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

Journal of Mathematics Research; Vol. 8, No. 5; October 2016 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education A Note on Relative (p,q) th Proximate Order of Entire Functions Luis M. S´ anchez Ruiz1, Sanjib Kumar Datta2, Tanmay Biswas3& Chinmay Ghosh4 1ETSID-Depto. de Matem´ atica Aplicada & CITG, Universitat Polit` ecnica de Val` encia, E-46022 Valencia, Spain. 2Department of Mathematics, University of Kalyani Kalyani, Dist-Nadia, PIN-741235, West Bengal, India. 3Rajbari, Rabindrapalli, R. N. Tagore Road P.O.- Krishnagar, Dist-Nadia, PIN741101, West Bengal, India. 4Guru Nanak Institute of Technology 157/F, Nilgunj Road, Panihati, Sodepur Kolkata-700114, West Bengal, India. Correspondence: Luis M. S´ anchez Ruiz, ETSID-Depto. de Matem´ atica Aplicada & CITG, Universitat Polit` ecnica de Val` encia, E-46022 Valencia, Spain. E-mail: [email protected].es Received: June 27, 2016 Accepted: July 12, 2016 Online Published: September 14, 2016 doi:10.5539/jmr.v8n5p1 URL: http://dx.doi.org/10.5539/jmr.v8n5p1 Abstract 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 pand 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. Keywords: entire function, index-pair, relative (p,q) th order, relative (p,q) th proximate order 1. 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 Mf, is defined on each non-negative real value rby Mf(r)=max |z|=r |f(z)|. With the aim of estimating the growth of a nonconstant entire function f,Boas (Boas, 1954) introduced the concept of order as the value ρfwhich is generally used in computational purpose and is defined in terms of the growth of frespect to the exp zfunction as ρf=lim sup r→∞ log log Mf(r) log log Mexp (r)=lim sup r→∞ log log Mf(r) log (r)(0≤ρf≤ ∞). Given another entire function g,the ratio Mf(r) Mg(r)as r→ ∞ is called the growth of fwith respect to gin terms of their maximum moduli. If this relative growth happens to be k∈R, then Mf(r)∝kMg(r)as r→ ∞. With the aim of knowing the relative growth of functions of the same nonzero finite order, the type of a given such funtion fwas introduced as τf=lim sup r→∞ log Mf(r) rρf(0≤τf≤ ∞). 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 1 http://jmr.ccsenet.org Journal of Mathematics Research Vol. 8, No. 5; 2016 used by Alburquerque et al. (Albuquerque, Bernal-Gonz´ alez, Pellegrino, & Seoane-Sep´ ulveda, 2014) who obtained new Peano type results by showing that the subset of continuous surjections from Rmto Cnsuch that each value ain Cnis assumed on an unbounded set of Rmis maximal strongly algebrable, i.e. there exists a c-generated free algebra contained in CS (Rm,Cn)∪ {0}, where CS (Rm,Cn) denotes the set of all continuous surjective mappings Rm→Cn. On the other hand, S´ anchez Ruiz et al. (S´ anchez Ruiz, Datta, Biswas, & Mondal, 2014) have introduced a new type of relative (p,q)th order of entire functions where p,qare 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 ρf(r)for an entire function fhaving finite order ρfwith the following properties: (i) ρf(r)is non-negative and continuous for r>r0, say, (ii) ρf(r)is differentiable for r≥r0except possibly at isolated points at which ρ′ f(r+0) and ρ′ f(r−0) exist, (iii) limr→∞ ρf(r)=ρf, (iv) limr→∞ rρ′ f(r) log r=0 and (v) lim supr→∞ log Mf(r) rρf(r)=1. Such a function is called a Lindel¨ of 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. 