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Suppose I have a function $f$ holomorphic in $\{z: \operatorname{Re}(z) > 0, \operatorname{Im}(z) > 0\}$. Let $C_{1}$ denote the circle centered at $2 + 10i$ of radius 1 and let $C_{2}$ denote the circle centered at $5 + 10i$ of radius 4.

In particular, the closed disk $D_{1}$ whose boundary is $C_{1}$ is contained in the disk closed $D_{2}$ with boundary $C_{2}$ and $C_{1}$ intersects $C_{2}$ at the point $1 + 10i$. Additionally, suppose I knew that $|f(z)| \leq M$ for all $z \in D_{1}$. Does this give any bounds for $|f(z)|$ on $D_{2}$? Is $|f(z)| \leq M$ for all $z \in D_{2}$?

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No, for example consider the function $f(z) = z$

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