Arithmetic mean function of the real part of entire Dirichlet series I
When treating spaces of holomorphic functions with growth conditions, one is led to introduce associated weights. In our main theorem we characterize, in terms of the sequence of associated weights, several properties of weighted (LB)-spaces of holomorphic functions on an open subset which play an important role in the projective description problem. A number of relevant examples are provided, and a “new projective description problem” is posed. The proof of our main result can also serve to characterize...
We extend and simplify results of [Din 2010] where the asymptotic behavior of the holomorphic sectional curvature of the Bergman metric in annuli is studied. Similarly to [Din 2010] the description enables us to construct an infinitely connected planar domain (in our paper it is a Zalcman type domain) for which the supremum of the holomorphic sectional curvature is two, whereas its infimum is equal to -∞ .
Schur introduced the problem on the smallest limit point for the arithmetic means of totally positive conjugate algebraic integers. This area was developed further by Siegel, Smyth and others. We consider several generalizations of the problem that include questions on the smallest limit points of symmetric means. The key tool used in the study is the asymptotic distribution of algebraic numbers understood via the weak* limits of their counting measures. We establish interesting properties of the...