Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 107 / 1 / c

Iteration gives the interior derivative estimate for a harmonic functionThe growth hypothesis bounds the right side by $C^{\prime }{R}^{\alpha -1}+o(1)$, which tends to zero as $R\to \infty$. Thus every derivative of order $k+1$ vanishes everywhere. The Taylor theorem makes $u$ a polynomial of degree at most $k$, proving the Polynomial-growth Liouville theorem for harmonic functions.

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