On the (𝑝,𝑞)th Relative Order Oriented Growth Properties of Entire Functions

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 (𝑝,𝑞)th order which improves the earlier definition of relative (𝑝,𝑞)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 (𝑝,𝑞)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, exp, sin, cos, and so forth are all entire functions. In 1926 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 𝑓 is a polynomial of degree 𝑛 with real or complex coefficients, then the equation 𝑓(𝑧)=0 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 𝑓, the so-called maximum modulus function, denoted by 𝑀_𝑓, is defined on each nonnegative real value 𝑟 as 𝑀_𝑓(𝑟)=max_|𝑧|=𝑟𝑓(𝑧). (1) And given two entire functions 𝑓 and 𝑔 the ratio 𝑀_𝑓(𝑟)/𝑀_𝑔(𝑟) as 𝑟→∞ is called the growth of 𝑓 with respect to 𝑔 in terms of their maximum moduli.

The order of an entire function 𝑓 which is generally used in computational purpose is defined in terms of the growth of 𝑓 with respect to the exponential function as 𝜌_𝑓=limsup_𝑟→∞loglog𝑀_𝑓(𝑟)loglog𝑀_exp𝑧(𝑟)=limsup_𝑟→∞loglog𝑀_𝑓(𝑟)(𝑟).log (2)

Bernal [1, 2] introduced the relative order between two entire functions to avoid comparing growth just with exp𝑧. 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 (𝑝,𝑞), 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 ℎ and whenever the corresponding domain and range allow it, we will use the notation ℎ^[0](𝑥)ℎ=𝑥,_[𝑘](𝑥)=ℎℎ^[𝑘−1](𝑥)for𝑘=1,2,3,… (3) omitting the parenthesis when ℎ happens to be the log or exp function. Taking this into account the order (resp., lower order) of an entire function 𝑓 is given by 𝜌_𝑓=limsup_𝑟→∞log^[2]𝑀_𝑓(𝑟),log𝑟resp.𝜆_𝑓=liminf_𝑟→∞log^[2]𝑀_𝑓(𝑟).log𝑟 (4) Let us recall that Juneja et al. [4] defined the order (𝑝,𝑞) and lower order (𝑝,𝑞) of an entire function 𝑓, respectively, as follows: 𝜌_𝑓𝑝,𝑞=limsup_𝑟→∞log^[𝑝]𝑀_𝑓(𝑟)log^[𝑞]𝑟,𝜆_𝑓𝑝,𝑞=liminf_𝑟→∞log^[𝑝]𝑀_𝑓(𝑟)log^[𝑞]𝑟, (5) where 𝑝,𝑞 are any two positive integers with 𝑝≥𝑞. These definitions extended the generalized order 𝜌_𝑓^[𝑙] and generalized lower order 𝜆_𝑓^[𝑙] of an entire function 𝑓 considered in [5] for each integer 𝑙≥2 since these correspond to the particular case 𝜌_𝑓^[𝑙]=𝜌_𝑓(𝑙,1) and 𝜆_𝑓^[𝑙]=𝜆_𝑓(𝑙,1). Clearly 𝜌_𝑓(2,1)=𝜌_𝑓 and 𝜆_𝑓(2,1)=𝜆_𝑓.

In this connection let us recall that if 0<𝜌_𝑓(𝑝,𝑞)<∞, then the following properties hold: 𝜌_𝑓𝑝−𝑛,𝑞=∞,for𝜌𝑛<𝑝,_𝑓𝑝,𝑞−𝑛=0,for𝜌𝑛<𝑞,_𝑓𝑝+𝑛,𝑞+𝑛=1,for𝑛=1,2,…. (6) Similarly for 0<𝜆_𝑓(𝑝,𝑞)<∞, one can easily verify that 𝜆_𝑓𝑝−𝑛,𝑞=∞,for𝜆𝑛<𝑝,_𝑓𝑝,𝑞−𝑛=0,for𝜆𝑛<𝑞,_𝑓𝑝+𝑛,𝑞+𝑛=1,for𝑛=1,2,…. (7)

Recalling that for any pair of integer numbers 𝑚,𝑛 the Kronecker function is defined by 𝛿_𝑚,𝑛=1 for 𝑚=𝑛 and 𝛿_𝑚,𝑛=0 for 𝑚≠𝑛, the aforementioned properties provide the following definition.

Definition 1 (see [4]). An entire function 𝑓 is said to have index-pair (1,1) if 0<𝜌_𝑓(1,1)<∞. Otherwise, 𝑓 is said to have index-pair (𝑝,𝑞)≠(1,1), 𝑝≥𝑞≥1, if 𝛿_{𝑝−𝑞,0}<𝜌_𝑓(𝑝,𝑞)<∞ and 𝜌_𝑓(𝑝−1,𝑞−1)∉ℝ⁺.

