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

Authors: Luis Manuel Sánchez Ruiz; Sanjib Kumar Datta; Tanmay Biswas; Golok Kumar Mondal
Journal: Abstract and Applied Analysis
 (2014)

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## 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 integersp 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.


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## Body

## 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 functionf, the so-called maximum modulus function, denoted by Mf, is defined on each nonnegative real value r as
(1)Mf(r)=max⁡|z|=r|f(z)|.
And given two entire functions f and g the ratio Mf(r)/Mg(r) as r→∞ is called the growth of f with respect to g in terms of their maximum moduli.Theorder 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
(2)ρf=lim⁡sup⁡r→∞log⁡log⁡Mf(r)log⁡log⁡Mexp⁡z(r)=lim⁡sup⁡r→∞log⁡log⁡Mf(r)log⁡(r).Bernal [1, 2] introduced the relative order between two entire functions to avoid comparing growth just with exp⁡z. 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 functionh and whenever the corresponding domain and range allow it, we will use the notation
(3)h[0](x)=x,h[k](x)=h(h[k-1](x))fork=1,2,3,…
omitting the parenthesis when h happens to be the log⁡ or exp⁡ function. Taking this into account theorder (resp.,lower order) of an entire function f is given by
(4)ρf=lim⁡sup⁡r→∞log⁡[2]Mf(r)log⁡r,(resp.λf=lim⁡inf⁡r→∞log⁡[2]Mf(r)log⁡r).
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:
(5)ρf(p,q)=lim⁡sup⁡r→∞log⁡[p]Mf(r)log⁡[q]r,λf(p,q)=lim⁡inf⁡r→∞log⁡[p]Mf(r)log⁡[q]r,
where p,q are any two positive integers with p≥q. These definitions extended the generalized order ρf[l] and generalized lower order λf[l] of an entire function f considered in [5] for each integer l≥2 since these correspond to the particular case ρf[l]=ρf(l,1) and λf[l]=λf(l,1). Clearly ρf(2,1)=ρf and λf(2,1)=λf.In this connection let us recall that if0<ρf(p,q)<∞, then the following properties hold:
(6)ρf(p-n,q)=∞,forn<p,ρf(p,q-n)=0,forn<q,ρf(p+n,q+n)=1,forn=1,2,….
Similarly for 0<λf(p,q)<∞, one can easily verify that
(7)λf(p-n,q)=∞,forn<p,λf(p,q-n)=0,forn<q,λf(p+n,q+n)=1,forn=1,2,….Recalling that for any pair of integer numbersm,n the Kronecker function is defined by δm,n=1 for m=n and δm,n=0 for m≠n, the aforementioned properties provide the following definition.Definition 1 (see [4]).
An entire functionf is said to have index-pair (1,1) if 0<ρf(1,1)<∞. Otherwise, f is said to have index-pair (p,q)≠(1,1), p≥q≥1, if δp-q,0<ρf(p,q)<∞ and ρf(p-1,q-1)∉ℝ+.Definition 2 (see [4]).
An entire functionf is said to have lower index-pair (1,1) if 0<λf(1,1)<∞. Otherwise, f is said to have lower index-pair (p,q)≠(1,1), p≥q≥1, if δp-q,0<λf(p,q)<∞ and λf(p-1,q-1)∉ℝ+.An entire functionf 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 functionf defined in the open complex plane ℂ its maximum modulus function Mf is strictly increasing and continuous. Hence there exists its inverse function Mf-1:(|f(0)|,∞)→(0,∞) with lim⁡s→∞Mf-1(s)=∞.Then Bernal [1, 2] introduced the definition of relative order of f with respect to g, denoted by ρg(f) as follows:
(8)ρg(f)=inf⁡{μ>0:Mf(r)<Mg(rμ),∀r>r0(μ)>0}=lim⁡sup⁡r→∞log⁡Mg-1Mf(r)log⁡r.This definition coincides with the classical one [6] if g=exp⁡z. Similarly one can define the relative lower order of f with respect to g denoted by λg(f) as
(9)λg(f)=lim⁡inf⁡r→∞log⁡Mg-1Mf(r)log⁡r.Lahiri and Banerjee [7] gave a more generalized concept of relative order in the following way.Definition 3 (see [7]).
