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

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1 Jounal o Mathematics Reseach; Vol. 8, No. 5; Octobe 2016 ISSN E-ISSN Published by Canadian Cente o Science and Education A Note on Relative (p, q) th Poximate Ode o Entie Functions Luis M. Sánchez Ruiz 1, Sanjib Kuma Datta 2, Tanmay Biswas 3 & Chinmay Ghosh 4 1 ETSID-Depto. de Matemática Aplicada & CITG, Univesitat Politècnica de València, E Valencia, Spain. 2 Depatment o Mathematics, Univesity o Kalyani Kalyani, Dist-Nadia, PIN , West Benal, India. 3 Rajbai, Rabindapalli, R. N. Taoe Road P.O.- Kishnaa, Dist-Nadia, PIN , West Benal, India. 4 Guu Nanak Institute o Technoloy 157/F, Nilunj Road, Panihati, Sodepu Kolkata , West Benal, India. Coespondence: Luis M. Sánchez Ruiz, ETSID-Depto. de Matemática Aplicada & CITG, Univesitat Politècnica de València, E Valencia, Spain. LMSR@mat.upv.es Received: June 27, 2016 Accepted: July 12, 2016 Online Published: Septembe 14, 2016 doi: /jm.v8n5p1 Abstact URL: Relative ode o unctions measues speciically how dieent in owth two iven unctions ae which helps to settle the exact physical state o a system. In this pape o any two positive intees p and q, we intoduce the notion o elative (p, q) th poximate ode o an entie unction with espect to anothe entie unction and pove its existence. Keywods: entie unction, index-pai, elative (p, q) th ode, elative (p, q) th poximate ode 1. Intoduction A sinle valued analytic unction in the inite complex plane is called an entie (o inteal) unction. It is well known that o example exp, sin, cos ae all entie unctions. In 1926 Rol Nevanlinna initiated the value distibution theoy o entie unctions which is a pominent banch o Complex Analysis and is the pime concen o this pape. In this line the value distibution theoy studies how an entie unction assumes some values and convesely, what is in some speciic manne the inluence on a unction o takin cetain values. It also deals with vaious aspects o the behaviou o entie unctions one o which is the study o compaative owth popeties o entie unctions. Fo any entie unction, the so called maximum modulus unction and denoted by M, is deined on each non-neative eal value by M () = max z = (z). With the aim o estimatin the owth o a nonconstant entie unction, Boas (Boas, 1954) intoduced the concept o ode as the value ρ which is eneally used in computational pupose and is deined in tems o the owth o espect to the exp z unction as lo lo M () lo lo M () ρ = lim sup = lim sup lo lo M exp () lo () ( 0 ρ ). Given anothe entie unction, the atio M () M () as is called the owth o with espect to in tems o thei maximum moduli. I this elative owth happens to be k R, then M () km () as. With the aim o knowin the elative owth o unctions o the same nonzeo inite ode, the type o a iven such untion was intoduced as lo M () ( τ = lim sup ρ 0 τ ). L. Benal (Benal, 1988) intoduced the elative ode between two entie unctions to avoid compain owth just with exp. Thus the owth o entie unctions may be studied in tems o its elative odes. In act, some woks on elative ode o entie unctions and the owth estimates o composite entie unctions on the basis o it have been exploed in (Chakaboty & Roy, 2006; Datta, Biswas, 2009; Datta, Biswas, 2010; Datta, Biswas, Biswas, 2013; Datta, Biswas & Biswas, 2013; Datta, Biswas, & Pamanick, 2012; Lahii & Banejee, 2005). This has dieent applications elated to entopy as this is the amount o additional inomation needed to speciy the exact physical state o a system, and elative ode o unctions measues how dieent in owth two iven unctions ae. Indeed vey ecently these ideas have been 1

