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I have a question about the proof of Morera's theorem as presented in Conway's text volume I.

Let $G$ be a region and let $ f:G \to \mathbb{C} $ be a continuous function.
Fix $ z_o$ in $G$. Then for any $z \in G$, let $[z_o,z]$ denote a line segment from $ z_o$ to $z$. Then, $$\bigg| \frac{1}{z-z_o} \int_{[z_o,z]} [f(w)-f(z_o)] dw \bigg|\leq |f(z)-f(z_o)|$$

I want to ask how to establish that inequality, my attempt is this, I know that $$\bigg| \frac{1}{z-z_o} \int_{[z_o,z]} [f(w)-f(z_o)] dw \bigg|\leq \frac{V([z_o,z])}{|z-z_o|}~ \text{sup}\{ ~|f(w)-f(z_o)| : w \in [z_o,z] ~\} $$ where $ V([z_o,z]) $ is the length of $ [z_o,z]$.
So, $$\bigg| \frac{1}{z-z_o} \int_{[z_o,z]} [f(w)-f(z_o)] dw \bigg| \leq \text{sup}\{ ~|f(w)-f(z_o)| : w \in [z_o,z] ~\} $$

I can't show that $$\text{sup}\{ ~|f(w)-f(z_o)| : w \in [z_o,z] ~\} \leq |f(z)-f(z_o)| $$ just by the continuity of $f$.
I feel that I'm just missing some simple detail to show this. thank you.

Khoa ta
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    Are you sure that you quoted the estimate from Conway's book correctly? What you need (in order to prove Morera) is to show that the left-hand side converges to zero for $ z \to z_0$, and that follows from your estimate (with the supremum on the right-hand side) and the continuity of $f$. – Martin R Jan 30 '16 at 09:21
  • @MartinR thank you for the response. Yes, I check and I did quote it correctly. that's why it keeps bugging me about that. I know we can bound sup{...} to complete the proof. But I was puzzle about that inequality. Do you think this might be a typo?? I thought that before but I just want to post it on here to see if I miss anything. – Khoa ta Jan 30 '16 at 22:24
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    I don't see how that inequality can be obtained, but perhaps I am overlooking something. I don't have Conway's book to check it there. – Martin R Jan 30 '16 at 23:33
  • Let $z_0=0,,z=1$ and $f(w)=w(1-w)$. Then\begin{align} &\left|\frac{1}{z-z_o} \int_{[z_o,z]} [f(w)-f(z_o)] dw\right| =\int_0^1 x(1-x) dx=\frac{1}{6},\ &|f(z)-f(z_o)|=0. \end{align} The inequality is not correct. – ts375_zk26 Jan 30 '16 at 23:40
  • @ts375_zk26 thank you for your response, it is clear now. – Khoa ta Jan 31 '16 at 00:26
  • I got super confused on his proof for hours also. So, in short, his proof was incorrect, and the usual way is to let $z$ approach the fixed point? – MonkeyKing Aug 03 '18 at 04:19
  • Having the same confusion as above. So, what was the conclusion? The final argument of his proof was incorrect? Or was his entire proof invalid? – math-physicist Aug 03 '22 at 02:35

1 Answers1

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He meant to use the Cauchy ML-inequality and then take the limit I believe.

$$\left| \dfrac{1}{z-z_0} \int_{[z_0,z]} [f(w)-f(z_0)]\ dw \right|\leq \max_{z \in [z,z_0]} |f(z)-f(z_0)| $$

and then take the $\lim$ as $z$ approaches $z_0$ to get the result.

math-physicist
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