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 hand whenever the corresponding domain and range allow it we will use the notation h[0] (x)=x,and h[k](x)=h(h[k−1] (x))for k=1,2,3, ... omitting the parenthesis when hhappens to be the log or exp function. Taking this into account the order (resp. lower order) of an entire function fis given by 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, Kapoor and Bajpai (Juneja, Kapoor, Bajpai, 1976) defined the (p,q)-th order (resp. (p,q)-th lower order) of an entire function fas follows: ρf(p,q)=lim sup r→∞ log[p]Mf(r) log[q]r (resp. λf(p,q)=lim inf r→∞ log[p]Mf(r) log[q]r ), where p,qare 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 fconsidered in (Sato, 1963) for each integer l≥2 since these correspond to the particular case ρ[l] f=ρf(l,1)(resp. λ[l] f=λf(l,1)).Clearly ρf(2,1)=ρfand λf(2,1)=λf. Related to this, let us recall the following properties. If 0 < ρf(p,q)<∞,then ρf(p−n,q)=∞for n<p, ρf(p,q−n)=0 for n<q, ρf(p+n,q+n)=1 for n=1,2, ... 2 http://jmr.ccsenet.org Journal of Mathematics Research Vol. 8, No. 5; 2016 Similarly for 0 < λf(p,q)<∞,one can easily verify that λf(p−n,q)=∞for n<p, λf(p,q−n)=0 for n<q, λf(p+n,q+n)=1 for n=1,2, ... Recalling that for any pair of integer numbers m,nthe Kroenecker function is defined by δm,n=1 for m=nand δm,n=0 for m,n, the aforementioned properties provide the following definition. Definition 1. (Juneja, Kapoor, Bajpai, 1976) An entire function f 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. (Juneja, Kapoor, Bajpai, 1976) An entire function f is said to have lower index-pair (1,1)if 0< λf(1,1)< ∞. Otherwise, f has lower index-pair (p,q),(1,1), p ≥q≥1, if δp−q,0< λf(p,q)<∞and λf(p−1,q−1)<R+. Given a non-constant entire function fdefined in the open complex plane, its maximum modulus function Mfis strictly increasing and continuous. Hence there exists its inverse function M−1 f:(|f(0)|,∞)→(0,∞)with lims→∞ M−1 f(s)=∞. Bernal (Bernal, 1988) introduced the definition of relative order of fwith respect to g, denoted by ρg(f),as follows: ρg(f)=inf {µ > 0 : Mf(r)<Mg(rµ)for all r>r0(µ)>0} =lim sup r→∞ log M−1 gMf(r) log r. 