2 Jounal o Mathematics Reseach Vol. 8, No. 5; 2016 used by Albuqueque et al. (Albuqueque, Benal-González, Pelleino, & Seoane-Sepúlveda, 2014) who obtained new Peano type esults by showin that the subset o continuous sujections om R m to C n such that each value a in C n is assumed on an unbounded set o R m is maximal stonly alebable, i.e. thee exists a c-eneated ee aleba contained in CS (R m, C n ) {0}, whee CS (R m, C n ) denotes the set o all continuous sujective mappins R m C n. On the othe hand, Sánchez Ruiz et al. (Sánchez Ruiz, Datta, Biswas, & Mondal, 2014) have intoduced a new type o elative (p, q)th ode o entie unctions whee p, q ae any two positive intees evisitin the ideas developed by a numbe o authos includin Lahii and Banejee (Lahii & D. Banejee, 2005). Howeve, these concepts ae not adequate o compain the owth o entie unctions with eithe zeo o ininite ode. Fo this eason Valion (Valion, 1949) intoduced the concept o a positive continuous unction ρ () o an entie unction havin inite ode ρ with the ollowin popeties: (i) ρ () is non-neative and continuous o > 0, say, (ii) ρ () is dieentiable o 0 except possibly at isolated points at which ρ ( + 0) and ρ ( 0) exist, (iii) lim ρ () = ρ, (iv) lim ρ () lo = 0 and (v) lim sup lo M () ρ () Such a unction is called a Lindelö poximate ode which makes unnecessay to conside unctions o minimal o maximal type, its existence bein established op. cit. It was simpliied by Shah (Shah, 1946), and Nandan et al. (Nandan, Doheey, & Sivastava, 1980) extended this notion o poximate ode o an entie unction o one complex vaiable with index-pai (p, q) with positive intees p q. Also Lahii (Lahii, 1989) enealised the idea o the poximate ode o a meomophic unction with inite enealised ode and poved its existence. As a consequence o the above it seems easonable o any two positive intees, p, q, to deine the elative (p, q)th poximate ode o an entie unction with espect to anothe entie unction. In this pape we do so and pove its existence. 2. Notation and Peliminay Remaks Ou notation is standad within the theoy o Nevanlinna s value distibution o entie unctions, Fo shot, iven a eal unction h and wheneve the coespondin domain and ane allow it we will use the notation h 0] (x) = x, and h k] (x) = h ( h k 1] (x) ) o k = 1, 2, 3,... omittin the paenthesis when h happens to be the lo o exp unction. Takin this into account the ode (esp. lowe ode) o an entie unction is iven by lo 2] M () ρ = lim sup (esp. λ = lim in lo lo 2] M () ). lo Let us ecall that Juneja, Kapoo and Bajpai (Juneja, Kapoo, Bajpai, 1976) deined the (p, q)-th ode (esp. (p, q)-th lowe ode) o an entie unction as ollows: lo p] M () ρ (p, q) = lim sup lo q] (esp. λ (p, q) = lim in lo p] M () lo q] ), whee p, q ae any two positive intees with p q. These deinitions extended the enealized ode ρ l] (esp. enealized lowe ode λ l] ) o an entie unction consideed in (Sato, 1963) o each intee l 2 since these coespond to the paticula case ρ l] = ρ (l, 1) (esp. λ l] = λ (l, 1)). Clealy ρ (2, 1) = ρ and λ (2, 1) = λ. Related to this, let us ecall the ollowin popeties. I 0 < ρ (p, q) <, then ρ (p n, q) = o n < p, ρ (p, q n) = 0 o n < q, ρ (p + n, q + n) = 1 o n = 1, 2,... 2

