Intereting Posts

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I don’t understand the final step of an argument I read.

Suppose $f$ is holomorphic in a neighborhood containing the closed unit disk, nonconstant, and $|f(z)|=1$ when $|z|=1$. There is some point $z_0$ in the unit disk such that $f(z_0)=0$.

By the maximum modulus principle, it follows that $|f(z)|<1$ in the open unit disk. Since the closed disk is compact, $f$ obtains a minimum on the closed disk, necessarily on the interior in this situation.

- Does a convergent power series on a closed disk always converge uniformly?
- Find a conformal map from the disc to the first quadrant.
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- Convergence of $\prod_{n=1}^\infty(1+a_n)$
- Is the reciprocal of an analytic function analytic?
- Residue theorem:When a singularity on the circle (not inside the circle)

But why does that imply that $f(z_0)=0$ for some $z_0$? I’m aware of the minimum modulus principle, that the modulus of a holomorphic, nonconstant, nonzero function on a domain does not obtain a minimum in the domain. But I’m not sure if that applies here.

- Show that this function is entire
- How is Cauchy's estimate derived?
- Prove the open mapping theorem by using maximum modulus principle
- Is there any connection between Green's Theorem and the Cauchy-Riemann equations?
- On functions with Fourier transform having compact support
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- Stereographic projection is conformal — from the line element
- Prove that the taylor series of cos(z) and sin(z) are holomorphic
- Show that an entire function bounded by $|z|^{10/3}$ is cubic

If not, consider $g(z)=\frac 1{f(z)}$ on the closure of the unit disc. We have $|g(z)|=1$ if $|z|=1$ and $|g(z)|>1$ if $|z|<1$. Since $g$ is holomorphic on the unit disk, the maximum modulus principle yields a contradiction.

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