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 λg(f),is defined as λg(f)=lim inf r→∞ log M−1 gMf(r) log r. Recently, S´ anchez Ruiz et al. (S´ anchez Ruiz, Datta, Biswas, & Mondal, 2014) have introduced a definition of relative (p,q)-th order ρ(p,q) g(f)of an entire function fwith 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) g(f)=ρk g(f)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 ρ(p,q) g(f)=lim sup r→∞ log[p]M−1 gMf(r) log[q]r . And the relative (p,q)-th lower order of f with respect to g is defined by λ(p,q) g(f)=lim inf r→∞ log[p]M−1 gMf(r) log[q]r . When (m,1)and (m,k)are the index-pairs of fand grespectively, then Definition 3 reduces to definition of generalized relative order (Lahiri & Banerjee, 2002). If the entire functions fand ghave 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 ρg(f)=ρ[m] fand ρ(p,q) g(f)= ρf(m,q).Also Definition 3 becomes the classical one given in (Titchmarsh, 1968) if fis 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+→Ris said to be a relative (p,q)th proximate order of f with respect to g if there is some r0>0so that it satisfies: 3 http://jmr.ccsenet.org Journal of Mathematics Research Vol. 8, No. 5; 2016 (i) ρ(p,q) g(f) (r)is non-negative and continuous for r>r0, (ii) ρ(p,q) g(f) (r)is differentiable for r≥r0except possibly at isolated points where ρ(p,q)′ g(f)(r+0) and ρ(p,q)′ g(f)(r−0) exist, (iii) limr→∞ ρ(p,q) g(f) (r)=ρ(p,q) g(f), (iv) limr→∞ ρ(p,q)′ g(r)∏max(p,q) i=0log[i]r=0, (v) lim supr→∞ log[p−1]M−1 gMf(r) [log[q−1]r]ρ(p,q) g(f)(r)=1. When (m,1)and (m,k)are the index-pairs of fand grespectively, Definition 4 reduces to definition of generalized relative proximate order. If the entire functions fand ghave the same index-pair (p,1), the above definition provides the relative proximate order ρg(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+→Ris said to be a relative (p,q)th lower proximate order of f with respect to g if there is some r0>0so that it satisfies: (i) λ(p,q) g(f) (r)is non-negative and continuous for r>r0, (ii) λ(p,q) g(f) (r)is differentiable for r≥r0except possibly at isolated points at which λ(p,q)′ g(f)(r+0) and λ(p,q)′ g(f)(r−0) exist, (iii) limr→∞ λ(p,q) g(f) (r)=λ(p,q) g(f), (iv) limr→∞ λ(p,q)′ g(r)∏max(p,q) i=0log[j]r=0, (v) lim infr→∞ log[p−1]M−1 gMf(r) [log[q−1]r]λ(p,q) g(f)(r)=1. 3. 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 σ(r)=log[p]M−1 gMf(r) log[q]r and it can be easily proved that σ(r)is continuous and lim sup r→∞ σ(r)=ρ(p,q) g(f). Now we consider the following three sub cases: Sub Case AI. Let σ(r)> ρ(p,q) g(f)for at least a sequence of values of rtending to infinity. Then we define the non increasing real function ϕ(r)=max x≥r{σ(x)}. 4 http://jmr.ccsenet.org Journal of Mathematics Research Vol. 8, No. 5; 2016 Now let us take R1>Rwith R1>exp[p+2]1 and σ(R)> ρ(p,q) g(f). Then for any given r≥R1, we obtain that σ(r)≤σ(R). As σ(r)is continuous, there exists r1∈[R,R1] such that σ(r1)=max R≤x≤R1 {σ(x)}. Clearly r1>exp[p+2]1 and ϕ(r1)=σ(r1), there being a sequence of such r1values tending to infinity. Let us now consider that ρ(p,q) g(f) (r1)=ϕ(r1) and let t1be the smallest integer not smaller than 1 +r1such that ϕ(r1)> ϕ(t1).Also we define ρ(p,q) g(f) (r)=ρ(p,q) g(f) (r1)for r1<r≤t1. Now we observe that: (i) ϕ(r) and ρ(p,q) g(f) (r1)−log[p+2]r+log[p+2]t1are continuous functions, (ii) ρ(p,q) g(f) (r1)−log[p+2]r+log[p+2]t1> ϕ(t1) for