3 Jounal o Mathematics Reseach Vol. 8, No. 5; 2016 Similaly o 0 < λ (p, q) <, one can easily veiy that λ (p n, q) = o n < p, λ (p, q n) = 0 o n < q, λ (p + n, q + n) = 1 o n = 1, 2,... Recallin that o any pai o intee numbes m, n the Koenecke unction is deined by δ m,n = 1 o m = n and δ m,n = 0 o m n, the aoementioned popeties povide the ollowin deinition. Deinition 1. (Juneja, Kapoo, Bajpai, 1976) An entie unction is said to have index-pai (1, 1) i 0 < ρ (1, 1) <. Othewise, is said to have index-pai (p, q) (1, 1), p q 1, i δ p q,0 < ρ (p, q) < and ρ (p 1, q 1) R +. Deinition 2. (Juneja, Kapoo, Bajpai, 1976) An entie unction is said to have lowe index-pai (1, 1) i 0 < λ (1, 1) <. Othewise, has lowe index-pai (p, q) (1, 1), p q 1, i δ p q,0 < λ (p, q) < and λ (p 1, q 1) R +. Given a non-constant entie unction deined in the open complex plane, its maximum modulus unction M is stictly inceasin and continuous. Hence thee exists its invese unction M 1 : ( (0), ) (0, ) with lim s M 1 (s) =. Benal (Benal, 1988) intoduced the deinition o elative ode o with espect to, denoted by ρ ( ), as ollows: ρ ( ) = in { µ > 0 : M () < M ( µ ) o all > 0 (µ) > 0 } = lim sup lo M 1 M (). lo This deinition coincides with the classical one (Titchmash, 1968) i = exp. Analoously, the elative lowe ode o with espect to, denoted by λ ( ), is deined as λ ( ) = lim in lo M 1 M (). lo Recently, Sánchez Ruiz et al. (Sánchez Ruiz, Datta, Biswas, & Mondal, 2014) have intoduced a deinition o elative (p, q)-th ode ( ) o an entie unction with espect to anothe entie unction, shapennin an ealie deiniton o elative (p, q)-th ode o Lahii and Banejee (Lahii & Banejee, 2005), om which the moe natual paticula case ρ (k,1) ( ) = ρ k ( ) aises. This is done as ollows. Deinition 3. Let, be two entie unctions with index-pais (m, q) and (m, p), espectively, whee p, q, m ae positive intees with m max(p, q). Then the elative (p, q)-th ode o with espect to is deined by ( ) = lim sup lo p] M 1 And the elative (p, q)-th lowe ode o with espect to is deined by λ ( ) = lim in M () lo q]. lo p] M 1 M () lo q]. When (m, 1) and (m, k) ae the index-pais o and espectively, then Deinition 3 educes to deinition o enealized elative ode (Lahii & Banejee, 2002). I the entie unctions and have the same index-pai (p, 1), we et the deinition o elative ode intoduced by Benal (Benal, 1988) and i = exp m 1], then ρ ( ) = ρ m] and ( ) = ρ (m, q). Also Deinition 3 becomes the classical one iven in (Titchmash, 1968) i is an entie unction with index-pai (2, 1) and = exp. In ode to eine the above owth scale, now we intend to intoduce the deinition o an intemediate compaison unction, called elative (p, q)th poximate ode o entie unction with espect to anothe entie unction in the liht o thei indexpai which is as ollows. Its consistency will be established in Section 3. Deinition 4. Let, be two entie unctions with index-pais (m, q) and (m, p) espectively whee p, q, m ae positive intees with m max(p, q). Fo a inite elative (p, q)-th ode ( ) o with espect to, then a unction ( ) () : R + R is said to be a elative (p, q)th poximate ode o with espect to i thee is some 0 > 0 so that it satisies: 3

