complex mean-value theorem

Theorem [1]Suppose $\mathrm{\Omega }$ is an open convex set in $ℂ$,suppose $f$ is a holomorphic function $f:\mathrm{\Omega }\to ℂ$, andsuppose $a,b$ are distinct points in $\mathrm{\Omega }$.Then there exist points $u,v$ on $L_{ab}$ (the straight lineconnecting $a$ 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 ofa complex number, respectively.