Ifk≥1 is a positive integer, then the kth generalized relative order of f with respect to g, denoted by ρfk(g), is defined by
(10)ρgk(f)=inf⁡{μ>0:Mf(r)<Mg(exp⁡[k-1]rμ),∀r>r0(μ)>0}=lim⁡sup⁡r→∞log⁡[k]Mg-1Mf(r)log⁡r.
Clearly, ρg1(f)=ρg(f) and ρexp⁡z1(f)=ρf.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]).
Letp 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
(11)ρg(p,q)(f)=inf⁡{μ>0:Mf(r)<Mg(exp⁡[p-1](μlog⁡[q]r)),∀r>r0(μ)>0(exp⁡[p-1](μlog⁡[q]r))}=lim⁡sup⁡r→∞log⁡[p-1]Mg-1Mf(r)log⁡[q]r.
Then ρexp⁡z(p,q)(f)=ρf(p,q) and ρg(k+1,1)(f)=ρgk(f) for any k≥1.In this paper we give an alternative definition of(p,q)th relative order ρg(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 ρg(k,1)(f)=ρgk(f).Definition 5.
Letf 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
(12)ρg(p,q)(f)=lim⁡sup⁡r→∞log⁡[p]Mg-1Mf(r)log⁡[q]r.
The (p,q)th relative lower order of f with respect to g is defined by
(13)λg(p,q)(f)=lim⁡inf⁡r→∞log⁡[p]Mg-1Mf(r)log⁡[q]r.The previous definitions are easily generated as particular cases; for example, iff 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 ρg(f)=ρf[m] and ρg(p,q)(f)=ρf(m,q). And if f is an entire function with index-pair (2,1) and g=exp⁡z, 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 ofexp⁡).
Given any natural numbern, the exponential function f(z)=exp⁡zn has got Mf(r)=exp⁡rn, and therefore log⁡[2]Mf(r)/log⁡r is constantly equal to n and, consequently,
(14)ρf=λf=n.Example 7 (generalized order).
Given any natural numbersk, n, the function f(z)=exp⁡[k]zn has got Mf(r)=exp⁡[k]rn. Therefore log⁡[k]Mf(r)/log⁡r is constantly equal to n for each natural k≥2, following that
(15)ρf[k+1]=λf[k+1]=n.
Note that ρf[l]=λf[l]=+∞ for 2≤l≤k and ρf[l]=λf[l]=0 for l>k+1.Example 8 (index-pair).
Given any four positive integersk, n, p, q with p≥q, then function f(z)=exp⁡[k]zn generates a constant quotient log⁡[p]Mf(r)/log⁡[q]r, and clearly
(16)ρf(p,q)=λf(p,q)=n,for(p,q)=(k+1,1)
but
(17)ρf(p,q)=λf(p,q)={1,∀(p,q)suchthatp=q+k,q>1,∞,∀(p,q)suchthatp<q+k,0,∀(p,q)suchthatp>q+k.
Thus f is a regular function with growth (k+1,1).Example 9 (regular function of growth (1,1)).
Given any positive integern, and nonnull real number a, the power function f(z)=azn generates a constant quotient log⁡[p]Mf(r)/log⁡[q]r, and clearly
(18)ρf(p,q)=λf(p,q)=n,for(p,q)=(1,1)
but
(19)ρf(p,q)=λf(p,q)={1,∀(p,q)suchthatp=q,q>1,∞,∀(p,q)suchthatp<q,0,∀(p,q)suchthatp>q.
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 numbersm, n the functions
(20)f(z)=exp⁡zm,g(z)=exp⁡zn
are of regular growth (2,1). In order to find their relative order of growth we evaluate
(21)log⁡Mg-1Mf(r)log⁡r=log⁡[log⁡[exp⁡rm]]1/nlog⁡r,
which happens to be constant. Its upper and lower limits provide
(22)ρg(f)=λg(f)=mn.Example 11 (relative order(p,q) between functions).
Letk, m, n be any three positive integers and let f(z)=exp⁡[k]zm and g=exp⁡[k]zn. Then f and g are regular functions with (k+1,1)-growth for which
(23)ρf(k+1,1)=m,ρg(k+1,1)=n.