4 Jounal o Mathematics Reseach Vol. 8, No. 5; 2016 (i) ( ) () is non-neative and continuous o > 0, (ii) exist, ( ) () is dieentiable o 0 except possibly at isolated points whee (iii) lim ( ) () = ( ), (iv) lim () max lo i] = 0, lo (v) lim sup p 1] M 1 M () ] lo q 1] ρ ( )() ( ) ( + 0) and ( ) ( 0) When (m, 1) and (m, k) ae the index-pais o and espectively, Deinition 4 educes to deinition o enealized elative poximate ode. I the entie unctions and have the same index-pai (p, 1), the above deinition povides the elative poximate ode ρ ( ) (). The elative (p, q)th lowe poximate ode o an entie unction with espect to anothe entie unction may analoously be deined, consistency bein held by vitue o Section 3, too. Deinition 5. Let and be any two entie unctions with index-pais (m, q) and (m, p) espectively whee p, q, m ae positive intees such that m max(p, q). Fo a inite elative (p, q)-th lowe ode o with espect to, λ ( ), then a unction λ ( ) () : R + R is said to be a elative (p, q)th lowe poximate ode o with espect to i thee is some 0 > 0 so that it satisies: (i) λ ( ) () is non-neative and continuous o > 0, (ii) λ exist, ( ) () is dieentiable o 0 except possibly at isolated points at which λ (iii) lim λ ( ) () = λ ( ), (iv) lim λ () max lo j] = 0, lo (v) lim in p 1] M 1 M () ] lo q 1] λ ( )() ( ) (+0) and λ ( ) ( 0) 3. Main Results In this section we state the main esults o the pape. We include the poo o the ist main Theoem 1 o the sake o completeness. The othes ae basically omitted since they ae easily poved with the same techniques o with some easy easonins. Theoem 1. Let, be any two entie unctions with index-pais (m, q) and (m, p) espectively whee p, q, m ae positive intees with m max(p, q). I the elative (p, q)-th ode ( ) is inite, then the elative (p, q)th poximate ode ( ) () o with espect to exists. Poo. We distinuish the ollowin two cases: Case I. Assume p q. Then we wite and it can be easily poved that σ () is continuous and σ () = lop] M 1 M () lo q] lim sup σ () = ρ ( ). Now we conside the ollowin thee sub cases: Sub Case A I. Let σ () > ρ ( ) o at least a sequence o values o tendin to ininity. Then we deine the non inceasin eal unction ϕ() = max{σ (x)}. x 4

5 Jounal o Mathematics Reseach Vol. 8, No. 5; 2016 Now let us take R 1 > R with R 1 > exp p+2] 1 and σ (R) > ( ). Then o any iven R 1, we obtain that σ () σ (R). As σ () is continuous, thee exists 1 R, R 1 ] such that σ ( 1 ) = max R x R 1 {σ (x)}. Clealy 1 > exp p+2] 1 and ϕ( 1 ) = σ ( 1 ), thee bein a sequence o such 1 values tendin to ininity. Let us now conside that ( ) ( 1 ) = ϕ( 1 ) and let t 1 be the smallest intee not smalle than such that ϕ( 1 ) > ϕ(t 1 ). Also we deine ( ) () = ρ ( ) ( 1 ) o 1 < t 1. Now we obseve that: (i) ϕ() and ( ) ( 1 ) lo p+2] + lo p+2] t 1 ae continuous unctions, (ii) ( ) ( 1 ) lo p+2] + lo p+2] t 1 > ϕ(t 1 ) o (> t 1 ) suiciently