Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 1 / d

Solution

Write

r_{t} = 2 t

. The hulls

A_{t} = r_{t} (H \cap D)

are nested, so property (i) holds. Scaling the given map gives

g_{t} (z) = z + \frac{r _{t}^{2}}{z} = z + \frac{2 t}{z},

hcap (A_{t}) = 2 t

and property (ii) holds.

Property (iii) fails. For

s > 0

, the image under

g_{s}

of the outer semicircle of

A_{t} ∖ A_{s}

g_{s} (r_{t} e^{i θ}) = r_{t} e^{i θ} + \frac{r _{s}^{2}}{r _{t} e ^{i θ}} .

t ↓ s

, this converges to

2 r_{s} cos θ

, which fills the real interval

[- 2 r_{s}, 2 r_{s}]

. Hence the diameter of the mapped new increment tends to

4 r_{s}

, rather than zero. Therefore (i) and (ii) hold, while (iii) does not.

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