Assume $Q$ is a positive definite random matrix such that $0 < \lambda_{\min}(Q)....\leq \lambda_{\max}(Q) \leq 1$ holds. I want to show that \begin{align} E\left[\frac{\lambda_{\min}(Q)}{\lambda_{\min}(Q)+\lambda_{\max}(Q)}\right] \geq \frac{\lambda_{\min}(E[Q])}{\lambda_{\min}(E[Q])+\lambda_{\max}(E[Q])} \end{align}

Or I want to show the following Kantorovich type inequality for expected values: \begin{align} E\left[\frac{(y^Ty)^2}{(y^TQy)(y^TQ^{-1}y)}\right] \geq \frac{4\lambda_{\min}(E[Q])\lambda_{\max}(E[Q])}{\left[\lambda_{\min}(E[Q])+\lambda_{\max}(E[Q])\right]^2} \end{align}