A Note on Relative ( p , q ) th Proximate Order of Entire Functions

Relative order of functions measures specifically how different in growth two given functions are which helps to settle the exact physical state of a system. In this paper for any two positive integers p and q, we introduce the notion of relative (p, q) th proximate order of an entire function with respect to another entire function and prove its existence.


Introduction
A single valued analytic function in the finite complex plane is called an entire (or integral) function.It is well known that for example exp, sin, cos 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.In this line the value distribution theory studies how an entire function assumes some values and conversely, what is in some specific manner the influence on a function of taking certain values.It also deals with various aspects of the behaviour of entire functions one of which is the study of comparative growth properties of entire functions.For any entire function f , the so called maximum modulus function and denoted by M f , is defined on each non-negative real value r by With the aim of estimating the growth of a nonconstant entire function f, Boas (Boas, 1954) introduced the concept of order as the value ρ f which is generally used in computational purpose and is defined in terms of the growth of f respect to the exp z function as ) .
Given another entire function g, the ratio M g (r) as r → ∞ is called the growth of f with respect to g in terms of their maximum moduli.If this relative growth happens to be k ∈ R, then With the aim of knowing the relative growth of functions of the same nonzero finite order, the type of a given such funtion f was introduced as L. Bernal (Bernal, 1988) introduced the relative order between two entire functions to avoid comparing growth just with exp .Thus the growth of entire functions may be studied in terms of its relative orders.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 (Chakraborty & Roy, 2006;Datta, Biswas, 2009;Datta, Biswas, 2010;Datta, Biswas, Biswas, 2013;Datta, Biswas & Biswas, 2013;Datta, Biswas, & Pramanick, 2012;Lahiri & Banerjee, 2005).This has different applications related to entropy as this is the amount of additional information needed to specify the exact physical state of a system, and relative order of functions measures how different in growth two given functions are.Indeed very recently these ideas have been used by Alburquerque et al. (Albuquerque, Bernal-González, Pellegrino, & Seoane-Sepúlveda, 2014) who obtained new Peano type results by showing that the subset of continuous surjections from R m to C n such that each value a in C n is assumed on an unbounded set of R m is maximal strongly algebrable, i.e. there exists a c-generated free algebra contained in CS (R m , C n ) ∪ {0}, where CS (R m , C n ) denotes the set of all continuous surjective mappings R m → C n .
On the other hand, Sánchez Ruiz et al. (Sánchez Ruiz, Datta, Biswas, & Mondal, 2014) have introduced a new type of relative (p, q)th order of entire functions where p, q are any two positive integers revisiting the ideas developed by a number of authors including Lahiri and Banerjee (Lahiri & D. Banerjee, 2005).
However, these concepts are not adequate for comparing the growth of entire functions with either zero or infinite order.For this reason Valiron (Valiron, 1949) introduced the concept of a positive continuous function ρ f (r) for an entire function f having finite order ρ f with the following properties: (i) ρ f (r) is non-negative and continuous for r > r 0 , say, (ii) ρ f (r) is differentiable for r ≥ r 0 except possibly at isolated points at which Such a function is called a Lindelöf proximate order which makes unnecessary to consider functions of minimal or maximal type, its existence being established op.cit.It was simplified by Shah (Shah, 1946), and Nandan et al. (Nandan, Doherey, & Srivastava, 1980) extended this notion of proximate order for an entire function of one complex variable with index-pair (p, q) with positive integers p ≥ q.Also Lahiri (Lahiri, 1989) generalised the idea of the proximate order for a meromorphic function with finite generalised order and proved its existence.
As a consequence of the above it seems reasonable for any two positive integers, p, q, to define the relative (p, q)th proximate order of an entire function with respect to another entire function.In this paper we do so and prove its existence.

