The Spaces of Entire Function of Finite Order

This paper is a continuation of the research of the first author. We consider the linear topology space of entire functions of a proximate order and normal type with respect to the proximate order. We obtain the form of continuous linear functional on this space.


Introduction
This paper is a continuation of the research [1] where the linear topology space of entire functions of a proximate order and normal type, less than or equal σ, with respect to the proximate order were considered. We introduce the necessary definitions. A function ρ(r), defined on the ray (0,∞) and satisfying the Lipschitz condition on any segment [a, b]⊂ (0,∞) that satisfies the conditions This is called a proximate order.
A detailed exposition of the properties of proximate order can be found [2,3]. In this paper we use the notation V (r)=r ρ(r) . We will assume that V (r) is an increasing function on (0,∞) and We now formulate some simple property of proximate order that we shall need frequently [2].
For r→∞ and 0 a k b < ≤ ≤ < ∞ asymptotic inequality holds uniformly in k.
If for the entire function f(z) the quantity Is different from zero and infinity, then ρ(r) is called of a proximate order of the given entire function f(z) and σf is called the type of the function f(z) with respect to the proximate order ρ(r).Let ρ(r) be a proximate order, lim ( ) 0 x r ρ ρ →∞ = ≥ . A single valued function f(z) of the complex variable z is said to belong to the space [ρ(r),¥) if f(z) has the order less than ρ(r) or equal ρ(r) but in this case type less than ¥. A sequence of functions {f n (z)} from [ρ(r),¥) converges in the sense of [ρ(r),¥) if (i) It converges uniformly on compacts, (ii) there exists β<1 such that where r 0 (β) does not depend on (n ≥ 1). For a suitable C(β), which does not depend on n, for all z The space [ρ(r),¥) is the linear topology space with sequence topology. Furthermore, a single valued function f(z) of the complex variable z is said to belong to the space [ρ(r), p] if f(z) has the order less than ρ(r) or equal ρ(r) but in this case type less than or equal p.
The space [ρ(r), p] is also the linear topology space with sequence topology. We introduce the function φ(t) defined to be the unique solution of the equation t=V (r). So

Theorem 1.1 ([2, Theorem 2', p.42])
The type σf of the entire function Let ρ > 0 It is regular, in any case in the circle |z| < 1 [1]. Fact mapping function f(z) of [ρ(r), p] to the function F(z) as indicated above will be celebrating a record f(z) ~F(z).
In [1] it is proved the following two theorems.

Theorem 1.2
In order to be a sequence {f n (z)} of functions from [ρ(r), p] to converge in the sense of [ρ(r), p] necessary and sufficient that the sequence {f n (z)} (fn(z)~Fn(z)) converges uniformly inside the disk |z| < 1. A sequence of functions {f n (z)} from E ρ(r) converges in the sense of E ρ(r) if (i) it converges uniformly on compacts, (ii) there exists proximate order ρ 1 (r), where r 0 (β) does not depend on (n ≥ 1), V 1 (r)=r r1(r) .The space E ρ(r) is the linear topology space with sequence topology. A continuous linear functional l(f) on the space E ρ(r) has the form (8). Let us find the conditions that satisfy the values a n . The functional l(f) is in particular continuous linear functional on the space [ρ1(r),¥) for all proximate order ρ 1 (r), where φ 1 (t) defined to be the unique solution of the equation t=V 1 (r). From this 0 log | | 1, ( ) n a n n n n < > 1 j So ρ 1 (r) is arbitrary less then ρ(r) that Contrary, let the condition (13) is true and ρ1(r) is arbitrary less than ρ(r). So φ 1 (n) > φ(n), n > n 0 , that 0 log | | 1, ( ) n a n n n n < > 1 j Therefor the condition (12) is true and l(f) is continuous linear functional on the space [ρ(r),¥). So ρ 1 (r) is arbitrary less then, ρ(r) that l(f) is continuous linear functional on the space E r(r) .

Conclusion
The linear topology space of entire functions of a proximate order and normal type with respect to the proximate order is considered. We obtain the form of continuous linear functional on this space through our work.
The following is our main result.