complex mean-value theorem

Theorem [1] Suppose $\mathrm{\Omega }$ is an open convex set in $ℂ$, suppose $f$ is a holomorphic function $f:\mathrm{\Omega }\to ℂ$, and suppose $a,b$ are distinct points in $\mathrm{\Omega }$. Then there exist points $u,v$ on $L_{ab}$ (the straight line connecting $a$ and $b$ not containing the endpoints), such that

$\mathrm{\Re }\{{\displaystyle \frac{f(b)-f(a)}{b-a}}\}=\mathrm{\Re }\{f^{\prime }(u)\},$

$\mathrm{\Im }\{{\displaystyle \frac{f(b)-f(a)}{b-a}}\}=\mathrm{\Im }\{f^{\prime }(v)\},$

where $\mathrm{\Re }$ and $\mathrm{\Im }$ are the real (http://planetmath.org/RealPart) and imaginary parts of a complex number, respectively.

Title	complex mean-value theorem
Canonical name	ComplexMeanvalueTheorem
Date of creation	2013-03-22 13:49:02
Last modified on	2013-03-22 13:49:02
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	7
Author	matte (1858)
Entry type	Theorem
Classification	msc 26A06