r(>t1)sufficiently close to t1and (iii) ϕ(r) is non increasing. Consequently we can define u1>t1as follows: ρ(p,q) g(f) (r)=ρ(p,q) g(f) (r1)−log[p+2]r+log[p+2]t1for t1≤r≤u1, ρ(p,q) g(f) (r)=ϕ(r) for r=u1and ρ(p,q) g(f) (r)> ϕ(r) for t1≤r<u1. Let now r2be the smallest value of rfor which r2≥u1and ϕ(r2)=σ(r2). If r2>u1then let ρ(p,q) g(f) (r)=ϕ(r) for u1≤r≤r2.Then it can be easily shown that ϕ(r) and ρ(p,q) g(f) (r)are both constant in u1≤r≤r2.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=0log[i]r)−1and ρ(p,q) g(f) (r)≥ϕ(r)≥σ(r)for all r≥r1. Also ρ(p,q) g(f) (r)=σ(r)for a sequence of values of rtending to infinity and ρ(p,q) g(f) (r)is non increasing for r≥r1. So ρ(p,q) g(f)=lim sup r→∞ σ(r)=lim r→∞ ϕ(r) i.e., lim sup r→∞ ρ(p,q) g(f) (r)=lim inf r→∞ ρ(p,q) g(f) (r) =lim r→∞ ρ(p,q) g(f) (r)=ρ(p,q) g(f) and lim r→∞ ρ(p,q)′ g(r) p ∏ i=0 log[i]r=0. Again we get that log[p−1]M−1 gMf(r)=[log[q−1]r]σ(r)=[log[q−1]r]ρ(p,q) g(f)(r) for a sequence of values of rtending to infinity and log[p−1]M−1 gMf(r)<[log[q−1]r]ρ(p,q) g(f)(r) for the remaning r’s. Hence lim sup r→∞ log[p−1]M−1 gMf(r) [log[q−1]r]ρ(p,q) g(f)(r)=1. 5 http://jmr.ccsenet.org Journal of Mathematics Research Vol. 8, No. 5; 2016 The continuity of ρ(p,q) g(f) (r)for r≥r1follows by construction. Sub Case BI. Let σ(r)< ρ(p,q) g(f)for all sufficiently large values of rtending to infinity. Now we define the real function ξ(r)=max X≤x≤r{σ(x)}, where X>exp[p+2]1 is such that σ(r)< ρ(p,q) g(f)whenever x≥X. Here we note that ξ(r) is non decreasing and the roots of ξ(x)=ρ(p,q) g(f)+log[p+2]x−log[p+2]r are smaller than rfor all sufficiently large values of r≥X. Now for a suitable large value v1>X,we define ρ(p,q) g(f) (v1)=ρ(p,q) g(f), ρ(p,q) g(f) (r)=ρ(p,q) g(f)+log[p+2]r−log[p+2]v1for s1≤r≤v1where s1<v1is such that ξ(s1)=ρ(p,q) g(f) (s1). In fact s1is given by the largest positive root of ξ(x)=ρ(p,q) g(f)+log[p+2]x−log[p+2]v1. If ξ(s1),σ(s1)let ω1be an upper bound of the ω < s1at which ξ(ω) is different from σ(ω).If we define ρ(p,q) g(f) (r)= ξ(r) for ω1≤r≤s1,it is clear that ξ(r) is constant in [ω1,s1], hence ρ(p,q) g(f) (r)is constant in [ω1,s1], too. If ξ(s1)=σ(s1)we take ω1=s1.Now we choose v2>v1suitably large and let ρ(p,q) g(f) (v1)=ρ(p,q) g(f)and ρ(p,q) g(f) (r)=ρ(p,q) g(f)+log[p+2]r−log[p+2]v2 for s2≤r≤v2where s2<v2is such that ξ(s2)=ρ(p,q) g(f) (s2). If ξ(s2),ρ(p,q) g(f) (s2)then suppose that ρ(p,q) g(f) (r)=ξ(r) for ω2≤r≤s2, with ω2mimicking the behavour of ω1. Hence ρ(p,q) g(f) (r)is constant in [ω2,s2]. If ξ(s2)=σ(s2)we take ω2=s2. Also suppose that ρ(p,q) g(f) (r)=ρ(p,q) g(f) (ω2)−log[p+2]r+log[p+2]ω2for q1≤r≤ω2where q1< ω2is the point of intersection of y=ρ(p,q) g(f)with y=ρ(p,q) g(f) (ω2)−log[p+2]x+log[p+2]ω2.Now it is also possible to choose v2so large that v1<q1and for the case under consideration, let us consider ρ(p,q) g(f) (r)=ρ(p,q) g(f)for v1≤r≤q1.Therefore if we repeat this process it can be shown that for all r≥v1, ρ(p,q) g(f)≥ρ(p,q) g(f) (r)≥ξ(r)≥σ(r)and ρ(p,q) g(f) (r)=σ(r)for r=ω1, ω2, ... Hence we obtain that lim sup r→∞ ρ(p,q) g(f) (r)=lim inf r→∞ ρ(p,q) g(f) (r)=lim r→∞ ρ(p,q) g(f) (r)=ρ(p,q) g(f) since log[p−1]M−1 gMf(r)=[log[q−1]r]σ(r)=[log[q−1]r]ρ(p,q) g(f)(r) for a sequence of values of rtending to infinity and log[p−1]M−1 gMf(r)<[log[q−1]r]ρ(p,q) g(f)(r) for remaning r’s. Therefore it follows that lim sup r→∞ log[p−1]M−1 gMf(r) [log[q−1]r]ρ(p,q) g(f)(r)=1. 