#### Abstract

The relative order of growth gives a quantitative assessment of how different functions scale each other and to what extent they are self-similar in growth. In this paper for any two positive integers *p* and *q*, we wish to introduce an alternative definition of relative th order which improves the earlier definition of relative th order as introduced by Lahiri and Banerjee (2005). Also in this paper we discuss some growth rates of entire functions on the basis of the improved definition of relative th order with respect to another entire function and extend some earlier concepts as given by Lahiri and Banerjee (2005), providing some examples of entire functions whose growth rate can accordingly be studied.

#### 1. Introduction

A single valued function of one complex variable which is analytic in the finite complex plane is called an integral (entire) function. For example, , , , and so forth are all entire functions. In Rolf Nevanlinna initiated the value distribution theory of entire functions which is a prominent branch of Complex Analysis and is the prime concern of this paper. Perhaps the Fundamental Theorem of Classical Algebra which states that “If is a polynomial of degree with real or complex coefficients, then the equation has at least one root” is the most well known value distribution theorem, and consequently any such given polynomial can take any given, real or complex, value. In the value distribution theory one studies how an entire function assumes some values and, conversely, what is the influence in some specific manner of taking certain values on a function. It also deals with various aspects of the behavior of entire functions, one of which is the study of their comparative growth.

For any entire function , the so-called maximum modulus function, denoted by , is defined on each nonnegative real value as And given two entire functions and the ratio as is called the growth of with respect to in terms of their maximum moduli.

The* order* of an entire function which is generally used in computational purpose is defined in terms of the growth of with respect to the exponential function as

Bernal [1, 2] introduced the relative order between two entire functions to avoid comparing growth just with . Extending the notion of relative order as cited in the reference, in this paper we extend some results related to the growth rates of entire functions on the basis of avoiding some restriction, introducing a new type of relative order , and revisiting ideas developed by a number of authors including Lahiri and Banerjee [3].

#### 2. Notation and Preliminary Remarks

Our notation is standard within the theory of Nevanlinna’s value distribution of entire functions. For short, given a real function and whenever the corresponding domain and range allow it, we will use the notation
omitting the parenthesis when happens to be the or function. Taking this into account the* order* (resp.,* lower order*) of an entire function is given by
Let us recall that Juneja et al. [4] defined the order and lower order of an entire function , respectively, as follows:
where are any two positive integers with . These definitions extended the generalized order and generalized lower order of an entire function considered in [5] for each integer since these correspond to the particular case and . Clearly and .

In this connection let us recall that if , then the following properties hold: Similarly for , one can easily verify that

Recalling that for any pair of integer numbers the Kronecker function is defined by for and for , the aforementioned properties provide the following definition.

*Definition 1 (see [4]). *An entire function is said to have index-pair if . Otherwise, is said to have index-pair , , if and .

*Definition 2 (see [4]). *An entire function is said to have lower index-pair if . Otherwise, is said to have lower index-pair , , if and .

An entire function of index-pair is said to be of regular -growth if its th order coincides with its th lower order; otherwise is said to be of irregular -growth.

Given a nonconstant entire function defined in the open complex plane its maximum modulus function is strictly increasing and continuous. Hence there exists its inverse function with .

Then Bernal [1, 2] introduced the definition of relative order of with respect to , denoted by as follows:

This definition coincides with the classical one [6] if . Similarly one can define the relative lower order of with respect to denoted by as

Lahiri and Banerjee [7] gave a more generalized concept of relative order in the following way.

*Definition 3 (see [7]). *If is a positive integer, then the th generalized relative order of with respect to , denoted by , is defined by
Clearly, and .

In the case of relative order, it was then natural for Lahiri and Banerjee [3] to define the relative th order of entire functions as follows.

*Definition 4 (see [3]). *Let and be any two positive integers with . The relative th order of with respect to is defined by
Then and for any .

In this paper we give an alternative definition of th relative order of an entire function with respect to another entire function , in the light of index-pair. Our next definition avoids the restriction and gives the more natural particular case .

*Definition 5. *Let and be any two entire functions with index-pair and , respectively, where , , are positive integers such that . Then the th relative order of with respect to is defined as
The th relative lower order of with respect to is defined by

The previous definitions are easily generated as particular cases; for example, if and have got index-pair and , respectively, then Definition 5 reduces to Definition 3. If the entire functions and have the same index-pair , where is any positive integer, we get the definition of relative order introduced by Bernal [1] and if , then and . And if is an entire function with index-pair and , then Definition 5 becomes the classical one given in [6].

#### 3. Some Examples

In this section we present some examples of entire functions in connection with definitions given in the previous section.

*Example 6 (order of ). *Given any natural number , the exponential function has got , and therefore is constantly equal to and, consequently,

*Example 7 (generalized order). *Given any natural numbers , , the function has got . Therefore is constantly equal to for each natural , following that
Note that for and for .

*Example 8 (index-pair). *Given any four positive integers , , , with , then function generates a constant quotient , and clearly
but
Thus is a regular function with growth .

*Example 9 (regular function of growth (1,1)). *Given any positive integer , and nonnull real number , the power function generates a constant quotient , and clearly
but
Thus is a regular function with growth .

*Example 10 (relative order between functions). *From the above examples it follows that given the natural numbers , the functions
are of regular growth . In order to find their relative order of growth we evaluate
which happens to be constant. Its upper and lower limits provide

*Example 11 (relative order between functions). *Let , , be any three positive integers and let and . Then and are regular functions with -growth for which
In order to find out their relative order we evaluate
which happens to be constant. By taking limits, we easily get that

The orders obtained in the last two examples will be easy consequences of the results given in Section 4.

