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^{1}Department of Mathematics, Southwest State University, Russia.

^{2}Department 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.

**Visit for more related articles at** Journal of Physical Mathematics

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.

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

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*.

(1)

Let . 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_{x→∞} *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 {*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*

(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 {*f*_{n}(*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 *r*_{0}(*ε*) does not depend on *n* such that

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 with the proximate order ρ(r) (ρ > 0) is given by the equation*

(4)

Let *ρ* > 0

For a function we associate the function

(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*

(6)

*Where the quantities an satisfy*

(7)

The following is our main result.

**Theorem 1.4**

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

(8)

*Where the quantities an satisfy*

(9)

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

Hence

(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

But *p* is arbitrary, hence,

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

And then the series (10) converges.

Let if *k*→*¥* and let *l* satisfies (9). By (2), there exists *β* > 0 such that {{*f _{k}*(

We now consider the space of entire functions *E _{ρ(r)}* which have a proximate orders less then

A sequence of functions {*f _{n}*(

(11)

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

(12)

where *φ*_{1}(*t*) defined to be the unique solution of the **equation** *t*=*V*_{1}(*r*). From this

So *ρ*_{1}(*r*) is arbitrary less then *ρ*(*r*) that

(13)

Contrary, let the condition (13) is true and *ρ*1(*r*) is arbitrary less than *ρ*(*r*). So *φ*_{1}(*n*) > *φ*(*n*), *n* > *n*_{0}, that

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)}.

**Theorem 3.1**

*Continuous linear functional l on the space E _{ρ(r)} has the form*

*Where the quantities an satisfy*

**Remark:** The case of the spaces [*ρ*,*¥*) and *E _{r}*, where

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.

- Malyutin KG, Malyutina TI (2015) Linear Functionals in Some Spaces of Entire Functions of Finite Order. Istanbul University Science Faculty the Journal of Mathematics, Physics and Astronomy 6: 1-6.
- Levin BIA (1980) Distribution of Zeros of Entire Functions. Amer Math Soc.
- Bingham NH, Goldie CM, Teugels JL (1987) Regular variation. Cambridge university press, London.
- Leont’ev AF (1981) Obobshcheniye ryadov e’ksponent, Nauka.

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