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Suppose $f$ is analytic in a domain $D$. Let $z_0\in D$ and $\overline{D}_R(z_0)\subset D$. What I need to prove is that if $f$ is analytic and $|f(z_0)|\geq |f(z)|$ for every $z$ such that $|z-z_0|=R$ then there is no point $z_1$ such that $|f(z_1)|<|f(z_0)|$. I just have no idea where to even begin so any hint will be much appreciated. I apologize for not showing any effort. Any help will be appreciated. Thanks

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  • $\begingroup$ Do the inequalities above go in the right direction? Should it perhaps be $|f(z_0)|\leq|f(z)|$? $\endgroup$ – mickep Dec 23 '14 at 17:33
  • $\begingroup$ Nope the given is correct $\endgroup$ – Heisenberg Dec 23 '14 at 17:33
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    $\begingroup$ Hint: Use Maximum Modulus Principle for the Disc $\overline{D}_R(z_0).$ $\endgroup$ – Krish Dec 23 '14 at 17:41
  • $\begingroup$ The Mean Value Property implies that $f$ is constant on $\partial D_R(z_0)$. Then Cauchy's Integral Formula implies that $f$ is constant on $\overline{D}_R(z_0)$. $\endgroup$ – robjohn Dec 23 '14 at 17:53
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Maximum Modulus Principle:

First Version: Let $G$ be a bounded domain and let $f : G \rightarrow \mathbb{C}$ be an analytic function. Suppose there exists an element $a \in G$ such that $|f(a)| \geq |f(z)|, \forall z \in G.$ Then $f$ is constant.

Second Version: Let $G$ be bounded open set in $\mathbb{C}$ and let $f : \overline{G} \rightarrow \mathbb{C}$ be a continuous function which is analytic on $G.$ Then max$\{|f(z)| : z \in \overline{G}\} = $ max$\{|f(z)| : z \in \partial G \}.$

(Ref: J. B. Conway: Functions of one complex variable, Chapter $VI.$)

Apply this to $G := \{z : |z - z_0| < R\}$ to conclude that $f$ is constant on $G.$ Now construct a function $g : D \rightarrow \mathbb{C}$ by $g(z) = f(z) - f(z_0).$ Then $g$ is an analytic function on $D$ and $D(z_0, R) \subseteq Z(g),$ the zero set of $g$. So $g = 0$ on $D.$

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