Is f(z) one-to-one on D?

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In summary, the problem at hand is to show that if a function f(z) is analytic on the unit disk D and has a one-to-one restriction on an annulus U, then it is also one-to-one on D. The attempt at a solution involves using the Swarz lemma and the Identity theorem, and considering a point z0 in D and the function g(z)=f(z)-f(z0). It is necessary to show that g'(z) is non-zero on U and therefore on D, which implies that f'(z) is non-zero on D.
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esisk
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1. if f(z) is analytic on the unit disk D. Suppose there is an annulus U={r<|z|<1} such that the restriction of f(z) to U is one-to-one. Show that f(z) is one-to-one on D.



Homework Equations



I am having trouble using the hypotheseis "f is one-to-one" I believe. I cannot think other than this implying that f'(z) is nonzero, but this does not seem to suffice





The Attempt at a Solution


I am trying to use Swarz lemma in conjunction with Identity theorem (that is showing that two functions agree on an infinite set), but I
 
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am still having trouble linking them. Let us consider a point $z_0$ in the unit disk D and let $g(z)=f(z)-f(z_0)$. The given hypothesis implies that $g(z)$ is one-to-one on U. Then by Schwarz lemma, it follows that $g'(z)\neq 0$ on U. Since $U$ is a subset of D, this implies that $g'(z)\neq 0$ on D. This further implies that $f'(z)\neq 0$ on D (since $g(z)=f(z)-f(z_0)$). Hence, $f(z)$ is one-to-one on D.
 

1. What is an annulus?

An annulus is a geometric shape that consists of a circular ring or disk with a hole in the center.

2. What is analytic in an annulus?

Analytic in an annulus refers to a function that is defined and has a continuous derivative within the annulus, meaning it is differentiable at every point in the annulus.

3. How is analyticity determined in an annulus?

Analyticity in an annulus is determined by the Cauchy-Riemann equations, which state that a function is analytic if it satisfies certain conditions related to its partial derivatives.

4. What is the importance of analyticity in an annulus?

Analyticity in an annulus is important in fields such as complex analysis and differential equations, as it allows for the use of powerful mathematical techniques and theorems to solve problems involving functions in an annulus.

5. Can a function be analytic in an annulus but not in its interior or exterior?

Yes, a function can be analytic in an annulus but not in its interior or exterior. This is because analyticity in an annulus only requires the function to be differentiable within the annulus, not necessarily in its interior or exterior. It is possible for a function to have discontinuities or singularities outside of the annulus, but still be analytic within it.

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