Definition 2 (see [4]). An entire function 𝑓 is said to have lower index-pair (1,1) if 0<𝜆_𝑓(1,1)<∞. Otherwise, 𝑓 is said to have lower index-pair (𝑝,𝑞)≠(1,1), 𝑝≥𝑞≥1, if 𝛿_{𝑝−𝑞,0}<𝜆_𝑓(𝑝,𝑞)<∞ and 𝜆_𝑓(𝑝−1,𝑞−1)∉ℝ⁺.

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

Given a nonconstant entire function 𝑓 defined in the open complex plane ℂ its maximum modulus function 𝑀_𝑓 is strictly increasing and continuous. Hence there exists its inverse function 𝑀_𝑓⁻¹∶(|𝑓(0)|,∞)→(0,∞) with lim_𝑠→∞𝑀_𝑓⁻¹(𝑠)=∞.

Then Bernal [1, 2] introduced the definition of relative order of 𝑓 with respect to 𝑔, denoted by 𝜌_𝑔(𝑓) as follows: 𝜌_𝑔𝑓=inf𝜇>0∶𝑀_𝑓(𝑟)<𝑀_𝑔𝑟^𝜇,∀𝑟>𝑟₀𝜇>0=limsup_𝑟→∞log𝑀_𝑔⁻¹𝑀_𝑓(𝑟).log𝑟 (8)

This definition coincides with the classical one [6] if 𝑔=exp𝑧. Similarly one can define the relative lower order of 𝑓 with respect to 𝑔 denoted by 𝜆_𝑔(𝑓) as 𝜆_𝑔𝑓=liminf_𝑟→∞log𝑀_𝑔⁻¹𝑀_𝑓(𝑟).log𝑟 (9)

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

Definition 3 (see [7]). If 𝑘≥1 is a positive integer, then the 𝑘th generalized relative order of 𝑓 with respect to 𝑔, denoted by 𝜌^𝑘_𝑓(𝑔), is defined by 𝜌^𝑘_𝑔𝑓=inf𝜇>0∶𝑀_𝑓(𝑟)<𝑀_𝑔exp^[𝑘−1]𝑟^𝜇,∀𝑟>𝑟₀𝜇>0=limsup_𝑟→∞log^[𝑘]𝑀_𝑔⁻¹𝑀_𝑓(𝑟).log𝑟 (10) Clearly, 𝜌¹_𝑔(𝑓)=𝜌_𝑔(𝑓) and 𝜌¹_exp𝑧(𝑓)=𝜌_𝑓.

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

Definition 4 (see [3]). Let 𝑝 and 𝑞 be any two positive integers with 𝑝>𝑞. The relative (𝑝,𝑞)th order of 𝑓 with respect to 𝑔 is defined by 𝜌_𝑔^(𝑝,𝑞)𝑓=inf𝜇>0∶𝑀_𝑓(𝑟)<𝑀_𝑔exp^[𝑝−1]𝜇log^[𝑞]𝑟,∀𝑟>𝑟₀𝜇>0=limsup_𝑟→∞log^[𝑝−1]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^[𝑞]𝑟. (11) Then 𝜌^(𝑝,𝑞)_exp𝑧(𝑓)=𝜌_𝑓(𝑝,𝑞) and 𝜌_𝑔^(𝑘+1,1)(𝑓)=𝜌^𝑘_𝑔(𝑓) for any 𝑘≥1.

In this paper we give an alternative definition of (𝑝,𝑞)th relative order 𝜌_𝑔^(𝑝,𝑞)(𝑓) of an entire function 𝑓 with respect to another entire function 𝑔, in the light of index-pair. Our next definition avoids the restriction 𝑝>𝑞 and gives the more natural particular case 𝜌_𝑔^(𝑘,1)(𝑓)=𝜌^𝑘_𝑔(𝑓).

