I encountered an inequality when reading a paper. Can someone help to show how to prove it?

Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of matrix }A\}$. For matrices $S$ and $T$ with positive spectral radii, and two arbitrary real positive numbers $a$ and $b$, such that $\rho(S) < a < b$ Is the following inequality true? $$b\rho((bI-S)^{-1}T) \leq a\rho((aI-S)^{-1}T)$$ If the above is not true in general, will it be true if $S$ and $T$ are non-negative matrices?