close to t 1 and (iii) ϕ() is non inceasin. Consequently we can deine u 1 > t 1 as ollows: ( ) () = ( ) ( 1 ) lo p+2] + lo p+2] t 1 o t 1 u 1, ( ) () = ϕ() o = u 1 and ( ) () > ϕ() o t 1 < u 1. ( ) () = ϕ() o ( ) () ae both constant in u 1 2. By epeatin this ( ) () is dieentiable in adjacent intevals. Let now 2 be the smallest value o o which 2 u 1 and ϕ( 2 ) = σ ( 2 ). I 2 > u 1 then let u 1 2. Then it can be easily shown that ϕ() and pocess, we obtain that Moeove () coincides with 0 o ( p+1 loi] ) 1 and ( ) () ϕ() σ () o all 1. Also ( ) () = σ () o a sequence o values o tendin to ininity and ( ) () is non inceasin o 1. So and Aain we et that ( ) = lim sup σ () = lim ϕ () i.e., lim sup ( ) () = lim in ρ ( ) () = lim ( ) () = ( ) lo p 1] M 1 lim o a sequence o values o tendin to ininity and ρ () p lo i] = 0. M () = lo q 1] ] σ() ] = lo q 1] ρ ( )() lo p 1] M 1 M () < lo q 1] ] ( )() o the emanin s. Hence lim sup lo p 1] M 1 M () ] lo q 1] ρ ( )() 5

6 Jounal o Mathematics Reseach Vol. 8, No. 5; 2016 The continuity o ρ ( ) () o 1 ollows by constuction. Sub Case B I. Let σ () < ( ) o all suiciently lae values o tendin to ininity. Now we deine the eal unction ξ() = max {σ (x)}, X x whee X > exp p+2] 1 is such that σ () < ( ) wheneve x X. Hee we note that ξ() is non deceasin and the oots o ae smalle than o all suiciently lae values o X. ξ(x) = ρ ( ) + lo p+2] x lo p+2] Now o a suitable lae value v 1 > X, we deine ( ) (v 1 ) = ( ), ( ) () = ( ) + lo p+2] lo p+2] v 1 o s 1 v 1 whee s 1 < v 1 is such that ξ(s 1 ) = ( ) (s 1 ). In act s 1 is iven by the laest positive oot o ξ(x) = ( ) + lo p+2] x lo p+2] v 1. I ξ(s 1 ) σ (s 1 ) let ω 1 be an uppe bound o the ω < s 1 at which ξ(ω) is dieent om σ (ω). I we deine ( ) () = ξ() o ω 1 s 1, it is clea that ξ() is constant in ω 1, s 1 ], hence ( ) () is constant in ω 1, s 1 ], too. I ξ(s 1 ) = σ (s 1 ) we take ω 1 = s 1. Now we choose v 2 > v 1 suitably lae and let ( ) () = ( ) + lo p+2] lo p+2] v 2 o s 2 v 2 whee s 2 < v 2 is such that ξ(s 2 ) = ( ) (s 2 ). ( ) (v 1 ) = ( ) and I ξ(s 2 ) ( ) (s 2 ) then suppose that ρ ( ) () = ξ() o ω 2 s 2, with ω 2 mimickin the behavou o ω 1. Hence ( ) () is constant in ω 2, s 2 ]. I ξ(s 2 ) = σ (s 2 ) we take ω 2 = s 2. Also suppose that intesection o y = ( ) () = ( ) (ω 2 ) lo p+2] + lo p+2] ω 2 o q 1 ω 2 whee q 1 < ω 2 is the point o ( ) with y = ( ) (ω 2 ) lo p+2] x + lo p+2] ω 2. Now it is also possible to choose v 2 so lae that v 1 < q 1 and o the case unde consideation, let us conside epeat this pocess it can be shown that o all v 1, = ω 1, ω 2,... Hence we obtain that since lim sup ( ) () = lim in lo p 1] M 1 o a sequence o values o tendin to ininity and o emanin s. Theeoe it ollows that ρ ( ) ( ) () = lim ( ) () = ( ) o v 1 q 1. Theeoe i we ( ) () ξ() σ () and ( ) () = σ () o ( ) () = ( ) M () = lo q 1] ] σ() ] = lo q 1] ρ ( )() lo p 1] M 1 lim sup M () < lo q 1] ] ( )() lo p 1] M 1 M () ] lo q 1] ρ ( )() 6

