SCHWEITZER 2019 Problem 10

Let $A$ and $B$ be positive self-adjoint operators on a complex Hilbert space $H$ . Prove that $$\limsup_{n \to \infty} \|A^n x\|^{1/n} \le \limsup_{n \to \infty} \|B^n x\|^{1/n}$$ holds for every $x \in H$ if and only if $A^n \le B^n$ for each positive integer $n$ .