A are known in the literature as generalized trapezoid inequalities and it has been shown in [2] that b (t − a) f (a) + (b − t) f (b) − f (s) ds

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Summary

Introduction

Inequalities providing upper bounds for the quantity b (t − a) f (a) + (b − t) f (b) − f (s) ds , t ∈ [a, b]. Suppose γ is a smooth path parametrized by z (t) , t ∈ [a, b] and f is a complex function which is continuous on γ. We observe that that the actual choice of parametrization of γ does not matter This definition immediately extends to paths that are piecewise smooth. Suppose γ is parametrized by z (t), t ∈ [a, b], which is differentiable on the intervals [a, c] and [c, b], assuming that f is continuous on γ we define f (z) dz := f (z) dz + f (z) dz γ u,w γ u,v γ v,w where v := z (c) This can be extended for a finite number of intervals. In this paper we extend the trapezoid inequality to the complex integral, by providing upper bounds for the quantity

Trapezoid Type Inequalities
Examples For Circular Paths
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