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This is a sequel of a recent article by Borichev-Golinskii-Kupin, where theauthors obtain Blaschke-type conditions for special classes of analyticfunctions in the unit disk which satisfy certain growth hypotheses. Theseresults were applied to get Lieb-Thirring inequalities for complex compactperturbations of a selfadjoint operator with a simply connected resolvent set.  The first result of the present paper is an appropriate local version of theBlaschke-type condition from Borichev-Golinskii-Kupin. We apply it to obtain asimilar condition for an analytic function in a finitely connected domain of aspecial type. Such condition is by and large the same as a Lieb-Thirring typeinequality for complex compact perturbations of a selfadjoint operator with afinite-band spectrum. A particular case of this result is the Lieb-Thirringinequality for a selfadjoint perturbation of the Schatten class of a periodic(or a finite-band) Jacobi matrix. The latter result seems to be new in suchgenerality even in this framework.

A Blaschke-type condition for analytic functions on finitely connected  domains. Applications to complex perturbations of a finite-band selfadjoint  operator