Research Article On Growth of Meromorphic Solutions of Complex Functional Difference Equations

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1 Abstract ad Applied Aalysis, Article ID , 6 pages Research Article O Growth of Meromorphic Solutios of Complex Fuctioal Differece Equatios Jig Li, 1,2 Jiaju Zhag, 3 ad Liagwe Liao 1 1 Departmet of Mathematics, Najig Uiversity, Najig , Chia 2 Nakai Uiversity Bihai College, Tiaji , Chia 3 Mathematics ad Iformatio Techology School, Jiagsu Secod Normal Uiversity, Najig , Chia Correspodece should be addressed to Jiaju Zhag; zhagjiaju1982@163.com Received 29 November 2013; Accepted 13 Jauary 2014; Published 25 February 2014 Academic Editor: Zog-Xua Che Copyright 2014 Jig Li et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. The mai purpose of this paper is to ivestigate the growth order of the meromorphic solutios of complex fuctioal differece equatio of the form ( λ I α λ (z( f(z + c ] l λ,] /( μ J β μ (z( f(z + c ] m μ,] = Q(z, f(p(z, wherei = {λ = (l λ,1,l λ,2,...,l λ, l λ,] N {0}, ] = 1,2,...,}ad J={μ=(m μ,1,m μ,2,...,m μ, m μ,] N {0}, ] = 1,2,...,}are two fiite idex sets, c ] (] = 1,2,...,are distict complex umbers, α λ (z (λ I ad β μ (z (μ J aresmallfuctiosrelativetof(z, ad Q(z, u is a ratioal fuctio i u with coefficiets which are small fuctios of f(z, p(z = p k z k +p k 1 z k 1 + +p 0 C[z] of degree k 1.Wealsogivesomeexamplestoshowthatourresultsaresharp. 1. Itroductio ad Mai Results Let f(z be a fuctio meromorphic i the complex plae C. We assume that the reader is familiar with the stadard otatios ad results i Nevalia s value distributio theory of meromorphic fuctios such as the characteristic fuctio T(r, f, proximity fuctio m(r, f, coutig fuctio N(r, f, ad the first ad secod mai theorems (see, e.g., [1 4].WealsouseN(r, f to deote the coutig fuctio of the poles of f(z whoseeverypoleiscoutedolyoce.the otatios ρ(f ad μ(f deote the order ad the lower order of f(z,respectively.s(r, f deotes ay quatity that satisfies the coditio: S(r, f = o(t(r, f as r possibly outside a exceptioal set of r of fiite liear measure. A meromorphic fuctio a(z is called a small fuctioof f(z or a small fuctio relative to f(z if ad oly if T(r, a(z = S(r, f. Recetly, some papers (see, e.g., [5 7]focusigo complex differece ad fuctioal differece equatios emerged. I 2005, Laie et al. [5] firstly cosidered the growth of meromorphic solutios of the complex fuctioal differece equatios by utilizig Nevalia theory. They obtaied the followig result. Theorem A. Suppose that f is a trascedetal meromorphic solutio of the equatio {J} α J (z ( j J f (z+c j =f(p (z, (1 where {J} is a collectio of all subsets of {1,2,...,}, c j s are distict complex costats, ad p(z is a polyomial of degree k 2. Moreover, we assume that the coefficiets α J (z are small fuctios relative to f ad that k.the where α=log /. T(r,f=O((log r α+ε, (2 I 2007, Rieppo [6] gaveaestimatioofgrowthof meromorphic solutios of complex fuctioal equatios as follows. Theorem B. Suppose that f is a trascedetal meromorphic fuctio. Let Q(z, f, R(z, f be ratioal fuctios i f with small meromorphic coefficiets relative to f such that 0<q:=deg f Qd:=deg f R ad p(z = p k z k +p k 1 z k 1 +

