 Solution

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.

Solved by gpt-5.6-sol high.