In order to find out their (1,1) relative order we evaluate
(24)log⁡Mg-1Mf(r)log⁡r=log⁡⁡(1/n){log⁡[k](exp⁡[k]rm)}1/nlog⁡r,
which happens to be constant. By taking limits, we easily get that
(25)ρg(1,1)(f)=λg(1,1)(f)=mn.The orders obtained in the last two examples will be easy consequences of the results given in Section4.


## 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.
Letf 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
(26)λf(m,q)ρg(m,p)≤λg(p,q)(f)≤min⁡{λf(m,q)λg(m,p),ρf(m,q)ρg(m,p)}≤max⁡{λf(m,q)λg(m,p),ρf(m,q)ρg(m,p)}≤ρg(p,q)(f)≤ρf(m,q)λg(m,p).Proof.
From the definitions ofρf(m,q) and λf(m,q) we have for all sufficiently large values of r that
(27)Mf(r)≤exp⁡[m]{(ρf(m,q)+ε)log⁡[q]r},(28)Mf(r)≥exp⁡[m]{(λf(m,q)-ε)log⁡[q]r}
and also for a sequence of values of r tending to infinity we get that
(29)Mf(r)≥exp⁡[m]{(ρf(m,q)-ε)log⁡[q]r},(30)Mf(r)≤exp⁡[m]{(λf(m,q)+ε)log⁡[q]r}.
Similarly from the definitions of ρg(m,p) and λf(m,q) it follows for all sufficiently large values of r that
(31)Mg(r)≤exp⁡[m]{(ρg(m,p)+ε)log⁡[p]r}i.e.,r≤Mg-1[exp⁡[m]{(ρg(m,p)+ε)log⁡[p]r}]i.e.,Mg-1(r)≥exp⁡[p][log⁡[m]r(ρg(m,p)+ε)],(32)Mg(r)≥exp⁡[m]{(λg(m,p)-ε)log⁡[p]r}i.e.,Mg-1(r)≤exp⁡[p][log⁡[m]r(λg(m,p)-ε)]
and for a sequence of values of r tending to infinity we obtain that
(33)Mg(r)≥exp⁡[m]{(ρg(m,p)-ε)log⁡[p]r}i.e.,Mg-1(r)≤exp⁡[p][log⁡[m]r(ρg(m,p)-ε)],(34)Mg(r)≤exp⁡[m]{(λg(m,p)+ε)log⁡[p]r}i.e.,Mg-1(r)≥exp⁡[p][log⁡[m]r(λg(m,p)+ε)].
Now from (29) and in view of (31), for a sequence of values of r tending to infinity we get that
(35)log⁡[p]Mg-1Mf(r)≥log⁡[p]Mg-1[exp⁡[m]{(ρf(m,q)-ε)log⁡[q]r}]i.e.,log⁡[p]Mg-1Mf(r)≥log⁡[p]exp⁡[p][log⁡[m]exp⁡[m]{(ρf(m,q)-ε)log⁡[q]r}(ρg(m,p)+ε)]=(ρf(m,q)-ε)(ρg(m,p)+ε)log⁡[q]ri.e.,log⁡[p]Mg-1Mf(r)log⁡[q]r≥(ρf(m,q)-ε)(ρg(m,p)+ε).
As ε(>0) is arbitrary, it follows that
(36)ρg(p,q)(f)≥ρf(m,q)ρg(m,p).
Analogously from (28) and in view of (34) it follows for a sequence of values of r tending to infinity that
(37)log⁡[p]Mg-1Mf(r)≥log⁡[p]Mg-1[exp⁡[m]{(λf(m,q)-ε)log⁡[q]r}]i.e.,log⁡[p]Mg-1Mf(r)≥log⁡[p]exp⁡[p][log⁡[m]exp⁡[m]{(λf(m,q)-ε)log⁡[q]r}(λg(m,p)+ε)]=(λf(m,q)-ε)(λg(m,p)+ε)log⁡[q]ri.e.,log⁡[p]Mg-1Mf(r)log⁡[q]r≥(λf(m,q)-ε)(λg(m,p)+ε).