2 2 Abstract ad Applied Aalysis + p 0 C[z] of degree k > 1.Iff isasolutioofthe fuctioal equatio R (z, f (z =Q(z,f(p(z, (3 the qkd,adforayε, 0 < ε < 1, thereexistpositivereal costats K 1 ad K 2 such that K 1 (log r α ε T(r,fK 2 (log r α+ε, whe r is large eough. log d log q α =, (4 Rieppo [6] also cosidered the growth order of meromorphic solutios of fuctioal equatio (3whe k=1ad got the followig. Theorem C. Suppose that f is a trascedetal meromorphic solutio of (3,wherep(z = az+b, a,b C, a =0ad a =1. The log d log q μ (f =ρ(f =. (5 Two years later, Zheg et al. [7] exteded Theorem A to more geeral type ad obtaied a similar result of Theorem C. I fact, they got the followig two results. Theorem D. Suppose that f is a trascedetal meromorphic solutio of the equatio {J} α J (z ( j J f(z+c j = Q (z, f (p (z, (6 where {J} is a collectio of all oempty subsets of {1,2,...,}, c j (j=1,...,are distict complex costats, p(z = p k z k + p k 1 z k 1 + +p 0 C[z] of degree k>1,adq(z, u is a ratioal fuctio i u of deg u Q=q(>0.Alsosupposethat all the coefficiets of (6 are small fuctios relative to f.the qk,ad where α=(log log q/. T(r,f=O((log r α+ε, (7 Theorem E. Suppose that f is a trascedetal meromorphic solutioof (6,where{J} is a collectio of all oempty subsets of {1,2,...,}, c j (j=1,...,are distict complex costats, p(z = az + b, a,b C,adQ(z, u is a ratioal fuctio i u of deg u Q=q(>0. Also suppose that all the coefficiets of (6 are small fuctios relative to f. (i If 0< a <1,thewehave (ii If a > 1,thewehaveqad log q log. (8 log log q. (9 (iii If a = 1, q >,thewehaveρ(f = μ(f =. I this paper, we will cosider a more geeral class of complex fuctioal differece equatios. We prove the followig results, which geeralize the above related results. Theorem 1. Suppose that f(z is a trascedetal meromorphic solutio of the fuctioal differece equatio λ I α λ (z ( f(z + c ] l λ,] μ J β μ (z ( f(z + c ] m μ,] =Q(z, f (p (z, (10 where c ] (] =1,...,aredistictcomplexcostats,I={λ= (l λ,1,l λ,2,...,l λ, l λ,] N {0}, ] = 1,2,...,} ad J= {μ = (m μ,1,m μ,2,...,m μ, m μ,] N {0}, ] = 1,2,...,} are two fiite idex sets, p(z = p k z k +p k 1 z k 1 + +p 0 C[z] of degree k>1,adq(z, u is a ratioal fuctio i u of deg u Q=q(>0. Also suppose that all the coefficiets of (10 are small fuctios relative to f.deotig σ ] = max λ,μ {l λ,],m μ,] } (] =1,2,...,, σ = The qkσ,ad where α=(/. σ ]. (11 T(r,f=O((log r α+ε, (12 Theorem 2. Suppose that f is a trascedetal meromorphic solutio of the equatio λ I α λ (z ( f(z + c ] l λ,] μ J β μ (z ( f(z + c ] m μ,] = Q (z, f (az+b, (13 where c ] (] =1,...,aredistictcomplexcostats,I={λ= (l λ,1,l λ,2,...,l λ, l λ,] N {0}, ] =1,2,...,}ad J={μ= (m μ,1,m μ,2,...,m μ, m μ,] N {0}, ] =1,2,...,}are two fiite idex sets, a,b C,adQ(z, u is a ratioal fuctio i u of deg u Q=q(>0. Also suppose that all the coefficiets of (10 are small fuctios relative to f.deotig σ ] = max λ,μ {l λ,],m μ,] } (] =1,2,...,, σ = (i If 0< a <1,thewehave (ii If a > 1,thewehaveqσad σ ]. (14. (15. (16 (iii If a = 1 ad q>σ,thewehaveμ(f = ρ(f =.

