# Maximum modulus theorem variant

Let $$f : \mathbb{C} \rightarrow \mathbb{R}$$ is analytic inside and on a simple closed curve $$C$$, then the maximum value of $$f(z)$$ occurs over $$C$$?

It's known the Maximum modulus theorem but in this case you take only the function $$f$$, is it true or not? I can´t prove it.

• If the codomain of your function is indeed $\mathbb{R}$ and analytic means complex-analytic, then $f$ is necessarily constant. If you mean to talk about an analytic function $f\colon\mathbb{C}\rightarrow\mathbb{C}$ and the maximum of its modulus $|f|$, then I don't see how the statement is different from the Maximum modulus principle. – Thorgott Dec 16 '19 at 17:01
• @Thorgott How it's different from MMT: Well of course it is a version of MMT. But getting this from, say. "an analytic function cannot have a local maximum" requires some highly non-trivial topology. (If you don't believe that, tell me, what's the definition of "on and inside $C$"?) – David C. Ullrich Dec 16 '19 at 17:27
• What is the meaning of : $f$ have maximum value? . I think that is the following definition. $f$ have maximum value in $z_0$ if $\vert f(z) \vert \leq \vert f(z_0) \vert$ for all $z$ – Juan Daniel Valdivia Fuentes Dec 18 '19 at 1:51

It's true (after fixing the typo; as already noted, the $$f:\Bbb C\to\Bbb R$$ should be $$f:\Bbb C\to\Bbb C$$. Proving it is non-trivial, because first you need a definition of "inside $$C$$".
This is exactly why you're more likely to find language like "on and inside $$C$$" in elementary books like say Brown&Churchill; they're willing to just wave their hands. In more rigorous/"advanced" texts these days they tend to avoid the notion of the inside of a curve... (what needs to be proved after the definitions are in place is that if $$V$$ is the bounded open set bounded by $$C$$ then $$C$$ is in fact the topological boundary of $$V$$, or equivalently $$\overline V = V\cup C$$. Once you know that, the version of MMT you know gives the current version.)