Definition 5. Let 𝑓 and 𝑔 be any two entire functions with index-pair (𝑚,𝑞) and (𝑚,𝑝), respectively, where 𝑝, 𝑞, 𝑚 are positive integers such that 𝑚≥max(𝑝,𝑞). Then the (𝑝,𝑞)th relative order of 𝑓 with respect to 𝑔 is defined as 𝜌_𝑔^(𝑝,𝑞)𝑓=limsup_𝑟→∞log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^[𝑞]𝑟. (12) The (𝑝,𝑞)th relative lower order of 𝑓 with respect to 𝑔 is defined by 𝜆_𝑔^(𝑝,𝑞)𝑓=liminf_𝑟→∞log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^[𝑞]𝑟. (13)

The previous definitions are easily generated as particular cases; for example, if 𝑓 and 𝑔 have got index-pair (𝑚,1) and (𝑚,𝑘), respectively, then Definition 5 reduces to Definition 3. If the entire functions 𝑓 and 𝑔 have the same index-pair (𝑝,1), where 𝑝 is any positive integer, we get the definition of relative order introduced by Bernal [1] and if 𝑔=exp^[𝑚−1]𝑧, then 𝜌_𝑔(𝑓)=𝜌_𝑓^[𝑚] and 𝜌_𝑔^(𝑝,𝑞)(𝑓)=𝜌_𝑓(𝑚,𝑞). And if 𝑓 is an entire function with index-pair (2,1) and 𝑔=exp𝑧, 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 exp). Given any natural number 𝑛, the exponential function 𝑓(𝑧)=exp𝑧^𝑛 has got 𝑀_𝑓(𝑟)=exp𝑟^𝑛, and therefore log^[2]𝑀_𝑓(𝑟)/log𝑟 is constantly equal to 𝑛 and, consequently, 𝜌_𝑓=𝜆_𝑓=𝑛. (14)

Example 7 (generalized order). Given any natural numbers 𝑘, 𝑛, the function 𝑓(𝑧)=exp^[𝑘]𝑧^𝑛 has got 𝑀_𝑓(𝑟)=exp^[𝑘]𝑟^𝑛. Therefore log^[𝑘]𝑀_𝑓(𝑟)/log𝑟 is constantly equal to 𝑛 for each natural 𝑘≥2, following that 𝜌_𝑓^[𝑘+1]=𝜆_𝑓^[𝑘+1]=𝑛. (15) Note that 𝜌_𝑓^[𝑙]=𝜆_𝑓^[𝑙]=+∞ for 2≤𝑙≤𝑘 and 𝜌_𝑓^[𝑙]=𝜆_𝑓^[𝑙]=0 for 𝑙>𝑘+1.

Example 8 (index-pair). Given any four positive integers 𝑘, 𝑛, 𝑝, 𝑞 with 𝑝≥𝑞, then function 𝑓(𝑧)=exp^[𝑘]𝑧^𝑛 generates a constant quotient log^[𝑝]𝑀_𝑓(𝑟)/log^[𝑞]𝑟, and clearly 𝜌_𝑓𝑝,𝑞=𝜆_𝑓𝑝,𝑞=𝑛,for𝑝,𝑞=(𝑘+1,1) (16) but 𝜌_𝑓𝑝,𝑞=𝜆_𝑓=𝑝,𝑞1,∀𝑝,𝑞suchthat𝑝=𝑞+𝑘,𝑞>1,∞,∀𝑝,𝑞suchthat𝑝<𝑞+𝑘,0,∀𝑝,𝑞suchthat𝑝>𝑞+𝑘. (17) Thus 𝑓 is a regular function with growth (𝑘+1,1).

Example 9 (regular function of growth (1,1)). Given any positive integer 𝑛, and nonnull real number 𝑎, the power function 𝑓(𝑧)=𝑎𝑧^𝑛 generates a constant quotient log^[𝑝]𝑀_𝑓(𝑟)/log^[𝑞]𝑟, and clearly 𝜌_𝑓𝑝,𝑞=𝜆_𝑓𝑝,𝑞=𝑛,for𝑝,𝑞=(1,1) (18) but 𝜌_𝑓𝑝,𝑞=𝜆_𝑓=𝑝,𝑞1,∀𝑝,𝑞suchthat𝑝=𝑞,𝑞>1,∞,∀𝑝,𝑞suchthat𝑝<𝑞,0,∀𝑝,𝑞suchthat𝑝>𝑞. (19) Thus 𝑓 is a regular function with growth (1,1).