7 Jounal o Mathematics Reseach Vol. 8, No. 5; 2016 Futhemoe, ( ) () is dieentiable in adjacent intevals and Consequently, () = 0 o lim ρ () p+1 lo i] 1 p lo i] = 0. Once aain, continuity o ρ ( ) () ollows by constuction. Sub Case C I. Let σ () = ( ) o at least a sequence o values o tendin to ininity. Now considein ( ) () = ( ) o all suiciently lae values o one can easily veiy the existance o the elative (p, q)th poximate ode o the case unde consideation. Case II. Assume q p. Now let us conside the ollowin unction Theeoe it can easily be shown that Now puttin x = lo q] and y = lo σ (), we obtain that So σ () = lo q 1] ] ( ) lo p 1] M 1 M (). lo σ () lim sup lo q] = 0. y = lo σ ( exp q] x ). lo σ ( exp q] x ) lo σ () lim sup = lim sup x lo q] = 0 which shows that o any abitay ε > 0 and o lae values o x, x x 0 (ε), the entie cuve y = lo σ ( exp q] x ) lies below the line y = εx and, on the othe hand, thee ae points on the cuve with abitaily lae abscissae lyin above the line y = εx. Now we conside the ollowin two sub cases: Sub Case A II. Let us conside that lim sup lo σ ( exp q] x ) = +. Now we constuct the smallest convex domain so that it contains the positive ay o the x axis and all the points o the cuve y = lo σ ( exp q] x ). Thus the bounday o newly omed domain lyin above the x-axis is a continuous cuve and we denote it as y = δ (x). This cuve must satisy the ollowin popeties: (I) The cuve is convex om the above, (II) lim x δ(x) x = 0, (III) lo σ ( exp q] x ) δ (x),. (IV) lo σ ( exp q] x ) = δ (x) at the exteme points o the cuve y = δ (x) and (V) The cuve y = δ (x) contains a sequence o exteme points tendin to ininity. Also the cuve y = δ (x) is made dieentiable in the neihbouhood o each anula point ( i necessay) by makin some unessential chanes. Thus it is assumed that the cuve y = δ (x) is dieentiable eveywhee. Hence om (I) and (II), above it ollows that lim x δ (x) = 0 and om (III) we have lo p 1] M 1 M () lo q 1] ] ( )() 7

8 Jounal o Mathematics Reseach Vol. 8, No. 5; 2016 whee Now om (II) it ollows that lim ρ ( ) () = lim ( ) () = ρ ( ) + δ ( ) lo q] lo q]. ( ) + δ ( ) lo q] lo q] = ρ ( ). Also in view o the popeties (IV) and (V) one can easily veiy that thee exists a sequence o values o tendin to ininity o which i.e., lim sup lo p 1] M 1 M () = lo q 1] ] ( )() lo p 1] M 1 M () ] lo q 1] ρ ( )() = 1 and lim () q loi] = 0 holds. Thus we have constucted the unction ( ) (). Sub Case B II. In ode to enealize the case, let us conside a concave unction β (x) which satisies the ollowin popeties: (I) lim x β (x) = 0, (II) lim x β(x) x = 0 and (III) lim sup lo σ ( exp q] x ) + β (x) ] =. With the oal o constuctin β (x) we o thouh the ollowin steps: Fist we conside a sement a 1 o the line y = ε 1 x om the oiin to a point x 1 whee lo σ ( exp q] x 1 ) > ε1 x Havin chosen a positive numbe ε 2 < ε 1 we daw a sement a 2 o the line y + ε 1 x 1 = ε 2 (x x 1 ) om the point (x 1, ε 1 x 1 ) to a point x 2 > x 1 satisyin lo σ ( exp q] x 2 ) > ε1 x 1 ε 2 (x 2 x 1 ) + 2. Then we choose a sement a 3 with slope ε 3 (0 < ε 3 < ε 2 ), etc. The selected {ε n } is stictly deceasin with ε n 0 but the sequence {x n } o points is stictly inceasin with x n. The polyonal unction y = β 1 (x) constucted in this manne satisies β 1 (x) lim = 0. x x The unction β 1 (x) can be made eveywhee dieentiable by chanin it in an unessential manne in the neihbouhood o each anula point. The unction β (x) deined as β (x) = β 1 (x) has the equied popeties. A convex majoant β 2 (x) o the unction lo σ ( exp q] x ) + β (x) is now consideed and witin yields Moeove, on some sequence { x n} 1 o exteme points, x n. Also i the unction ρ ( ) () is deined as δ (x) = β 2 (x) β (x) lo σ ( exp q] x ) δ (x). lo σ ( exp q] x n ( ) () = ) = δ ( x n ) ( ) + δ ( ) lo q] lo q], 8

