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ISSN: 2090-0902
Journal of Physical Mathematics
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The Spaces of Entire Function of Finite Order

Malyutin KG1* and Studenikina IG2

1Department of Mathematics, Southwest State University, Russia.

2Department of Mathematics, Sumy State University, Ukraine.

*Corresponding Author:
Malyutin KG
Professor
Department of Mathematics
Southwest State University, Russia
Tel: 800-642-0684
E-mail: [email protected]

Received Date: June 03, 2016; Accepted Date: July 26, 2016; Published Date: August 01, 2016

Citation: Malyutin KG, Studenikina IG (2016) The Spaces of Entire Function of Finite Order. J Phys Math 7:190. doi:10.4172/2090-0902.1000190

Copyright: © 2016 Malyutin KG, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

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.

Keywords

Entire function; Proximate order; Normal type; Continuous linear functional

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

Equation

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)=rρ(r). We will assume that V (r) is an increasing function on (0,∞) and Equation.

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.

Equation    (1)

Let Equation. If for the entire function f(z) the quantity

Equation

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, limx→∞ ρ(r)= ρ ≥0. 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 {fn(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

Equation

where r0(β) does not depend on (n ≥ 1). For a suitable C(β), which does not depend on n, for all z

Equation     (2)

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. A sequence of functions {fn(z)} from [ρ(r), p] converges in the sense of [ρ(r), p] if (i) it converges uniformly on compacts, (ii) for all ε > 0 there exists r0(ε) does not depend on n such that

Equation

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

φ(V (t))=t. (3)

Theorem 1.1 ([2, Theorem 2', p.42])

The type σf of the entire function Equation with the proximate order ρ(r) (ρ > 0) is given by the equation

Equation     (4)

Let ρ > 0

Equation

For a function Equation we associate the function

Equation    (5)

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 {fn(z)} of functions from [ρ(r), p] to converge in the sense of [ρ(r), p] necessary and sufficient that the sequence {fn(z)} (fn(z)~Fn(z)) converges uniformly inside the disk |z| < 1.

Theorem 1.3

Continuous linear functional l on the space [ρ(r), p] has the form

Equation    (6)

Where the quantities an satisfy

Equation    (7)

The following is our main result.

Theorem 1.4

Continuous linear functional l on the space [ρ(r),¥ has the form

Equation    (8)

Where the quantities an satisfy

Equation     (9)

The Space of Entire Functions [ρ(r),¥)

We now prove the theorem 1.4. Let l(f) be a continuous linear functional on the space [ρ(r),¥). Set Equation Let Equation be a function in [ρ(r),¥). Since the series converges in the sense of [ρ(r),¥) then, by continuous of the functional,

Equation

Hence

Equation    (10)

Take an arbitrary finite p>0. Functional l(f) is, in particular, continuous linear functional on the space [ρ(r), p]. By theorem 1.3, the condition

Equation

But p is arbitrary, hence,

Equation

We now verify that if the condition (9) then the functional (10) is continuous linear functional on the space [ρ(r),¥). Let Equation By theorem 1.1, Equation. Then Equation

And then the series (10) converges.

Let Equation if k¥ and let l satisfies (9). By (2), there exists β > 0 such that {{fk(z)}, f(z)}Ì[r(r), β] in the sense [ρ(r), β]. By (9) and (8) l is continuous linear functional on the space [ρ(r), β]. Then l(fk)→l l(f) if k¥. Therefore l is continuous linear functional on the space [ρ(r),¥).

Space of Entire Functions Eρ(r)

We now consider the space of entire functions Eρ(r) which have a proximate orders less then ρ(r). A proximate order ρ1(r) less then ρ(r) if Equation aingle valued function f(z) of the complex variable z is said to belong to the space Eρ(r) if f(z) has the order less than ρ(r).

A sequence of functions {fn(z)} from Eρ(r) converges in the sense of Eρ(r) if (i) it converges uniformly on compacts, (ii) there exists proximate order ρ1(r), Equation such that

Equation     (11)

where r0(β) does not depend on (n ≥ 1), V1(r)=rr1(r).The space Eρ(r) is the linear topology space with sequence topology. A continuous linear functional l(f) on the space Eρ(r) has the form (8). Let us find the conditions that satisfy the values an. The functional l(f) is in particular continuous linear functional on the space [ρ1(r),¥) for all proximate order ρ1(r), Equation. Therefore, by theorem

Equation    (12)

where φ1(t) defined to be the unique solution of the equation t=V1(r). From this

Equation

So ρ1(r) is arbitrary less then ρ(r) that

Equation    (13)

Contrary, let the condition (13) is true and ρ1(r) is arbitrary less than ρ(r). So φ1(n) > φ(n), n > n0, that

Equation

Therefor the condition (12) is true and l(f) is continuous linear functional on the space [ρ(r),¥). So ρ1(r) is arbitrary less then, ρ(r) that l(f) is continuous linear functional on the space Er(r).

Theorem 3.1

Continuous linear functional l on the space E ρ(r) has the form

Equation

Where the quantities an satisfy

Equation

Remark: The case of the spaces [ρ,¥) and Er, where ρ(r)=ρ > 0, considered A.F Leont’ev [4].

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.

References

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