3 Abstract ad Applied Aalysis 3 Next we will give some examples to show that our results are best i some extet. Example 3. Let c 1 = arcta 2, c 2 = π/4.theitiseasyto check that f(z = ta z solves the followig equatio: f(z + c 1 2 f(z+c 2 f(z+c 1 +f(z+c 2 2 =( 4f( z f( z f( z f( z f( z f( z f( z 2 2 8f( z 2 4 (3f( z f( z f( z f( z 2 5 6f( z f( z f( z f ( z Obviously, we have μ(f=ρ(f=1= where q=8, σ=4ad a=1/2. (17, (18 Example 3 shows that the estimate i Theorem 2(i is sharp. Example 4. It is easy to check that f(z = ta z satisfies the equatio Clearly, we have f(z+(π/3 2 f (z+(π/6 f(z+(π/6 f (z+(π/3 f(z+(π/6 2 f(z+(π/3 = 3f(2z 2 +4f(2z + 3 3f(2z 2 +4f(2z 3. μ(f=ρ(f=1= where σ=4, q=2ad a=2. (19, (20 Example 4 shows that the estimate i Theorem 2(ii is sharp. Example 5. f(z = ta z satisfies the equatio of the form f(z+(π/4 2 f (z+(π/4 +f(z (π/4 2 (f(z/2 2 2f(z/2 1 3 = 8f (z/2 (f(z/2 2 1(f(z/2 2 +2f(z/2 1, (21 where σ = 4, q = 6,ada = 1/2. ρ(f = μ(f = 1 > log(3/2/ log 2=(/. Example 5 shows that the strict iequality i Theorem 2 may occur. Therefore, we do ot have the same estimatio as i Theorem C for the growth order of meromorphic solutios of (13. The followig Example shows that the restrictio q>σ i case (iii i Theorem 2 is ecessary. Example 6. Meromorphic fuctio f(z = ta z solves the followig equatio: f(z+(π/4 2 f (z+(π/4 +f(z (π/4 2 = (f (z +1 4f (z (1 f (z, (22 where a=1ad 4=σ>q=3,butρ(f = μ(f = 1. Next, we give a example to show that case (iii i Theorem 2 may hold. Example 7. Fuctio f(z = ze ez satisfies the followig equatio: (z + log 6 (z + log 2 5 [f(z + log 4 4 +f(z+log 4] (z + log 4 f (z + log 6 = (z + log 43 f(z + log 2 6 +(z+log 2 6, f(z+log 2 3 (23 where a=1ad q=6>5=σ.obviously,ρ(f = μ(f =. 2. Mai Lemmas I order to prove our results, we eed the followig lemmas. Lemma 1 (see [4, 8]. Let f(z be a meromorphic fuctio. The for all irreducible ratioal fuctios i f, P (z, f R (z, f = Q (z, f = p i=0 a i (z f i q j=0 b, (24 j (z fj such that the meromorphic coefficiets a i (z, b j (z satisfy the oe has T(r,a i =S(r,f, T(r,b j =S(r,f, i=0,1,...,p, j=0,1,...,q; (25 T(r,R(z,f=max {p,q} T(r,f+S(r,f. (26 From the proof of Theorem 1 i [9], we have the followig estimate for the Nevalia characteristic. Lemma 2. Let f 1,f 2,...,f be distict meromorphic fuctios ad F (z = P (z Q (z = λ I α l λ (z f λ,1 1 f l λ,2 2...f l λ, μ J β μ (z f m μ,1 1 f m μ,2 2...f m μ,. (27

4 4 Abstract ad Applied Aalysis The T (r, F (z σ ] T(r,f ] +S(r,f, (28 where I = {λ = (l λ,1,l λ,2,...,l λ, l λ,] N {0}, ] = 1,2,...,} ad J = {μ = (m μ,1,m μ,2,...,m μ, m μ,] N {0}, ] = 1,2,...,} are two fiite idex sets, σ ] = max λ,μ {l λ,],m μ,] }(] = 1,2,...,. α λ (z = o(t(r, f ] (λ I ad β μ (z = o(t(r, f ] (μ J hold for all ] {1,2,...,}ad satisfy T(r, α λ = S(r, f (λ I ad T(r, β μ = S(r, f (μ J. Lemma 3 (see [7]. Let c be a complex costat. Give ε>0 ad a meromorphic fuctio f,oe has T(r,f(z±c (1+ε T(r+ c,f, (29 for all r>r 0,wherer 0 is some positive costat. Lemma 4 (see [4]. Let g : (0, + R, h : (0, + R be mootoe icreasig fuctios such that g(r h(r outside of a exceptioal