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 and 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 Let us recall that Juneja, Kapoor and Bajpai (Juneja, Kapoor, Bajpai, 1976) defined the (p, q)-th order (resp.(p, q)-th lower order) of an entire function f as follows: where p, q are any two positive integers with p ≥ q.These definitions extended the generalized order ρ [l]  f (resp.generalized lower order λ [l]  f ) of an entire function f considered in (Sato, 1963) for each integer l ≥ 2 since these correspond to the particular case ρ [l]  f = ρ f (l, 1) (resp.λ [l]  f = λ f (l, 1)).Clearly ρ f (2, 1) = ρ f and λ f (2, 1) = λ f .Related to this, let us recall the following properties.If 0 < ρ f (p, q) < ∞, then Recalling that for any pair of integer numbers m, n the Kroenecker 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. (Juneja, Kapoor, Bajpai, 1976) Definition 2. (Juneja, Kapoor, Bajpai, 1976) An entire function f is said to have lower index-pair Given a non-constant entire function f defined in the open complex plane, its maximum modulus function M f is strictly increasing and continuous.Hence there exists its inverse function (Bernal, 1988) introduced the definition of relative order of f with respect to g, denoted by ρ g ( f ) , as follows: This definition coincides with the classical one (Titchmarsh, 1968) if g = exp.Analogously, the relative lower order of f with respect to g, denoted by λ g ( f ) , is defined as Recently, Sánchez Ruiz et al. (Sánchez Ruiz, Datta, Biswas, & Mondal, 2014) have introduced a definition of relative (p, q)-th order ρ (p,q) g ( f ) of an entire function f with respect to another entire function g, sharpenning an earlier definiton of relative (p, q)-th order of Lahiri and Banerjee (Lahiri & Banerjee, 2005), from which the more natural particular case ρ (k,1) arises.This is done as follows.Definition 3. Let f, g be two entire functions with index-pairs (m, q) and (m, p), respectively, where p, q, m are positive integers with m ≥ max(p, q).Then the relative (p, q)-th order of f with respect to g is defined by And the relative (p, q)-th lower order of f with respect to g is defined by When (m, 1) and (m, k) are the index-pairs of f and g respectively, then Definition 3 reduces to definition of generalized relative order (Lahiri & Banerjee, 2002).If the entire functions f and g have the same index-pair (p, 1), we get the definition of relative order introduced by Bernal (Bernal, 1988) and if g = exp [m−1] , then ρ g ( f ) = ρ [m]  f and ρ (p,q) g ( f ) = ρ f (m, q).Also Definition 3 becomes the classical one given in (Titchmarsh, 1968) if f is an entire function with index-pair (2, 1) and g = exp.
In order to refine the above growth scale, now we intend to introduce the definition of an intermediate comparison function, called relative (p, q)th proximate order of entire function with respect to another entire function in the light of their indexpair which is as follows.Its consistency will be established in Section 3. Definition 4. Let f, g be two entire functions with index-pairs (m, q) and (m, p) respectively where p, q, m are positive integers with m ≥ max(p, q).For a finite relative (p, q)-th order ρ (p,q) g ( f ) of f with respect to g, then a function ρ (p,q) g ( f ) (r) : R + → R is said to be a relative (p, q)th proximate order of f with respect to g if there is some r 0 > 0 so that it satisfies: (i) ρ (p,q) g ( f ) (r) is non-negative and continuous for r > r 0 , (ii) ρ (p,q) g ( f ) (r) is differentiable for r ≥ r 0 except possibly at isolated points where ρ (p,q)′ g ( f ) (r + 0) and ρ When (m, 1) and (m, k) are the index-pairs of f and g respectively, Definition 4 reduces to definition of generalized relative proximate order.If the entire functions f and g have the same index-pair (p, 1), the above definition provides the relative proximate order ρ g ( f ) (r) .
The relative (p, q)th lower proximate order of an entire function with respect to another entire function may analogously be defined, consistency being held by virtue of Section 3, too.
Definition 5. Let f and g be any two entire functions with index-pairs (m, q) and (m, p) respectively where p, q, m are positive integers such that m ≥ max(p, q).For a finite relative (p, q)-th lower order of f with respect to g, λ (p,q) g ( f ) , then a function λ (p,q) g ( f ) (r) : R + → R is said to be a relative (p, q)th lower proximate order of f with respect to g if there is some r 0 > 0 so that it satisfies:

