IMC 2016 Problem 10

Let $A$ be a $n\times n$ complex matrix whose eigenvalues have absolute value at most $1$ . Prove that $$\|A^n\|\le \dfrac{n}{\ln 2} \|A\|^{n-1}.$$ (Here $\|B\|=\sup\limits_{\|x\|\leq 1} \|Bx\|$ for every $n\times n$ matrix $B$ and $\|x\|=\sqrt{\sum\limits_{i=1}^n |x_i|^2}$ for every complex vector $x\in\mathbb{C}^n$ .)

(Proposed by Ian Morris and Fedor Petrov, St. Petersburg State University)