set E of fiite liear measure. The, for ay α>1, there exists r 0 >0such that g(r h(αr for all r>r 0. Lemma 5 (see [10]. Let f be a trascedetal meromorphic fuctio, ad p(z = a k z k +a k 1 z k 1 + +a 1 z+a 0,a k =0, be a ocostat polyomial of degree k. Give0<δ< a k, deote λ= a k +δad μ= a k δ.thegiveε>0ad a C { },oehas k (μr k,a,f(r,a,f(p(z k (λr k,a,f N(μr k,a,f+o(log r N (r, a, f (p (z N(λr k,a,f+o(log r (1 ε T (μr k,ft(r,f(p(z (1+ε T(λr k,f, (30 for all r large eough. Lemma 6 (see [11]. Let φ : [r 0,+ (0,+ be positive ad bouded i every fiite iterval, ad suppose that φ(μr m Aφ(r + B holds for all r large eough, where μ>0, m>1, A>1ad B are real costats. The where α=log A/ log m. φ (r =O((log r α, (31 Lemma 7 (see [6]. Let φ:(r 0, (1,,wherer 0 1, be a mootoe icreasig fuctio. If for some real costat α>1, there exists a real umber K>1such that φ(αr Kφ(r,the log φ (r log K lim r log r log α. (32 Lemma 8 (see [12]. Let φ : (1, (0, be a mootoe icreasig fuctio ad let f be a ocostat meromorphic fuctio. If, for some real costat α (0, 1, thereexistreal costats K 1 >0ad K 2 1such that the T (r, f K 1 φ (αr +K 2 T (αr, f +S(αr, f, (33 3. Proof of Theorems log K 2 log α + lim log φ (r r log r. (34 Proof of Theorem 1. We assume f(z is a trascedetal meromorphic solutio of (10. Deotig C = max{ c 1, c 2,..., c }. Accordig to Lemmas 1, 2, ad 3 adthelastassertiooflemma 5,wegetthatforayε 1 >0, q(1 ε 1 T (μr k,f+s(r,f qt(r,f(p(z + S (r, f = T (r, Q (z, f (p (z =T(r, λ I α λ (z ( f(z + c ] l λ,] μ J β μ (z ( f(z + c ] m μ,] σ ] T(r,f(z+c ] + S (r, f σ ] (1 + ε 1 T(r+C,f(z+S(r,f =( σ ] (1+ε 1 T(r+C,f(z+S(r,f =σ(1+ε 1 T(r+C,f(z+S(r,f, (35 where r is large eough ad μ= p k δfor some 0<δ< p k. Sice T(r + C, f T(βr, f holds for r large eough for β>1, we may assume r tobelargeeoughtosatisfy q(1 ε 1 T(μr k,fσ(1+ε 1 T(βr,f (36 outside a possible exceptioal set of fiite liear measure. By Lemma 4, we kow that wheever γ>1, q (1 ε 1 T (μr k,f σ(1+ε 1 T (γβr, f (37 holds for all r large eough. Deote t = γβr;thusthe iequality (37maybewritteitheform By Lemma 6,wehave T( μ (γβ k tk,f σ(1+ε 1 T(t,f. (38 q(1 ε 1 T(r,f=O((log r α 1, (39

5 Abstract ad Applied Aalysis 5 where α 1 = log (σ (1 + ε 1/q(1 ε 1 = + log ((1 + ε 1/(1 ε 1. (40 Deotig ow α = (/ ad ε = log((1 + ε 1 /(1 ε 1 / ;thusweobtaitherequiredform. Fially, we show that qk σ. Ifqk > σ, thewehave α<1.forsufficietlysmallε>0,wehaveα+ε<1,which cotradicts with the trascedecy of f. Thus Theorem 1 is proved. Proof of Theorem 2. Suppose f(z is a trascedetal meromorphic solutio of (13. Deotig C = max{ c 1, c 2,..., c }. (i 0< a <1. We may assume that q>σ,sicethecase qσis trivial by the fact that μ(f 0.ByLemmas 1 3,we have forayε>0ad β>1, qt (r, f (p (z + S (r, f = T (r, Q (z, f (p (z =T(r, λ I α λ (z ( f(z + c ] l λ,] μ J β μ (z ( f(z + c ] m μ,] σ ] T(r,f(z+c ] + S (r, f σ ] (1+ε T(r+C,f(z+S(r,f =( σ ] (1+ε T(r+C,f(z+S(r,f =σ(1+ε T(r+C,f(z+S(r,f σ(1+ε T(βr,f+S(r,f, (41 where r is large eough. By the last assertio of Lemma 5 ad (41, we obtai that, for μ = a δ (0 < δ < a, 0 < μ < 1, thefollowig iequality T (μr, f