Example 10 (relative order between functions). From the above examples it follows that given the natural numbers 𝑚, 𝑛 the functions 𝑓(𝑧)=exp𝑧^𝑚,𝑔(𝑧)=exp𝑧^𝑛 (20) are of regular growth (2,1). In order to find their relative order of growth we evaluate log𝑀_𝑔⁻¹𝑀_𝑓(𝑟)=log𝑟loglogexp𝑟^𝑚1/𝑛,log𝑟 (21) which happens to be constant. Its upper and lower limits provide 𝜌_𝑔𝑓=𝜆_𝑔𝑓=𝑚𝑛. (22)

Example 11 (relative order (𝑝,𝑞) between functions). Let 𝑘, 𝑚, 𝑛 be any three positive integers and let 𝑓(𝑧)=exp^[𝑘]𝑧^𝑚 and 𝑔=exp^[𝑘]𝑧^𝑛. Then 𝑓 and 𝑔 are regular functions with (𝑘+1,1)-growth for which 𝜌_𝑓(𝑘+1,1)=𝑚,𝜌_𝑔(𝑘+1,1)=𝑛. (23) In order to find out their (1,1) relative order we evaluate log𝑀_𝑔⁻¹𝑀_𝑓(𝑟)=(log𝑟log1/𝑛)log^[𝑘]exp^[𝑘]𝑟^𝑚1/𝑛,log𝑟 (24) which happens to be constant. By taking limits, we easily get that 𝜌_𝑔^(1,1)𝑓=𝜆_𝑔^(1,1)𝑓=𝑚𝑛. (25)

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 𝑓 and 𝑔 be any two entire functions with index-pair (𝑚,𝑞) and (𝑚,𝑝), respectively, where 𝑝, 𝑞, 𝑚 are all positive integers such that 𝑚≥𝑝 and 𝑚≥𝑞. Then 𝜆_𝑓𝑚,𝑞𝜌_𝑔𝑚,𝑝≤𝜆_𝑔^(𝑝,𝑞)𝑓𝜆≤min^𝑓𝑚,𝑞𝜆_𝑔,𝜌𝑚,𝑝^𝑓𝑚,𝑞𝜌_𝑔𝜆𝑚,𝑝≤max^𝑓𝑚,𝑞𝜆_𝑔,𝜌𝑚,𝑝^𝑓𝑚,𝑞𝜌_𝑔𝑚,𝑝≤𝜌_𝑔^(𝑝,𝑞)𝑓≤𝜌_𝑓𝑚,𝑞𝜆_𝑔.𝑚,𝑝 (26)

