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Year 4 of the mathematics course of the University of Oxford

Complex analysis: conformal maps and geometry course of the University of Oxford

Past exam of the complex analysis: conformal maps and geometry course of the University of Oxford

2023

The <Koebe growth theorem> states that every <normalized univalent function> $\phi\in\mathcal S$ satisfies
$$
\boxed{\frac{|z|}{(1+|z|)^2}\leq|\phi(z)|\leq\frac{|z|}{(1-|z|)^2}\qquad(z\in\mathbb D)}.
$$
Both bounds are sharp. Rotations of the <Koebe function>, $\phi_\theta(z)=z/(1-e^{i\theta}z)^2$, attain them along suitable radial directions.

Solution

Past exam of the complex analysis: conformal maps and geometry course of the University of Oxford / 2023 / 1 / b / i / Solution

Past exam of the complex analysis: conformal maps and geometry course of the University of Oxford / 2023 / 1 / b / i / Solution

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Past exam of the complex analysis: conformal maps and geometry course of the University of Oxford / 2023 / 1 / b / i / Solution

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