σ(1+ε T (βr, f (42 holds, where r is large eough outside of a possible set of fiite liear measure. By Lemma 4, we get that for ay γ>1ad sufficietly large r, Therefore, T(μr,fσ(1+ε T (γβr, f. (43 T (r, f T(γβ r, f. (44 μ Sice β>1, γ>1, 0<μ<1ad q>σ,wehaveβγ/μ > 1 ad q(1 ε/σ(1 + ε > 1 whe ε is small eough. Usig Lemma 7,weseethat log log. (45 log γβ log μ Lettig ε 0, δ 0, β 1ad γ 1,wehave. (46 (ii a > 1. By the similar reasoig as is (i, we easily obtai that T(μr,fqT(r,f(p(z σ(1+ε T(r+C,f(z+S(r,f (47 for all r large eough. We may select sufficietly small umbers δ>0ad ε>0,such that μ= a δ>1ad (1/μ+ε<1.Thuswehave amely, T(μr,f T(r+C,f(z+S(r,f; (48 T(μr,f T(r+C,f(z, (49 where r is large eough possibly outside of a set of fiite liear measure. By Lemma4, wehaveforay1<γ<μ, that is, T(μr,f T(γr,f(z; (50 T(r,f T(γ r, f (z (51 μ holds for all sufficietly large r.bylemma 8,weobtai +log (1+ε log (1 ε. (52 log (γ/μ Lettig ε 0, δ 0ad γ 1,wehave. (53 (iii a = 1 ad q>σ. The proof of this case is completely similar as i the case i (i. I fact, we set μ= a δ= 1 δ(0<δ<1,0<μ<1. Similarly, we ca get Sice a = 1,wehaveμ(f = ρ(f =.. (54

6 6 Abstract ad Applied Aalysis Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmets The authors would like to thak the aoymous referees for their valuable commets ad suggestios. The research was supported by Coloel-level topics (JSNU-ZY- 01, (Jsie2012zd01, ad NSF of Chia ( Refereces [1] W. Cherry ad Z. Ye, Nevalia s Theory of Value Distributio, Spriger Moographs i Mathematics, Spriger, Berli, Germay, [2] W. K. Hayma, Meromorphic Fuctios, OxfordMathematical Moographs, Claredo Press, Oxford, UK, [3] Y. Z. He ad X. Z. Xiao, AlgebroidFuctiosadOrdiary Differetial Equatios, Beijig, Chia, [4] I. Laie, Nevalia Theory ad Complex Differetial Equatios, vol. 15 of de Gruyter Studies i Mathematics,Walter de Gruyter, Berli, Germay, [5] I. Laie, J. Rieppo, ad H. Silveoie, Remarks o complex differece equatios, ComputatioalMethodsadFuctio Theory,vol.5,o.1,pp.77 88,2005. [6] J. Rieppo, O a class of complex fuctioal equatios, Aales Academiæ Scietiarum Feicæ,vol.32,o.1,pp ,2007. [7] X.-M. Zheg, Z.-X. Che, ad J. Tu, Growth of meromorphic solutios of some differece equatios, Applicable Aalysis ad Discrete Mathematics,vol.4,o.2,pp ,2010. [8] A. Z. Mokho ko, The Nevalia characteristics of certai meromorphic fuctios, Teorija Fukciĭ, Fukcioal yĭ Aaliz iihpriložeija,vol.14,pp.83 87,1971(Russia. [9]A.A.Mokho koadv.d.mokho ko, Estimatesofthe Nevalia characteristics of certai classes of meromorphic fuctios, ad their applicatios to differetial equatios, Akademija Nauk SSSR,vol.15,pp ,1974. [10] R. Goldstei, Some results o factorisatio of meromorphic fuctios, Joural of the Lodo Mathematical Society, vol.4, pp , [11] R. Goldstei, O meromorphic solutios of certai fuctioal equatios, Aequatioes Mathematicae, vol.18,o.1-2,pp , [12] G. G. Guderse, J. Heittokagas, I. Laie, J. Rieppo, ad D. Yag, Meromorphic solutios of geeralized Schröder equatios, Aequatioes Mathematicae,vol.63,o.1-2,pp , 2002.

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