We consider an operator function F defined on the interval \documentclass[12pt]{minimal}
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$$\user2{[}\sigma \user2{,}\tau \user2{]} \subset \mathbb{R}$$
\end{document} whose values are semibounded self-adjoint operators in the Hilbert space \documentclass[12pt]{minimal}
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$$\mathfrak{H}$$
\end{document}. To the operator function F we assign quantities \documentclass[12pt]{minimal}
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$$\mathcal{N}_\user1{F}$$
\end{document} and νF(λ) that are, respectively, the number of eigenvalues of the operator function F on the half-interval [σ,τ) and the number of negative eigenvalues of the operator F(λ) for an arbitrary λ ∈ [σ,τ]. We present conditions under which the estimate \documentclass[12pt]{minimal}
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$$\mathcal{N}_\user1{F} \geqslant \nu _\user1{F} \user2{(}\tau \user2{)} - \nu _\user1{F} \user2{(}\sigma \user2{)}$$
\end{document} holds. We also establish conditions for the relation \documentclass[12pt]{minimal}
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$$\mathcal{N}_\user1{F} \geqslant \nu _\user1{F} \user2{(}\tau \user2{)} - \nu _\user1{F} \user2{(}\sigma \user2{)}$$
\end{document} to hold. The results obtained are applied to ordinary differential operator functions on a finite interval.
