For an $n \times n$ matrix $M$ with real eigenvalues, let $\lambda(M)$ be the largest eigenvalue of $M$. Prove that for any positive integer $r$ and positive semidefinite matrices $A, B$,

$$[\lambda(A^m B^m)]^{1/m} \leq [\lambda(A^{m+1} B^{m+1})]^{1/(m+1)}.$$