#### 4. Results

In this section we state the main results of the paper. We include the proof of the first main theorem for the sake of completeness. The others are basically omitted since they are easily proven with the same techniques or with some easy reasoning.

Theorem 12. *Let and be any two entire functions with index-pair and , respectively, where , , are all positive integers such that and . Then
*

*Proof. *From the definitions of and we have for all sufficiently large values of that
and also for a sequence of values of tending to infinity we get that
Similarly from the definitions of and it follows for all sufficiently large values of that
and for a sequence of values of tending to infinity we obtain that
Now from (29) and in view of (31), for a sequence of values of tending to infinity we get that
As is arbitrary, it follows that
Analogously from (28) and in view of (34) it follows for a sequence of values of tending to infinity that
Since is arbitrary, we get from above that
Again in view of (32) we have from (27) for all sufficiently large values of that
Since is arbitrary, we obtain that
Again from (28) and in view of (31) with the same reasoning we get that
Also in view of (33), we get from (27) for a sequence of values of tending to infinity that
Since is arbitrary, we get from above that
Similarly from (30) and in view of (32) it follows for a sequence of values of tending to infinity that
As is arbitrary, from above we obtain that
The theorem follows from (36), (38), (40), (41), (43), and (45).

Corollary 13. *Let be an entire function with index-pair and let be an entire of regular -growth, where , , are all positive integers such that and . Then
**
In addition, if , then
*

*Remark 14. *The first part of Corollary 13 improves [8, Theorem 2.1 and Theorem 2.2].

Corollary 15. *Let and be any two entire functions with regular -growth and regular -growth, respectively, where , , are all positive integers with . Then
*

Corollary 16. *Let and be any two entire functions with regular th growth and regular th growth, respectively, where , , are all positive integers with and . Also suppose that . Then
*

Corollary 17. *Let and be any two entire functions with regular growth and , respectively, where , , are all positive integers such that . Then
*

Corollary 18. *Let and be any two entire functions with index-pair and , respectively, where , , are all positive integers such that and . If either is not of regular th growth or is not of regular th growth, then
*

*Remark 19. *Corollaries 17 and 18 can be regarded as an extension of the Corollaries of [8, Theorems 2.1 and 2.2].

Corollary 20. *Let be an entire function with index-pair , where , are positive integers with . Then for any entire function ,*(i)* when ,*(ii) * when ,*(iii) * when ,*(iv) * when ,**where is any positive integer with .*

*Remark 21. *The first part of Corollary 20 improves [8, Theorem 2.3].

Corollary 22. *Let be an entire function with index-pair , where , are positive integers with . Then for any entire function ,*(i) * when ,*(ii) * when ,*(iii) * when ,*(iv) * when ,**where is any positive integer such that .*

*Example 23 (relative order between polynomials). *To simplify let us consider any two given natural numbers and and , , so that
Then
Now

*Example 24 (relative order between exponentials of the same order). *Let be any natural number and any positive real number and consider
In this case and are two entire functions with regular growth; thus
Clearly

*Example 25 (relative order between exponential and power function). *Let , be any two natural numbers and consider
Then
Now

When and are any two entire functions with index-pair and , respectively, where , , , are all positive integers such that and , but , the next definition enables studying their relative order.

*Definition 26. *Let and be any two entire functions with index-pair and , respectively, where , , , are all positive integers such that and . If , then the relative th order (resp., relative th lower) of with respect to is defined as*(i)*
If , then the relative th order (resp., relative th lower) of with respect to is defined as*(ii)*

The following result is easy to check.

Theorem 27. *Under the hypothesis of Definition 26, for :**(i)**
and for :**(ii)*

The next example will make an alternative use of Theorem 27.

*Example 28 (relative order between exponentials of different order). *Let
In this case and are entire functions of regular growth and , respectively, with
Now
and by taking and , we get
Obviously, the same limit is achieved if, by using Theorem 27, we consider the quotient

Reciprocally, in order to evaluate and , we would take limits in either
obtaining that

#### 5. Conclusion

The main aim of the paper is to extend and modify the notion of order to relative order of higher dimensions in case of entire functions as the relative order of growth gives a quantitative assessment of how different functions scale each other and to what extent they are self-similar in growth, and in this connection we have established some theorems. In fact, some works on relative order of entire functions and the growth estimates of composite entire functions on the basis of it have been explored in [8–15]. Actually we are trying to generalize the growth properties of composite entire functions on the basis of relative th order and relative th lower order and, analogously, we may also define relative th order of meromorphic functions in order to establish related growth properties, improving the results of [16–18]. For any two positive integers and , we are trying to establish the concepts of relative th type and relative th weak type of entire and meromorphic functions, too, in order to determine the relative growth of two entire or meromorphic functions having the same nonzero finite relative th order or relative th lower order with respect to another entire function, respectively. Moreover, the notion of relative order, relative type, and relative weak type of higher dimensions may also be applied in the field of slowly changing functions and also in case of entire or meromorphic functions of several complex variables.

The results of this paper in connection with Nevanlinna’s value distribution theory of entire functions on the basis of relative th order and relative th lower order may have a wide range of applications in complex dynamics, factorization theory of entire functions of single complex variable, the solution of complex differential equations, and so forth. In fact complex dynamics is a thrust area in modern function theory and it is solely based on the study of fixed points of entire functions as well as the normality of them. For further details in the progress of research in complex dynamics via Nevanlinna’s value distribution theory one may see [19–24]. Factorization theory of entire functions is another branch of applications of Nevanlinna’s theory which actually deals with how a given entire function can be factorized into other simpler entire functions in the sense of composition. Also Nevanlinna’s value distribution theory has immense applications into the study of the properties of the solutions of complex differential equations and is still an active area of research.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.