sharper interlacing

The usual interlacing inequalities say that if $M$ is a Hermitian $n \times n$ matrix and $\hat{M}$ is a principal submatrix of order $n-1$, then $\lambda_{\min}(M) \leq \lambda_{\min}(\hat{M})$. I would like to know if there are some results of the form $\lambda_{\min}(M)+f(M) \leq \lambda_{\min}(\hat{M})$, where $f$ is a function of the entries of $M$.

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