# maximum modulus principle

maximum modulus principle:

If a function $$f$$ is analytic and not constant in a given domain $$D$$, then $$|f(z)|$$ has no maximum value in $$D$$. That is, there is no point $$z_0$$ in the domain such that $$|f(z)|≤|f(z_0)|$$ for all points $$z$$ in it.

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Corollary:

Suppose that a function $$f$$ is continuous on a closed bounded region $$R$$ and that it is analytic and not constant in the interior of $$R$$. Then the maximum value of $$|f(z)|$$ in $$R$$, which is always reached, occurs somewhere on the boundary of $$R$$ and never in the interior.

Now let $$f$$ be an analytic function and $$z_0\in D$$. Let $$C_1=\{z: |z|=r_1\}$$ be a circle that passes through $$z_0$$ and assume $$|f(z)|\leq|f(z_0)|$$ for all $$z\in D_1=\{z: |z|\leq r_1\}$$.

Let $$C_2=\{z: |z|=r_2\}$$ be a circle that passes through $$z_0$$ and assume $$|f(z)|\leq|f(z_0)|$$ for all $$z\in D_2=\{z: |z|\leq r_2\}$$.

Constructing these disks which cover the domain $$D$$, we have that for any $$z\in D_1\bigcup\cdots\bigcup D_k$$, we have $$|f(z)|\leq|f(z_0)|$$, so $$z_0$$ is a point in domain $$D$$ that the maximum of $$|f|$$ in $$D$$ occur there. this is contradiction to maximum modulus principle.

you may see the picture here:

http://www.freeimagehosting.net/32879

I think somewhere in this constructing is wrong, but I don't know where.

Could anyone help to find the mistake.

thanks.

• Can you copy/paste the question in order to see the question when we type comments? Dec 14, 2011 at 18:27
• I copied the source over. Feel free to edit, though, asd. Dec 14, 2011 at 18:33
• It kind of seems like you're just assuming that $|f(z_0)|$ is as much of a maximum as you need it to be at every step of the construction. Can you do this for a particular $f$ and $D$? Otherwise it's hard to call it a construction. Dec 14, 2011 at 18:42
• @Dylan Moreland. thanks for copying.
– asd
Dec 14, 2011 at 19:15

There is no reason why your $z_0$ should be the maximum of $f$ along the boundary of any circle you construct. The maximum modulus principle just says the maximum of $f$ on a disc occurs at the boundary. If $z_0$ is a point on the boundary of a disc $B$, there may be $z_1$ on the boundary of $B$ such that $f(z_1) > f(z_0)$. If that is true then, of course, you may and will find $z_2$ in the interior of $B$ with $f(z_2) > f(z_0)$ but with $f(z_1) > f(z_2)$. There is no way to construct an inductive contradiction as you suggest.
• Exactly! My question is why $z_0$ can not be a maximum of $|f|$ in the $D_i$'s. Is there any lemma or theorem that discard this case.