Theorem [1] Suppose $\Omega$ is an open convex set in $\sC$ , suppose $f$ is a holomorphic function $f:\Omega\to \sC$ , 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 \begin{eqnarray*} \Re\{ \frac{f(b)-f(a)}{b-a} \} = \Re\{ f'(u) \}, \\ \Im\{ \frac{f(b)-f(a)}{b-a} \} = \Im\{ f'(v) \}, \end{eqnarray*}where $\Re$ and $\Im$ are the real and imaginary parts of a complex number, respectively.

References

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J.-Cl. Evard, F. Jafari, A Complex Rolle's Theorem, American Mathematical Monthly, Vol. 99, Issue 9, (Nov. 1992), pp. 858-861.