Holomorphic Injection and the its derivatives

By | June 12, 2015

Stein 책의 증명을 웬만해서는 좋아하지만, 몇몇 개의 증명은 너무 말로만 넘긴 느낌이 들어서, 논리를 채워야 하는 경우가 종종 발생한다.
복소변수 함수를 공부하다보면, 실변수 함수와는 다른 성질을 갖는 경우가 종종 발생한다. 대표적인 예가 다음과 같은 경우가 아닐까 싶다.

Theorem. If f:U\rightarrow V is holomorphic and injective, then f^{\prime}\left(z\right)\neq0 for all z\in U.

Proof. Suppose f^{\prime}\left(z_{0}\right)=0 for some z_{0}\in U. Then note that f\left(z\right)-f\left(z_{0}\right) and f^{\prime} are analytic at z_{0}. So there is \delta>0 such that f\left(z\right)-f\left(z_{0}\right)\neq0 and f^{\prime}\left(z\right)\neq0 for all z\in D_{\delta}\left(z_{0}\right)\setminus\left\{ z_{0}\right\}.
Let \varepsilon=\inf_{\left|z-z_{0}\right|=\delta}\left|f\left(z\right)-f\left(z_{0}\right)\right|>0. Then for w\in D\left(f\left(z_{0}\right),\varepsilon\right), we note that

    \[ f\left(z\right)-w=f\left(z\right)-f\left(z_{0}\right)+f\left(z_{0}\right)-w. \]

By definition of \varepsilon, for all z with \left|z-z_{0}\right|=\delta, we have

    \[ \left|f\left(z_{0}\right)-w\right|<\varepsilon\le\left|f\left(z\right)-w\right|. \]

So by Rouché’s theorem, \left|f\left(z\right)-w\right| and \left|f\left(z\right)-f\left(z_{0}\right)\right| have same number of zeros counting with multiplicities inside \left|z-z_{0}\right|=\delta.

Write f\left(z\right)-f\left(z_{0}\right)=\sum_{k=1}^{\infty}a_{k}\left(z-z_{0}\right)^{k} in D\left(z_{0},\delta\right). Then f^{\prime}\left(z\right)=\sum_{k=1}^{\infty}ka_{k}\left(z-z_{0}\right)^{k-1} in D\left(z_{0},\delta\right). Since f^{\prime}\left(z_{0}\right)=0, this shows that f\left(z\right)-f\left(z_{0}\right)=a\left(z-z_{0}\right)^{k}G\left(z\right)
where k\ge2 and G is holomorphic at z_{0} and G\left(z_{0}\right)\neq0.
So f\left(z\right)-w has k zeros by definition of \delta and the previous observation. Since f\left(z\right)-w\neq0 and f^{\prime}\left(z\right)\neq0 for all z\in D\left(z_{0},\delta\right)\setminus\left\{ z_{0}\right\}, the zeros of f\left(z\right)-w are all simple. But f is injective, a contradiction.

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