9 Jounal o Mathematics Reseach Vol. 8, No. 5; 2016 it can easily be seen that Hence lim ( ) () = ( ) and lim x δ (x) = 0, δ (x) lim = 0. x x Moeove and o some sequence { n }, n. Theeoe lim ρ lo p 1] M 1 () q lo i] = 0. M () lo q 1] ] ( )() lo p 1] M 1 M ( n ) = lo q 1] ] ρ ( )( n ) n lim sup lo p 1] M 1 M () ] lo q 1] ρ ( )() = 1, and the poo is complete. The ollowin theoem s poo can be obtained in the line o Theoem 1. Theoem 2. Let, be any two entie unctions with index-pais (m, q) and (m, p), espectively whee p, q, m ae positive intees with m max(p, q). I the elative (p, q)-th lowe ode λ ( ) o with espect to is inite and non zeo, then the elative (p, q)th lowe poximate ode λ ( ) () o with espect to exists. Now we ecall the that a positive unction η () is called slowly inceasin (Sivastava & Kuma, 2009), i lim η(n) η() We will say that η () is uniom slowly inceasin i the aoementioned limit happens to exist uniomly in m on each inteval 0 < b n < m <. The poos o the ollowin coollay can be caied out usin the same techniques involved in (Nandan, Doheey, & Sivastava, 1980). Coollay 1. Let ( ) () and λ ( ) () be espectively the elative (p, q)th poximate ode and the elative (p, q)th lowe poximate ode o with espect to, and let ρ (p, q)-th lowe ode o with espect to o any positive intees p and q. Then: ( ) and λ ( ) be the elative (p, q)-th ode and elative 1. The unctions ] lo q 1] ρ ( )() lo q 1] ] ρ ( ) and ] lo q 1] λ ( )() lo q 1] ] λ ( ) ae uniom slowly inceasin. 2. The unctions lo q 1] ] ( )() ] and lo q 1] λ ( )() ae monotone inceasin o suiciently lae values o. 3. Fo 0 < l k m < and, we have that ] lo q 1] ρ ( )(k) ] (k) lo q 1] ρ ( ) lo q 1] (k) ] ρ ( ) ] lo q 1] ρ ( )() ] lo q 1] λ ( )(k) ] (k) lo q 1] λ ( ) lo q 1] (k) ] λ ( ) ] lo q 1] λ ( )() 1, 1 hold uniomly in k. 9

10 Jounal o Mathematics Reseach Vol. 8, No. 5; Fo γ < min {( 1 + ( ) ), ( 1 + λ ( ) )}, we have 0 lo q 1] x ] ρ 1 ( ) γ+1 ( )(x) γ lo q 1] ] ρ dx q 2 loi] x = ( )() γ+1 + o lo q 1] ] ρ ( )() γ+1 and 0 lo q 1] x ] λ λ 1 ( ) γ+1 ( )(x) γ lo q 1] ] λ dx q 2 loi] x = ( )() γ+1 + o lo q 1] ] λ ( )() γ Conclusions The main aim o the pape is to extend and modiy the notion o poximate ode (lowe poximate ode) to elative poximate ode (elative lowe poximate ode) o hihe dimentions in case o entie unctions. The esults o this pape, in connection with Nevanlinna s Value Distibutibution theoy o entie unctions on the basis o elative (p, q)th poximate ode and elative (p, q)th poximate lowe ode, may have a wide ane o applications in Complex Dynamics, Factoization Theoy o entie unctions o sinle complex vaiable, the solution o complex dieential equations etc. In act, Complex Dynamics is a thust aea in moden unction theoy and it is solely based on the study o ixed points o entie unctions as well as the nomality o them. Factoization theoy o entie unctions is anothe banch o applications o Nevanlinna s theoy which deals on how a iven entie