Proof. From the definitions of 𝜌_𝑓(𝑚,𝑞) and 𝜆_𝑓(𝑚,𝑞) we have for all sufficiently large values of 𝑟 that 𝑀_𝑓(𝑟)≤exp^[𝑚]𝜌_𝑓𝑚,𝑞+𝜀log^[𝑞]𝑟,𝑀_𝑓(𝑟)≥exp^[𝑚]𝜆_𝑓𝑚,𝑞−𝜀log^[𝑞]𝑟 (27)(28) and also for a sequence of values of 𝑟 tending to infinity we get that 𝑀_𝑓(𝑟)≥exp^[𝑚]𝜌_𝑓𝑚,𝑞−𝜀log^[𝑞]𝑟,𝑀_𝑓(𝑟)≤exp^[𝑚]𝜆_𝑓𝑚,𝑞+𝜀log^[𝑞]𝑟. (29)(30) Similarly from the definitions of 𝜌_𝑔(𝑚,𝑝) and 𝜆_𝑓(𝑚,𝑞) it follows for all sufficiently large values of 𝑟 that 𝑀_𝑔(𝑟)≤exp^[𝑚]𝜌_𝑔𝑚,𝑝+𝜀log^[𝑝]𝑟i.e.,𝑟≤𝑀_𝑔⁻¹exp^[𝑚]𝜌_𝑔𝑚,𝑝+𝜀log^[𝑝]𝑟i.e.,𝑀_𝑔⁻¹(𝑟)≥exp^[𝑝]log^[𝑚]𝑟𝜌_𝑔,𝑚,𝑝+𝜀𝑀_𝑔(𝑟)≥exp^[𝑚]𝜆_𝑔𝑚,𝑝−𝜀log^[𝑝]𝑟i.e.,𝑀_𝑔⁻¹(𝑟)≤exp^[𝑝]log^[𝑚]𝑟𝜆_𝑔𝑚,𝑝−𝜀 (31)(32) and for a sequence of values of 𝑟 tending to infinity we obtain that 𝑀_𝑔(𝑟)≥exp^[𝑚]𝜌_𝑔𝑚,𝑝−𝜀log^[𝑝]𝑟i.e.,𝑀_𝑔⁻¹(𝑟)≤exp^[𝑝]log^[𝑚]𝑟𝜌_𝑔,𝑚,𝑝−𝜀𝑀_𝑔(𝑟)≤exp^[𝑚]𝜆_𝑔𝑚,𝑝+𝜀log^[𝑝]𝑟i.e.,𝑀_𝑔⁻¹(𝑟)≥exp^[𝑝]log^[𝑚]𝑟𝜆_𝑔.𝑚,𝑝+𝜀 (33)(34) Now from (29) and in view of (31), for a sequence of values of 𝑟 tending to infinity we get that log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)≥log^[𝑝]𝑀_𝑔⁻¹exp^[𝑚]𝜌_𝑓𝑚,𝑞−𝜀log^[𝑞]𝑟i.e.,log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)≥log^[𝑝]exp^[𝑝]log^[𝑚]exp^[𝑚]𝜌_𝑓𝑚,𝑞−𝜀log^[𝑞]𝑟𝜌_𝑔=𝜌𝑚,𝑝+𝜀_𝑓𝑚,𝑞−𝜀𝜌_𝑔𝑚,𝑝+𝜀log^[𝑞]𝑟i.e.,log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^[𝑞]𝑟≥𝜌_𝑓𝑚,𝑞−𝜀𝜌_𝑔.𝑚,𝑝+𝜀 (35) As 𝜀(>0) is arbitrary, it follows that 𝜌_𝑔^(𝑝,𝑞)𝑓≥𝜌_𝑓𝑚,𝑞𝜌_𝑔.𝑚,𝑝 (36) Analogously from (28) and in view of (34) it follows for a sequence of values of 𝑟 tending to infinity that log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)≥log^[𝑝]𝑀_𝑔⁻¹exp^[𝑚]𝜆_𝑓𝑚,𝑞−𝜀log^[𝑞]𝑟i.e.,log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)≥log^[𝑝]exp^[𝑝]log^[𝑚]exp^[𝑚]𝜆_𝑓𝑚,𝑞−𝜀log^[𝑞]𝑟𝜆_𝑔=𝜆𝑚,𝑝+𝜀_𝑓𝑚,𝑞−𝜀𝜆_𝑔𝑚,𝑝+𝜀log^[𝑞]𝑟i.e.,log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^[𝑞]𝑟≥𝜆_𝑓𝑚,𝑞−𝜀𝜆_𝑔.𝑚,𝑝+𝜀 (37) Since 𝜀(>0) is arbitrary, we get from above that 𝜌_𝑔^(𝑝,𝑞)𝑓≥𝜆_𝑓𝑚,𝑞𝜆_𝑔.𝑚,𝑝 (38) Again in view of (32) we have from (27) for all sufficiently large values of 𝑟 that log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)≤log^[𝑝]𝑀_𝑔⁻¹exp^[𝑚]𝜌_𝑓𝑚,𝑞+𝜀log^[𝑞]𝑟i.e.,log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)≤log^[𝑝]exp^[𝑝]log^[𝑚]exp^[𝑚]𝜌_𝑓𝑚,𝑞+𝜀log^[𝑞]𝑟𝜆_𝑔=𝜌𝑚,𝑝−𝜀_𝑓𝑚,𝑞+𝜀𝜆_𝑔𝑚,𝑝−𝜀log^[𝑞]𝑟i.e.,log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^[𝑞]𝑟≤𝜌_𝑓𝑚,𝑞+𝜀𝜆_𝑔.𝑚,𝑝−𝜀 (39) Since 𝜀(>0) is arbitrary, we obtain that 𝜌_𝑔^(𝑝,𝑞)𝑓≤𝜌_𝑓𝑚,𝑞𝜆_𝑔.𝑚,𝑝 (40) Again from (28) and in view of (31) with the same reasoning we get that 𝜆_𝑔^(𝑝,𝑞)𝑓≥𝜆_𝑓𝑚,𝑞𝜌_𝑔.𝑚,𝑝 (41) Also in view of (33), we get from (27) for a sequence of values of 𝑟 tending to infinity that log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)≤log^[𝑝]𝑀_𝑔⁻¹exp^[𝑚]𝜌_𝑓𝑚,𝑞+𝜀log^[𝑞]𝑟i.e.,log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)≤log^[𝑝]exp^[𝑝]log^[𝑚]exp^[𝑚]𝜌_𝑓𝑚,𝑞+𝜀log^[𝑞]𝑟𝜌_𝑔=𝜌𝑚,𝑝−𝜀_𝑓𝑚,𝑞+𝜀𝜌_𝑔𝑚,𝑝−𝜀log^[𝑞]𝑟i.e.,log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^[𝑞]𝑟≤𝜌_𝑓𝑚,𝑞+𝜀𝜌_𝑔.𝑚,𝑝−𝜀 (42) Since 𝜀(>0) is arbitrary, we get from above that 𝜆_𝑔^(𝑝,𝑞)𝑓≤𝜌_𝑓𝑚,𝑞𝜌_𝑔.𝑚,𝑝 (43) Similarly from (30) and in view of (32) it follows for a sequence of values of 𝑟 tending to infinity that log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)≤log^[𝑝]𝑀_𝑔⁻¹exp^[𝑚]𝜆_𝑓𝑚,𝑞+𝜀log^[𝑞]𝑟i.e.,log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)≤log^[𝑝]exp^[𝑝]log^[𝑚]exp^[𝑚]𝜆_𝑓𝑚,𝑞+𝜀log^[𝑞]𝑟𝜆_𝑔=𝜆𝑚,𝑝−𝜀_𝑓𝑚,𝑞+𝜀𝜆_𝑔𝑚,𝑝−𝜀log^[𝑞]𝑟i.e.,log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^[𝑞]𝑟≤𝜆_𝑓𝑚,𝑞+𝜀𝜆_𝑔.𝑚,𝑝−𝜀 (44) As 𝜀(>0) is arbitrary, from above we obtain that 𝜆_𝑔^(𝑝,𝑞)𝑓≤𝜆_𝑓𝑚,𝑞𝜆_𝑔.𝑚,𝑝 (45) The theorem follows from (36), (38), (40), (41), (43), and (45).