6 http://jmr.ccsenet.org Journal of Mathematics Research Vol. 8, No. 5; 2016 Furthermore, ρ(p,q) g(f)(r) is differentiable in adjacent intervals and ρ(p,q)′ g(r)=0 or         p+1 ∏ i=0 log[i]r        −1 . Consequently, lim r→∞ ρ(p,q)′ g(r) p ∏ i=0 log[i]r=0. Once again, continuity of ρ(p,q) g(f)(r) follows by construction. Sub Case CI. Let σ(r)=ρ(p,q) g(f)for at least a sequence of values of rtending to infinity. Now considering ρ(p,q) g(f) (r)= ρ(p,q) g(f)for all sufficiently large values of rone 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 σ(r)=[log[q−1]r]−ρ(p,q) g(f)·log[p−1]M−1 gMf(r). Therefore it can easily be shown that lim sup r→∞ log σ(r) log[q]r =0. Now putting x=log[q]rand y=log σ(r), we obtain that y=log σ(exp[q]x). So lim sup r→∞ log σ(exp[q]x) x=lim sup r→∞ log σ(r) log[q]r =0 which shows that for any abritrary ε > 0 and for large values of x,x≥x0(ε),the entire curve y=log σ(exp[q]x)lies below the line y=εxand, 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 AII. Let us consider that lim supr→∞ log σ(exp[q]x)= +∞. Now we construct the smallest convex domain so that it contains the positive ray of the xaxis and all the points of the curve y=log σ(exp[q]x).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: (I) The curve is convex from the above, (II) limx→∞ δ(x) x=0, (III) log σ(exp[q]x)≤δ(x), (IV) log σ(exp[q]x)=δ(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. 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 limx→∞ δ′(x)=0 and from (III) we have log[p−1]M−1 gMf(r)≤[log[q−1]r]ρ(p,q) g(f)(r) 7 http://jmr.ccsenet.org Journal of Mathematics Research Vol. 8, No. 5; 2016 where ρ(p,q) g(f) (r)=ρ(p,q) g(f)+ δ(log[q]r) log[q]r . Now from (II) it follows that lim r→∞ ρ(p,q) g(f) (r)=lim r→∞          ρ(p,q) g(f)+ δ(log[q]r) log[q]r          =ρ(p,q) g(f). Also in view of the properties (IV) and (V) one can easily verify that there exists a sequence of values of rtending to infinity for which log[p−1]M−1 gMf(r)=[log[q−1]r]ρ(p,q) g(f)(r) i.e., lim sup r→∞ log[p−1]M−1 gMf(r) [log[q−1]r]ρ(p,q) g(f)(r)=1 and limr→∞ ρ(p,q)′ g(r)∏q i=0log[i]r=0 holds. Thus we have constructed the function ρ(p,q) g(f) (r). Sub Case BII. In order to generalize the case, let us consider a concave function β(x)which satisfies the following properties: (I) limx→∞ β′(x)=0, (II) limx→∞ β(x) x=0 and (III) lim supr→∞ [log σ(exp[q]x)+β(x)]=∞. With the goal of constructing β(x)we go through the following steps: First we consider a segment a1of the line y=−ε1xfrom the origin to a point x1where log σ(exp[q]x1)>−ε1x1+1. Having chosen a