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                \begin{document}$$ \lambda_{i} \bigl((A+H)\widetilde{B}^{-1} \bigr) \ge \textstyle\begin{cases} \frac{\lambda_{\ell}(AB^{-1} )+\lambda_{\hbar} (HB^{-1} )}{1+\mu}, &\lambda_{i}(A+H)\ge0;\\ \frac{\lambda_{j} (AB^{-1} )+\lambda_{k} (HB^{-1} )}{1-\mu}, &\lambda_{i}(A+H)< 0. \end{cases} $$\end{document}$$ The proof of Theorem [2.1](#FPar8){ref-type="sec"} is complete. □

Corollary 2.1 {#FPar10}
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                \begin{document}$A,B,H,E \in\mathbb{C}^{n\times n}$\end{document}$ *be Hermitian matrices*, *B* *be a positive definite Hermitian matrix*, *and* $\documentclass[12pt]{minimal}
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                \begin{document}$\widetilde{B}=B+E$\end{document}$. *If* $\documentclass[12pt]{minimal}
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                \begin{document}$\mu =\frac{\|E\|_{2}}{\lambda_{n}(B)}<1$\end{docume