An extension of trapezoid inequality to the complex integral

Authors : Sever DRAGOMIR

Pages : 1113-1130

Doi:10.31801/cfsuasmas.798863

View : 18 | Download : 18

Publication Date : 2021-12-31

Article Type : Research Paper

Abstract :In this paper we extend the trapezoid inequality to the complex integral by providing upper bounds for the quantity | insert ignore into journalissuearticles values( v − u ); f insert ignore into journalissuearticles values( u ); + insert ignore into journalissuearticles values( w − v ); f insert ignore into journalissuearticles values( w ); − ∫ γ f insert ignore into journalissuearticles values( z ); d z | |insert ignore into journalissuearticles values(v−u);finsert ignore into journalissuearticles values(u);+insert ignore into journalissuearticles values(w−v);finsert ignore into journalissuearticles values(w);−∫γfinsert ignore into journalissuearticles values(z);dz|   under the assumptions that $γ$ is a smooth path parametrized by z insert ignore into journalissuearticles values( t ); , t ∈ [ a , b ] , u = z insert ignore into journalissuearticles values( a ); , v = z insert ignore into journalissuearticles values( x ); zinsert ignore into journalissuearticles values(t);,t∈[a,b],u=zinsert ignore into journalissuearticles values(a);,v=zinsert ignore into journalissuearticles values(x); with x ∈ insert ignore into journalissuearticles values( a , b ); x∈insert ignore into journalissuearticles values(a,b); and w = z insert ignore into journalissuearticles values( b ); w=zinsert ignore into journalissuearticles values(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. 
Keywords : Complex integral, continuous functions, holomorphic functions, trapezoid inequality