Corollary 13. Let 𝑓 be an entire function with index-pair (𝑚,𝑞) and let 𝑔 be an entire of regular (𝑚,𝑝)-growth, where 𝑝, 𝑞, 𝑚 are all positive integers such that 𝑚≥𝑝 and 𝑚≥𝑞. Then 𝜆_𝑔^(𝑝,𝑞)𝑓=𝜆_𝑓𝑚,𝑞𝜌_𝑔𝑚,𝑝,𝜌_𝑔^(𝑝,𝑞)𝑓=𝜌_𝑓𝑚,𝑞𝜌_𝑔.𝑚,𝑝 (46) In addition, if 𝜌_𝑓(𝑚,𝑞)=𝜌_𝑔(𝑚,𝑝), then 𝜌_𝑔^(𝑝,𝑞)𝑓=𝜆_𝑓^(𝑞,𝑝)𝑔=1. (47)

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

Corollary 15. Let 𝑓 and 𝑔 be any two entire functions with regular (𝑚,𝑞)-growth and regular (𝑚,𝑝)-growth, respectively, where 𝑝, 𝑞, 𝑚 are all positive integers with 𝑚≥max{𝑝,𝑞}. Then 𝜆_𝑔^(𝑝,𝑞)𝑓=𝜌_𝑔^(𝑝,𝑞)𝑓=𝜌_𝑓𝑚,𝑞𝜌_𝑔.𝑚,𝑝 (48)

Corollary 16. Let 𝑓 and 𝑔 be any two entire functions with regular (𝑚,𝑞)th growth and regular (𝑚,𝑝)th growth, respectively, where 𝑝, 𝑞, 𝑚 are all positive integers with 𝑚≥𝑝 and 𝑚≥𝑞. Also suppose that 𝜌_𝑓(𝑚,𝑞)=𝜌_𝑔(𝑚,𝑝). Then 𝜆_𝑔^(𝑝,𝑞)𝑓=𝜌_𝑔^(𝑝,𝑞)𝑓=𝜆_𝑓^(𝑞,𝑝)𝑔=𝜌_𝑓^(𝑞,𝑝)𝑔=1. (49)

Corollary 17. Let 𝑓 and 𝑔 be any two entire functions with regular growth (𝑚,𝑞) and (𝑚,𝑝), respectively, where 𝑝, 𝑞, 𝑚 are all positive integers such that 𝑚≥max{𝑝,𝑞}. Then 𝜌_𝑔^(𝑝,𝑞)𝑓⋅𝜌_𝑓^(𝑞,𝑝)𝑔=𝜆_𝑔^(𝑝,𝑞)𝑓⋅𝜆_𝑓^(𝑞,𝑝)𝑔=1. (50)

Corollary 18. Let 𝑓 and 𝑔 be any two entire functions with index-pair (𝑚,𝑞) and (𝑚,𝑝), respectively, where 𝑝, 𝑞, 𝑚 are all positive integers such that 𝑚≥𝑝 and 𝑚≥𝑞. If either 𝑓 is not of regular (𝑚,𝑞)th growth or 𝑔 is not of regular (𝑚,𝑝)th growth, then 𝜆_𝑔^(𝑝,𝑞)𝑓⋅𝜆_𝑓^(𝑞,𝑝)𝑔<1<𝜌_𝑔^(𝑝,𝑞)𝑓⋅𝜌_𝑓^(𝑞,𝑝)𝑔. (51)

