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Let $f$ be a holomorphic function on the open unitary disk $\mathbb{D}$ and continuous on $\mathbb{\overline{D}}$. If $f(\frac{z}{2})= \frac{1}{2}f(z)$ for all $z\in \mathbb{\overline{D}}$ and $f(1)=1$, then $f(z)=z$ for all $z\in \mathbb{\overline{D}}$.

Got this as homework. Any hints would be highly appreciated.

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    "Complex Analysis" in unsuitable as a title.2011-02-15

2 Answers 2

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Since $f$ is holomorphic it has a unique power series representation about origin, $ f(z)=\sum_{k=0}^{\infty} c_k z^k. $ Since $f\left(\dfrac{z}{2}\right)=\dfrac{1}{2} f(z)$ we obtain $ \frac{c_k}{2^k}=\frac{1}{2}c_k,\quad k=0,1,\ldots. $ From this we obtain $c_k=0$ if $k\neq 1$. Therefore $f(z)=c_1z$. We know that $f(1)=1$. Thus $c_1=1$ and $f(z)=z$.

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    It was really a wonderful solution.2012-05-16
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Another way of looking at this is that you are given that $f(z) - z = 0$ for $z = 1$. Then use the condition that $f(z/2) = {1 \over 2}f(z)$ to inductively show that $f(z) - z$ also has zeroes at $z = 2^{-n}$ for all positive integers $n$. Thus the zeroes of $f(z) - z$ have an accumulation point at $z = 0$, which is only possible if $f(z) - z = 0$ for all $z$ since nonzero analytic functions can only have isolated zeroes.