Main Results
In this section we state the main results of the paper.We include the proof of the first main Theorem 1 for the sake of completeness.The others are basically omitted since they are easily proved with the same techniques or with some easy reasonings.
Theorem 1.Let f, g be any two entire functions with index-pairs (m, q) and (m, p) respectively where p, q, m are positive integers with m ≥ max(p, q).If the relative (p, q)-th order ρ (p,q) g ( f ) is finite, then the relative (p, q)th proximate order ρ (p,q) g ( f ) (r) of f with respect to g exists.
Proof.We distinguish the following two cases: Case I. Assume p ≥ q.Then we write and it can be easily proved that σ (r) is continuous and Now we consider the following three sub cases: Sub Case A I .Let σ (r) > ρ (p,q) g ( f ) for at least a sequence of values of r tending to infinity.Then we define the non increasing real function Clearly r 1 > exp [p+2] 1 and ϕ(r 1 ) = σ (r 1 ), there being a sequence of such r 1 values tending to infinity.Let us now consider that ρ (p,q) g ( f ) (r 1 ) = ϕ(r 1 ) and let t 1 be the smallest integer not smaller than 1 + r 1 such that ϕ(r 1 ) > ϕ(t 1 ).Also we define ρ (p,q) g ( f ) (r) = ρ (p,q) g ( f ) (r 1 ) for r 1 < r ≤ t 1 .Now we observe that: (i) ϕ(r) and ρ for r (> t 1 ) sufficiently close to t 1 and (iii) ϕ(r) is non increasing.
Consequently we can define u 1 > t 1 as follows: Let now r 2 be the smallest value of r for which r 2 ≥ u 1 and ϕ(r Then it can be easily shown that ϕ(r) and ρ (p,q) g ( f ) (r) are both constant in u 1 ≤ r ≤ r 2 .By repeating this process, we obtain that ρ (p,q) g ( f ) (r) is differentiable in adjacent intervals.Moreover ρ (p,q)′ g (r) coincides with 0 or ( ∏ p+1 i=0 log [i] r ) −1 and ρ (p,q) g ( f ) (r) ≥ ϕ(r) ≥ σ (r) for all r ≥ r 1 .

Also ρ
(p,q) g ( f ) (r) = σ (r) for a sequence of values of r tending to infinity and ρ for a sequence of values of r tending to infinity and log for the remaning r's.Hence Once again, continuity of ρ (p,q) g ( f ) (r) follows by construction.Sub Case C I .Let σ (r) = ρ (p,q) g ( f ) for at least a sequence of values of r tending to infinity.Now considering ρ (p,q) g ( f ) (r) = ρ (p,q) g ( f ) for all sufficiently large values of r one can easily verify the existance of the relative (p, q)th proximate order for the case under consideration.
Case II.Assume q ≥ p.Now let us consider the following function Therefore it can easily be shown that ) . Thus the boundary of newly formed domain lying above the x-axis is a continuous curve and we denote it as y = δ (x).This curve must satisfy the following properties: Also the curve y = δ (x) is made differentiable in the neighbourhood of each angular point ( if necessary) by making some unessential changes.Thus it is assumed that the curve y = δ (x) is differentiable everywhere.Hence from (I) and (II), above it follows that lim x→∞ δ ′ (x) = 0 and from (III) we have putting x = log [q] r and y = log σ (r), we obtain that y for any abritrary ε > 0 and for large values of x, x ≥ x 0 (ε) , the entire curve y = log σ( exp [q] x) lies below the line y = εx and, on the other hand, there are points on the curve with arbitrarily large abscissae lying above the line y = −εx.Now we consider the following two sub cases:Sub Case A II .Let us consider that lim sup r→∞ log σ( exp [q] x )= +∞.Now we construct the smallest convex domain so that it contains the positive ray of the x axis and all the points of the curve y = log σ( exp [q] x

(
x) at the extreme points of the curve y = δ (x) and (V) The curve y = δ (x) contains a sequence of extreme points tending to infinity.