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As seen from Figure 1, Flett’s Mean Value Theorem claims that the curve seen the picture, $y = f(x)$ has a tangent at the two points $[a, b]$ and if the points at $(a,f(a))$ and $(b,f(b))$ are parallel then that means that the intermediate point, $n$ must have a tangent that passes through point $(a,f(a))$. [3 - Flett’s Mean Value Theorem]
Although this theorem works on the real plane, the complex plane, however, does not prove to do so. Given the function, $f(z) = ez − z
$, one can see that with Flett’s Mean Value Theorem, there is no solution. This provides an extension for holomorphic functions.
begin{theorem}
(Flett’s Mean Value Theorem for Holomorphic Functions) If $f : [a,b] rightarrow R$ is differentiable, then there exists $n subset (a,b)$ such that
$$f(n) - f(a) = (n - a)f’(n) - dfrac{1}{2} dfrac{f’(b) - f’(a)}{b-a}(n-a)^2$$
end{theorem}
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