Let us consider a symmetrix $n \times n$ - matrix $A$ with non-negative elements $a_{ij} \geq 0$. Furthermore, we look at a non-negative vector $x \in \mathbb{R}^n$ with $x_i \geq 0$. Then we want to prove that $$ \left(\frac{x^TAx}{x^Tx}\right)^m \leq \frac{x^TA^mx}{x^Tx} \quad \quad \forall m \in \mathbb{N}_{\geq1} .$$
As a hint, we should use induction over the matrix size $n$. I have tried this, but I am not sure how to properly reduce the case for $n+1$ to a smaller matrix of size $n$. Thanks for any help!