Since ε(>0) is arbitrary, we get from above that
(38)ρg(p,q)(f)≥λf(m,q)λg(m,p).
Again in view of (32) we have from (27) for all sufficiently large values of r that
(39)log⁡[p]Mg-1Mf(r)≤log⁡[p]Mg-1[exp⁡[m]{(ρf(m,q)+ε)log⁡[q]r}]i.e.,log⁡[p]Mg-1Mf(r)≤log⁡[p]exp⁡[p][log⁡[m]exp⁡[m]{(ρf(m,q)+ε)log⁡[q]r}(λg(m,p)-ε)]=(ρf(m,q)+ε)(λg(m,p)-ε)log⁡[q]ri.e.,log⁡[p]Mg-1Mf(r)log⁡[q]r≤(ρf(m,q)+ε)(λg(m,p)-ε).
Since ε(>0) is arbitrary, we obtain that
(40)ρg(p,q)(f)≤ρf(m,q)λg(m,p).
Again from (28) and in view of (31) with the same reasoning we get that
(41)λg(p,q)(f)≥λf(m,q)ρg(m,p).
Also in view of (33), we get from (27) for a sequence of values of r tending to infinity that
(42)log⁡[p]Mg-1Mf(r)≤log⁡[p]Mg-1[exp⁡[m]{(ρf(m,q)+ε)log⁡[q]r}]i.e.,log⁡[p]Mg-1Mf(r)≤log⁡[p]exp⁡[p][log⁡[m]exp⁡[m]{(ρf(m,q)+ε)log⁡[q]r}(ρg(m,p)-ε)]=(ρf(m,q)+ε)(ρg(m,p)-ε)log⁡[q]ri.e.,log⁡[p]Mg-1Mf(r)log⁡[q]r≤(ρf(m,q)+ε)(ρg(m,p)-ε).
Since ε(>0) is arbitrary, we get from above that
(43)λg(p,q)(f)≤ρf(m,q)ρg(m,p).
Similarly from (30) and in view of (32) it follows for a sequence of values of r tending to infinity that
(44)log⁡[p]Mg-1Mf(r)≤log⁡[p]Mg-1[exp⁡[m]{(λf(m,q)+ε)log⁡[q]r}]i.e.,log⁡[p]Mg-1Mf(r)≤log⁡[p]exp⁡[p][log⁡[m]exp⁡[m]{(λf(m,q)+ε)log⁡[q]r}(λg(m,p)-ε)]=(λf(m,q)+ε)(λg(m,p)-ε)log⁡[q]ri.e.,log⁡[p]Mg-1Mf(r)log⁡[q]r≤(λf(m,q)+ε)(λg(m,p)-ε).
As ε(>0) is arbitrary, from above we obtain that
(45)λg(p,q)(f)≤λf(m,q)λg(m,p).
The theorem follows from (36), (38), (40), (41), (43), and (45).Corollary 13.
Letf 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
(46)λg(p,q)(f)=λf(m,q)ρg(m,p),ρg(p,q)(f)=ρf(m,q)ρg(m,p).
In addition, if ρf(m,q)=ρg(m,p), then
(47)ρg(p,q)(f)=λf(q,p)(g)=1.Remark 14.
The first part of Corollary13 improves [8, Theorem 2.1 and Theorem 2.2].Corollary 15.
Letf 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
(48)λg(p,q)(f)=ρg(p,q)(f)=ρf(m,q)ρg(m,p).Corollary 16.
Letf 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 ρf(m,q)=ρg(m,p). Then
(49)λg(p,q)(f)=ρg(p,q)(f)=λf(q,p)(g)=ρf(q,p)(g)=1.Corollary 17.
Letf 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
(50)ρg(p,q)(f)·ρf(q,p)(g)=λg(p,q)(f)·λf(q,p)(g)=1.Corollary 18.
Letf 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
(51)λg(p,q)(f)·λf(q,p)(g)<1<ρg(p,q)(f)·ρf(q,p)(g).Remark 19.
Corollaries17 and 18 can be regarded as an extension of the Corollaries of [8, Theorems 2.1 and 2.2].Corollary 20.
