Generalized Convexity, Blaschke-type Condition in Unbounded Domains, and Application in Spectral Perturbation Theory of Linear Operators

We introduce a notion of r-convexity for subsets of the complex
plane. It is a pure geometric characteristic that generalizes the        
usual notion of convexity. Next, we investigate subharmonic              
functions that grow near the boundary in unbounded domains with          
r-convex compact complement. We obtain the Blaschke-type bounds          
for its Riesz measure and, in particular, for zeros of unbounded         
analytic functions  in unbounded domains. These results are based        
on a certain estimates for Green functions on complements of some        
neighborhoods of $r$-convex compact set. Also, we apply our              
results in perturbation theory of linear operators in a Hilbert          
space. More precisely, we find quantitative estimates for the rate       
of condensation of the discrete spectrum of a perturbed operator         
near its the essential spectrum.