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 𝑓 be an entire function with index-pair (𝑚,𝑞), where 𝑚, 𝑞 are positive integers with 𝑚≥𝑞. Then for any entire function 𝑔,(i)𝜆_𝑔^(𝑝,𝑞)(𝑓)=∞ when 𝜌_𝑔(𝑚,𝑝)=0,(ii) 𝜌_𝑔^(𝑝,𝑞)(𝑓)=∞ when 𝜆_𝑔(𝑚,𝑝)=0,(iii) 𝜆_𝑔^(𝑝,𝑞)(𝑓)=0 when 𝜌_𝑔(𝑚,𝑝)=∞,(iv) 𝜌_𝑔^(𝑝,𝑞)(𝑓)=0 when 𝜆_𝑔(𝑚,𝑝)=∞,where 𝑝 is any positive integer with 𝑚≥𝑝.

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

Corollary 22. Let 𝑔 be an entire function with index-pair (𝑚,𝑝), where 𝑚, 𝑝 are positive integers with 𝑚≥𝑝. Then for any entire function 𝑓,(i) 𝜌_𝑔^(𝑝,𝑞)(𝑓)=0 when 𝜌_𝑓(𝑚,𝑞)=0,(ii) 𝜆_𝑔^(𝑝,𝑞)(𝑓)=0 when 𝜆_𝑓(𝑚,𝑞)=0,(iii) 𝜌_𝑔^(𝑝,𝑞)(𝑓)=∞ when 𝜌_𝑓(𝑚,𝑞)=∞,(iv) 𝜆_𝑔^(𝑝,𝑞)(𝑓)=∞ when 𝜆_𝑓(𝑚,𝑞)=∞,where 𝑞 is any positive integer such that 𝑚≥𝑞.

Example 23 (relative order between polynomials). To simplify let us consider any two given natural numbers 𝑚 and 𝑛 and 𝑎∈ℝ, 𝑎≠0, so that 𝑓(𝑧)=𝑧^𝑚,𝑔(𝑧)=𝑎𝑧^𝑛. (52) Then 𝜌_𝑓(1,1)=𝜆_𝑓(1,1)=𝑚,𝜌_𝑔(1,1)=𝜆_𝑔(1,1)=𝑛. (53) Now 𝜌_𝑔^(1,1)𝑓=𝜆_𝑔^(1,1)𝑓=𝜌_𝑓(1,1)𝜌_𝑔()=𝑚1,1𝑛. (54)

Example 24 (relative order between exponentials of the same order). Let 𝑛 be any natural number and 𝑎 any positive real number and consider 𝑓(𝑧)=exp𝑧^𝑛,𝑔(𝑧)=exp(𝑎𝑧)^𝑛. (55) In this case 𝑓 and 𝑔 are two entire functions with regular (2,1) growth; thus 𝜆_𝑔^(1,1)𝑓=𝜌_𝑔^(1,1)𝑓=𝜌_𝑓(2,1)𝜌_𝑔()=𝑛2,1𝑛=1. (56) Clearly 𝜌_𝑓^(1,1)𝑔=𝜆_𝑓^(1,1)𝑔=1. (57)

Example 25 (relative order between exponential and power function). Let 𝑚, 𝑛 be any two natural numbers and consider 𝑓=exp𝑧^𝑚,𝑔=𝑧^𝑛. (58) Then 𝜌_𝑓=𝜆_𝑓=𝑚,𝜌_𝑔=𝜆_𝑔=0. (59) Now 𝜌_𝑔^(1,1)𝑓=𝜆_𝑔^(1,1)𝑓𝜌=∞,_𝑓(1,1)𝑔=𝜆_𝑓^(1,1)𝑔=0. (60)

When 𝑓 and 𝑔 are any two entire functions with index-pair (𝑚,𝑞) and (𝑛,𝑝), respectively, where 𝑝, 𝑞, 𝑚, 𝑛 are all positive integers such that 𝑚≥𝑞 and 𝑛≥𝑝, but 𝑚≠𝑛, the next definition enables studying their relative order.

