Let $A$ a square real matrix such that the largest singular value $\sigma_\text{max}(A) = \sigma < 1$. I want to find a lower bound on $\langle (I + A)^{-1}x, x\rangle$ where $x$ is a vector of euclidean norm $1$: $\langle x, x\rangle=1$.

I empirically find that a seemingly tight lower bound is $$ \langle (I + A)^{-1}x, x\rangle \geq \frac{1}{1+\sigma} $$ which is reached for $A= \sigma I$. I cannot prove the above result.

Note that it is pretty straightforward to prove that $\sigma_\text{min}((I + A)^{-1}) \geq \frac1{1+\sigma}$ but that does not suffice to conclude, since I do not assume that $A$ is symmetric.

PS: $\sigma_\text{max}$ and $\sigma_\text{min}$ are the largest and smallest singular values: $\sigma_\text{max} = \sqrt{\lambda_\text{max}(AA^T)}$ is the operator norm of $A$.