Abstract
In this chapter the Polya theorem about the relationship between the indicator diagram and the conjugate diagram of an entire function of finite power (see the Introduction, Theorems A 1, A 2; Theorem 1.10.8) is extended in various ways to entire functions of order ρ >0, ρ ≠ 1. We also consider applications of these extensions. The key notion is that of the generalized Borel transformation associated with the polynomial
various properties of which were examined in Chapter 4. We consider the operator B α that takes an entire function
of order ρ and of finite type to the function
holomorphic in a certain neighborhood G of the point ∞. Here u(ζ), ζ ∈ G, is a one-sheeted and nonvanishing branch of the multivalued function [α(ζ)]1/ρ such that
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Maergoiz, L.S. (2003). Spaces of Entire Functions of Order ρ > 0 with Restrictions on the Indicator. In: Asymptotic Characteristics of Entire Functions and Their Applications in Mathematics and Biophysics. Mathematics and Its Applications, vol 559. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0807-4_5
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