Definition 26. Let 𝑓 and 𝑔 be any two entire functions with index-pair (𝑚,𝑞) and (𝑛,𝑝), respectively, where 𝑝, 𝑞, 𝑚, 𝑛 are all positive integers such that 𝑚≥𝑞 and 𝑛≥𝑝. If 𝑚>𝑛, then the relative (𝑝+𝑚−𝑛,𝑞)th order (resp., relative (𝑝+𝑚−𝑛,𝑞)th lower) of 𝑓 with respect to 𝑔 is defined as
(i)𝜌_𝑔^{(𝑝+𝑚−𝑛,𝑞)}𝑓=limsup_𝑟→∞log^{[𝑝+𝑚−𝑛]}𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^[𝑞]𝑟,resp.𝜆_𝑔^{(𝑝+𝑚−𝑛,𝑞)}𝑓=liminf_𝑟→∞log^{[𝑝+𝑚−𝑛]}𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^[𝑞]𝑟. (61) If 𝑚<𝑛, then the relative (𝑝,𝑞+𝑛−𝑚)th order (resp., relative (𝑝,𝑞+𝑛−𝑚)th lower) of 𝑓 with respect to 𝑔 is defined as
(ii)𝜌_𝑔^{(𝑝,𝑞+𝑛−𝑚)}𝑓=limsup_𝑟→∞log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^{[𝑞+𝑛−𝑚]}𝑟,resp.𝜆_𝑔^{(𝑝,𝑞+𝑛−𝑚)}𝑓=liminf_𝑟→∞log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^{[𝑞+𝑛−𝑚]}𝑟. (62)

The following result is easy to check.

Theorem 27. Under the hypothesis of Definition 26, for 𝑚>𝑛:
(i)𝜌_𝑔^{(𝑝+𝑚−𝑛,𝑞)}𝑓=limsup_𝑟→∞log^[𝑚]𝑀_𝑓(𝑟)log^[𝑞]𝑟,𝜆_𝑔^{(𝑝+𝑚−𝑛,𝑞)}𝑓=liminf_𝑟→∞log^[𝑚]𝑀_𝑓(𝑟)log^[𝑞]𝑟, (63) and for 𝑚<𝑛:
(ii)𝜌_𝑔^{(𝑝,𝑞+𝑛−𝑚)}𝑓=limsup_𝑟→∞log^[𝑝]𝑟log^[𝑛]𝑀_𝑔(𝑟),𝜆_𝑔^{(𝑝,𝑞+𝑛−𝑚)}𝑓=liminf_𝑟→∞log^[𝑝]𝑟log^[𝑛]𝑀_𝑔(𝑟). (64)

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

Example 28 (relative order between exponentials of different order). Let 𝑓(𝑧)=exp^[27]𝑧⁵,𝑔(𝑧)=exp^[50]𝑧¹⁷. (65) In this case 𝑓 and 𝑔 are entire functions of regular growth (𝑚,𝑝)=(28,1) and (𝑛,𝑞)=(51,1), respectively, with 𝜌_𝑓(28,1)=𝜆_𝑓(28,1)=5,𝜌_𝑔(51,1)=𝜆_𝑔(51,1)=17. (66) Now log^[𝑝]𝑀_𝑔⁻¹𝑀_𝑓(𝑟)log^{[𝑞+𝑛−𝑚]}𝑟=loglog^[50]exp^[27]𝑟^51/17log^[24]𝑟 (67) and by taking limsup and liminf, we get 𝜌_𝑔^{(𝑝,𝑞+𝑛−𝑚)}𝑓=117=𝜆_𝑔^{(𝑝,𝑞+𝑛−𝑚)}𝑓. (68) Obviously, the same limit is achieved if, by using Theorem 27, we consider the quotient log^[𝑝]𝑟log^[𝑛]𝑀_𝑔(𝑟)=log𝑟log^[51]𝑀_𝑔(𝑟). (69)
Reciprocally, in order to evaluate 𝜌_𝑓^{(𝑝+𝑚−𝑛,𝑞)}(𝑔) and 𝜆_𝑓^{(𝑝+𝑚−𝑛,𝑞)}(𝑔), we would take limits in either log^[24]𝑀_𝑓⁻¹𝑀_𝑔(𝑟)=log𝑟log^[24]log^[27]exp^[50]𝑟^171/5log𝑟orlog^[51]𝑀_𝑔(𝑟),log𝑟 (70) obtaining that 𝜌_𝑓^{(𝑝+𝑚−𝑛,𝑞)}𝑔=17=𝜆_𝑓^{(𝑝+𝑚−𝑛,𝑞)}𝑔. (71)

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 (𝑝,𝑞)th order and relative (𝑝,𝑞)th lower order and, analogously, we may also define relative (𝑝,𝑞)th order of meromorphic functions in order to establish related growth properties, improving the results of [16–18]. For any two positive integers 𝑝 and 𝑞, we are trying to establish the concepts of relative (𝑝,𝑞)th type and relative (𝑝,𝑞)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 (𝑝,𝑞)th order or relative (𝑝,𝑞)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 (𝑝,𝑞)th order and relative (𝑝,𝑞)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.