Letf 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)
λg(p,q)(f)=∞ when ρg(m,p)=0,(ii)
ρg(p,q)(f)=∞ when λg(m,p)=0,(iii)
λg(p,q)(f)=0 when ρg(m,p)=∞,(iv)
ρg(p,q)(f)=0 when λg(m,p)=∞,where p is any positive integer with m≥p.Remark 21.
The first part of Corollary20 improves [8, Theorem 2.3].Corollary 22.
Letg 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)
ρg(p,q)(f)=0 when ρf(m,q)=0,(ii)
λg(p,q)(f)=0 when λf(m,q)=0,(iii)
ρg(p,q)(f)=∞ when ρf(m,q)=∞,(iv)
λg(p,q)(f)=∞ when λf(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 numbersm and n and a∈ℝ, a≠0, so that
(52)f(z)=zm,g(z)=azn.
Then
(53)ρf(1,1)=λf(1,1)=m,ρg(1,1)=λg(1,1)=n.
Now
(54)ρg(1,1)(f)=λg(1,1)(f)=ρf(1,1)ρg(1,1)=mn.Example 24 (relative order between exponentials of the same order).
Letn be any natural number and a any positive real number and consider
(55)f(z)=exp⁡zn,g(z)=exp⁡(az)n.
In this case f and g are two entire functions with regular (2,1) growth; thus
(56)λg(1,1)(f)=ρg(1,1)(f)=ρf(2,1)ρg(2,1)=nn=1.
Clearly
(57)ρf(1,1)(g)=λf(1,1)(g)=1.Example 25 (relative order between exponential and power function).
Letm, n be any two natural numbers and consider
(58)f=exp⁡zm,g=zn.
Then
(59)ρf=λf=m,ρg=λg=0.
Now
(60)ρg(1,1)(f)=λg(1,1)(f)=∞,ρf(1,1)(g)=λf(1,1)(g)=0.Whenf 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.
Letf 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)
(61)
ρ
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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)
(62)
ρ
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(
p
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Under the hypothesis of Definition26, for m>n:
(i)
(63)
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∞
log
⁡
[
m
]
M
f
(
r
)
log
⁡
[
q
]
r
,
λ
g
(
p
+
m
-
n
,
q
)
(
f
)
=
lim
⁡
inf
⁡
r
→
∞
log
⁡
[
m
]
M
f
(
r
)
log
⁡
[
q
]
r
,
and for m<n:
(ii)
(64)
ρ
g
(
p
,
q
+
n
-
m
)
(
f
)
=
lim
⁡
sup
⁡
r
→
∞
log
⁡
[
p
]
r
log
⁡
[
n
]
M
g
(
r
)
,
λ
g
(
p
,
q
+
n
-
m
)
(
f
)
=
lim
⁡
inf
⁡
r
→
∞
log
⁡
[
p
]
r
log
⁡
[
n
]
M
g
(
r
)
.The next example will make an alternative use of Theorem27.Example 28 (relative order between exponentials of different order).
Let(65)f(z)=exp⁡[27]z5,g(z)=exp⁡[50]z17.
In this case f and g are entire functions of regular growth (m,p)=(28,1) and (n,q)=(51,1), respectively, with
(66)ρf(28,1)=λf(28,1)=5,ρg(51,1)=λg(51,1)=17.
Now
(67)log⁡[p]Mg-1Mf(r)log⁡[q+n-m]r=log⁡[log⁡[50](exp⁡[27]r5)]1/17log⁡[24]r
and by taking limsup⁡ and liminf⁡, we get
(68)ρg(p,q+n-m)(f)=117=λg(p,q+n-m)(f).
Obviously, the same limit is achieved if, by using Theorem 27, we consider the quotient
(69)log⁡[p]rlog⁡[n]Mg(r)=log⁡rlog⁡[51]Mg(r).
Reciprocally, in order to evaluateρf(p+m-n,q)(g) and λf(p+m-n,q)(g), we would take limits in either
(70)log⁡[24]Mf-1Mg(r)log⁡r=log⁡[24][log⁡[27](exp⁡[50]r17)]1/5log⁡rorlog⁡[51]Mg(r)log⁡r,
obtaining that
(71)ρf(p+m-n,q)(g)=17=λf(p+m-n,q)(g).


## 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.



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*Source: 826137-2014-03-11.xml*