# Spaces of Entire Functions of Order ρ > 0 with Restrictions on the Indicator

• L. S. Maergoiz
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 559)

## 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
$$\alpha (z) = {z^\rho }[1 + \frac{{{a_1}}}{z} + \ldots + \frac{{{a_n}}}{{{z^n}}}],$$
various properties of which were examined in Chapter 4. We consider the operator B α that takes an entire function
$$f(z) = \sum\limits_{k = 0}^\infty {\frac{{{b_k}{z^k}}}{{\left( {\frac{{k + 1}}{\rho }} \right)}}}$$
(5.0.1)
of order ρ and of finite type to the function
$$\hat f(\zeta ) = [{B_a}f](\zeta ) = \sum\limits_{k = 0}^\infty {\frac{{{b_k}}}{{{{[u(\zeta )]}^{k + 1}}}}} ,$$
(5.0.2)
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
$$\mathop {\lim }\limits_{\zeta \to \infty } {\zeta ^{ - 1}}u(\zeta ) = 1.$$
(5.0.3)

## Keywords

Saddle Point Entire Function Support Function Convex Compact Convex Domain
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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