IMC 2016 · Problem 5

Let $A$ be a $n \times n$ complex matrix whose eigenvalues have absolute value at most $1$. Prove that

$$\|A^n\| \le \frac{n}{\ln 2}\|A\|^{n-1}.$$

(Here $\|B\| = \sup\limits_{\|x\| \le 1} \|Bx\|$ for every $n \times n$ matrix $B$ and $\|x\| = \sqrt{\sum\limits_{i=1}^{n} |x_i|^2}$ for every complex vector $x \in \mathbb{C}^n$.)