3

Let $f\in H(\Omega)$ for some open set $\Omega\subset\mathbb{C}$. Suppose $z\in\Omega$ and prove that there exists two distinct complex numbers $s,t\in\Omega$ such that $$f'(z)=\frac{f(s)-f(t)}{s-t}.$$

I'm not really sure how to prove this, though I suspect the formula $$f(s)-f(t)=\int_{[s,t]}f'(z)\;dz$$ will be useful, perhaps along with Cauchy's formula. I'm going to keep working towards a solution and post if it I come up with one, but any help is appreciated.

Sargera
  • 4,184
  • 21
  • 36
  • Just as with the mean value theorem, you might start by reducing it to the case where $f'(z)=0$. At which point it essentially becomes a duplicate of this question. – Micah Nov 18 '13 at 05:13

1 Answers1

2

Note that $g: w \mapsto f(w) - f'(z) w$ is holomorphic and $g'(z) = 0$. What does that imply in a neighborhood of $z$?

WimC
  • 32,192
  • 2
  • 48
  • 88
  • It implies the existence of some neighborhood $V$ of $z$ such that $g$ fails to be one to one and this implies the existence of distinct $s,t$ such that $g(s)=g(t)$, which is the desired conclusion. I'm thoroughly amused by the fact that all three questions I posted had solutions which depended on some suitably chosen auxiliary function and an application of some major theorem. – Sargera Nov 18 '13 at 21:45