Jounal o Mathematics Reseach; Vol. 8, No. 5; Octobe 2016 ISSN 1916-9795 E-ISSN 1916-9809 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-46022 Valencia, Spain. 2 Depatment o Mathematics, Univesity o Kalyani Kalyani, Dist-Nadia, PIN-741235, West Benal, India. 3 Rajbai, Rabindapalli, R. N. Taoe Road P.O.- Kishnaa, Dist-Nadia, PIN- 741101, West Benal, India. 4 Guu Nanak Institute o Technoloy 157/F, Nilunj Road, Panihati, Sodepu Kolkata-700114, West Benal, India. Coespondence: Luis M. Sánchez Ruiz, ETSID-Depto. de Matemática Aplicada & CITG, Univesitat Politècnica de València, E-46022 Valencia, Spain. E-mail: LMSR@mat.upv.es Received: June 27, 2016 Accepted: July 12, 2016 Online Published: Septembe 14, 2016 doi:10.5539/jm.v8n5p1 Abstact URL: http://dx.doi.o/10.5539/jm.v8n5p1 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
http://jm.ccsenet.o 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
http://jm.ccsenet.o 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
http://jm.ccsenet.o 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
http://jm.ccsenet.o 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 1 + 1 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
http://jm.ccsenet.o 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
http://jm.ccsenet.o 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
http://jm.ccsenet.o 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 1 + 1. 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
http://jm.ccsenet.o 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
http://jm.ccsenet.o Jounal o Mathematics Reseach Vol. 8, No. 5; 2016 4. 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+1. 4. 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 ESP2013-41078-R. Reeences Albuqueque, N., Benal-González, L., Pelleino, D., & Seoane-Sepúlveda, J. B. (2014). Peano cuves on topoloical vecto spaces. Linea Aleba Appl., 460, 81-96. http://dx.doi.o/10.1016/j.laa.2014.07.029 Benal, L. (1988). Oden elativo de cecimiento de unciones enteas. Collect. Math., 39, 209-229. 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, 151-158. Datta, S. K., & Biswas, T. (2009). Gowth o entie unctions based on elative ode. Int. J. Pue Appl. Math.,51(1), 49-58. Datta, S. K., & Biswas, T. (2010). Relative ode o composite entie unctions and some elated owth popeties. Bull. Cal. Math. Soc., 102(3), 259-266. 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), 59-67. 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, 60-68. 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, 53-67. Lahii, I. (1989). Genealised poximate ode o meomophic unctions. Mat. Vesnik, 41, 9-16. Lahii, B. K., & Banejee, D. (2002). Genealised elative ode o entie unctions. Poc. Nat. Acad. Sci., 72(A)(IV), 351-371. Lahii, B. K., & Banejee, D. (2005). A note on elative ode o entie unctions. Bull. Cal. Math. Soc., 97(3), 201-206. 10
http://jm.ccsenet.o 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), 497-513. 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), 33-39. Roy, C. (2010). On the elative ode and lowe elative ode o an entie unction. Bull. Cal. Math. Soc., 102(1), 17-26. 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 826137, 8 paes. http://dx.doi.o/10.1155/2014/826137 Sato, D. (1963). On the ate o owth o entie unctions o ast owth. Bull. Ame. Math. Soc., 69, 411-414. http://dx.doi.o/10.1090/s0002-9904-1963-10951-9 Shah, S. M. (1946). On poximate odes o inteal unctions, Bull. Ame. Math. Soc., 52, 326-328. http://dx.doi.o/10.1090/s0002-9904-1946-08572-9 Sivastava, G. S., Kuma, S. (2009). Mathematicum, 45(2), 137-146. 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 (http://ceativecommons.o/licenses/by/4.0/). 11