unction can be actoized into simple entie unctions as well as in the study o the popeties o the solutions o complex dieential equations. Competin Inteests Section The authos declae that they have no competin inteests. Acknowledements Suppoted by Spanish Ministeio de Economía y Competitividad. Secetaía Geneal de Ciencia y Tecnoloía e Innovación ESP R. Reeences Albuqueque, N., Benal-González, L., Pelleino, D., & Seoane-Sepúlveda, J. B. (2014). Peano cuves on topoloical vecto spaces. Linea Aleba Appl., 460, Benal, L. (1988). Oden elativo de cecimiento de unciones enteas. Collect. Math., 39, Boas, R. P. (1954). Entie unctions. Academic Pess, New Yok. Chakaboty, B. C., & Roy, C. (2006). Relative ode o an entie unction. J. Pue Math., 23, Datta, S. K., & Biswas, T. (2009). Gowth o entie unctions based on elative ode. Int. J. Pue Appl. Math.,51(1), Datta, S. K., & Biswas, T. (2010). Relative ode o composite entie unctions and some elated owth popeties. Bull. Cal. Math. Soc., 102(3), Datta, S. K., Biswas, T., & Biswas, R. (2013). On elative ode based owth estimates o entie unctions. Intenational J. o Math. Sci. & En. Appls. (IJMSEA), 7(II), Datta, S. K., Biswas, T., & Pamanick, D. C. (2012). On elative ode and maximum tem-elated comaative owth ates o entie unctions. Jounal Ti. Math. Soc., 14, Datta, S. K., Biswas, T., & Biswas, R. (2013). Compaative owth popeties o composite entie unctions in the liht o thei elative ode. The Mathematics Student, 82(1-4), 1-8. Juneja, O. P., Kapoo, G. P., & Bajpai, S. K. (1976). On the (p, q)-ode and lowe (p, q)-ode o an entie unction. J. Reine Anew. Math., 282, Lahii, I. (1989). Genealised poximate ode o meomophic unctions. Mat. Vesnik, 41, Lahii, B. K., & Banejee, D. (2002). Genealised elative ode o entie unctions. Poc. Nat. Acad. Sci., 72(A)(IV), Lahii, B. K., & Banejee, D. (2005). A note on elative ode o entie unctions. Bull. Cal. Math. Soc., 97(3),

11 Jounal o Mathematics Reseach Vol. 8, No. 5; 2016 Lahii, B. K., & Banejee, D. (2005). Entie unctions o elative ode (p, q). Soochow Jounal o Mathematics, 31(4), Nandan, K., Doheey, R. P., & Sivastava, R. S. L. (1980). Poximate ode o an entie unction with index pai (p, q). Indian J. Pue Appl. Math., 11(1), Roy, C. (2010). On the elative ode and lowe elative ode o an entie unction. Bull. Cal. Math. Soc., 102(1), Sánchez Ruiz, L. M., Datta, S. K., Biswas, T., & Mondal, G. K. (2014). On the (p, q)-th Relative Ode Oiented Gowth Popeties o Entie Functions. Abstact and Applied Analysis, Aticle ID , 8 paes. Sato, D. (1963). On the ate o owth o entie unctions o ast owth. Bull. Ame. Math. Soc., 69, Shah, S. M. (1946). On poximate odes o inteal unctions, Bull. Ame. Math. Soc., 52, Sivastava, G. S., Kuma, S. (2009). Mathematicum, 45(2), Appoximation o entie unctions o slow owth on compact sets, Achivum Titchmash, E. C. (1968). The theoy o unctions (2nd ed.). Oxod Univesity Pess, Oxod. Valion, G. (1949). Lectues on the Geneal Theoy o Inteal Functions. Chelsea Publishin Company, N.Y. Copyihts Copyiht o this aticle is etained by the autho(s), with ist publication ihts anted to the jounal. This is an open-access aticle distibuted unde the tems and conditions o the Ceative Commons Attibution license ( 11