complex mean-value theorem

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

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

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

References

• 1 J.-Cl. Evard, F. Jafari, A Complex Rolle’s Theorem, American Mathematical Monthly, Vol. 99, Issue 9, (Nov. 1992), pp. 858-861.
Title complex mean-value theorem ComplexMeanvalueTheorem 2013-03-22 13:49:02 2013-03-22 13:49:02 matte (1858) matte (1858) 7 matte (1858) Theorem msc 26A06