positive number ε2< ε1we draw a segment a2of the line y+ε1x1=−ε2(x−x1)from the point (x1,−ε1x1)to a point x2>x1satisfying log σ(exp[q]x2)>−ε1x1−ε2(x2−x1)+2.Then we choose a segment a3with slope −ε3(0< ε3< ε2),etc. The selected {εn}is strictly decreasing with εn→0 but the sequence {xn}of points is strictly increasing with xn→ ∞.The polygonal function y=β1(x)constructed in this manner satisfies lim x→∞ β1(x) x=0. The function β1(x)can be made everywhere differentiable by changing it in an unessential manner in the neighbourhood of each angular point. The function β(x)defined as β(x)=−β1(x)has the required properties. A convex majorant β2(x)for the function log σ(exp[q]x)+β(x)is now considered and writing δ(x)=β2(x)−β(x) yields log σ(exp[q]x)≤δ(x). Moreover, log σ(exp[q]x′ n)=δ(x′ n) on some sequence {x′ n}∞ 1of extreme points, x′ n→ ∞. Also if the function ρ(p,q) g(f) (r)is defined as ρ(p,q) g(f) (r)=ρ(p,q) g(f)+ δ(log[q]r) log[q]r , 8 http://jmr.ccsenet.org Journal of Mathematics Research Vol. 8, No. 5; 2016 it can easily be seen that lim x→∞ δ′(x)=0,lim x→∞ δ(x) x=0. Hence limr→∞ ρ(p,q) g(f) (r)=ρ(p,q) g(f)and lim r→∞ ρ(p,q)′ g(r) q ∏ i=0 log[i]r=0. Moreover log[p−1]M−1 gMf(r)≤[log[q−1]r]ρ(p,q) g(f)(r) and log[p−1]M−1 gMf(rn)=[log[q−1]rn]ρ(p,q) g(f)(rn) for some sequence {rn},rn→ ∞. Therefore lim sup r→∞ log[p−1]M−1 gMf(r) [log[q−1]r]ρ(p,q) g(f)(r)=1, and the proof is complete.  The following theorem’s proof can be obtained in the line of Theorem 1. Theorem 2. 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 lower order λ(p,q) g(f)of f with respect to g is finite and non zero, then the relative (p,q)th lower proximate order λ(p,q) g(f) (r)of f with respect to g exists. Now we recall the that a positive function η(r)is called slowly increasing (Srivastava & Kumar, 2009), if limr→∞ η(nr) η(r)=1. We will say that η(r)is uniform slowly increasing if the aforementioned limit happens to exist uniformly in mon each interval 0 <b≤n<m<∞. The proofs of the following corollary can be carried out using the same techniques involved in (Nandan, Doherey, & Srivastava, 1980). Corollary 1. Let ρ(p,q) g(f) (r)and λ(p,q) g(f) (r)be respectively the relative (p,q)th proximate order and the relative (p,q)th lower proximate order of f with respect to g, and let ρ(p,q) g(f)and λ(p,q) g(f)be the relative (p,q)-th order and relative (p,q)-th lower order of f with respect to g for any positive integers p and q.Then: 1. The functions [log[q−1]r]ρ(p,q) g(f)(r) [log[q−1]r]ρ(p,q) g(f)and [log[q−1]r]λ(p,q) g(f)(r) [log[q−1]r]λ(p,q) g(f)are uniform slowly increasing. 2. The functions [log[q−1]r]ρ(p,q) g(f)(r)and [log[q−1]r]λ(p,q) g(f)(r)are monotone increasing for sufficiently large values of r. 3. For 0 <l≤k≤m<∞and r→ ∞, we have that [log[q−1](kr)]ρ(p,q) g(f)(kr)·[log[q−1]r]ρ(p,q) g(f) [log[q−1](kr)]ρ(p,q) g(f)·[log[q−1]r]ρ(p,q) g(f)(r) ∝1, [log[q−1](kr)]λ(p,q) g(f)(kr)·[log[q−1]r]λ(p,q) g(f) [log[q−1](kr)]λ(p,q) g(f)·[log[q−1]r]λ(p,q) g(f)(r) ∝1 hold uniformly in k. 9