Journal of Mathematical Analysis and Applications

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1 J. Math. Anal. Appl ) Contents lists available at ScienceDirect Journal o Mathematical Analysis Applications The growth, oscillation ixed points o solutions o complex linear dierential equations in the unit disc Ting-Bin Cao Department o Mathematics, Nanchang University, Nanchang 33003, Jiangxi, China article ino abstract Article history: Received 5 January 2008 Available online 9 November 2008 Submitted by Steven G. Krantz Keywords: Dierential equation Meromorphic unction Order o the growth Convergence exponent o zero points Unit disc We consider the complex dierential equations o the orm A k z) k) A k z) k ) A z) A 0 z) = F z), where A 0 0), A,...,A k F are analytic unctions in the unit disc D ={z C: z < }. Some results on the inite iterated order the inite iterated convergence exponent o zero points in D o meromorphic analytic) solutions are obtained. The ixed points o solutions o dierential equations are also investigated in this paper Elsevier Inc. All rights reserved.. Introduction main results In this paper, we assume that the reader is amiliar with the undamental results the stard notations o the Nevanlinna s value distribution theory o meromorphic unctions on the complex plane C in the unit disc D ={z C: z < } see [5,3]). Many authors investigated the growth oscillation o solutions o complex linear dierential equation in C, see [2,3,7,3,23,32] others. The eicient tools are the Wiman Valiron theory Nevanlinna theory. G.G. Gundersen [3] studied the case the coeicients, hence the solutions, are entire unctions, obtained the ollowing result or second order equations. The improvements extensions o it can be ound in [0,4,22,25,32] others. Theorem.. See [3].) Let Az) Bz) 0 be entire unctions, let α, β, θ θ 2 be real numbers with α > 0, β>0 θ <θ 2.I Bz) exp { o) ) α z β } Az) exp { o) z β } as z with θ arg z θ 2, then every solution 0 o the equation Az) Bz) = 0 ) has ininite order. This work was supported by the NNSF o China No. 0772), the SRFDP o China No ) the NNSF o Jiangxi o China No. 2007GQS2063). addresses: ctb97@63.com, tbcao@ncu.edu.cn X/$ see ront matter 2008 Elsevier Inc. All rights reserved. doi:0.06/j.jmaa

2 740 T.-B. Cao / J. Math. Anal. Appl ) Recently, there has been an increasing interest in studying the growth o analytic solutions o linear dierential equations in the unit disc D by making use o Nevanlinna theory. The analysis o slowly growing solutions have been studied in [0,6,7,9,20,27]. Fast growth o solutions are considered by [4 6,9,6,2]. To make the introduction short clear, the deinitions o the iterated n-order σ n )σ M,n )) the iterated n-convergence exponent λ n ) o zero points o a meromorphic analytic) unction in D are given in Section 2. In a recent paper [6], the present author H.-X. Yi obtained some results on the solutions o second order linear dierential equations in D analogous as Theorem.. It is natural to ask how about the properties o solutions o arbitrary order linear dierential equations in D? One o our main purposes o this paper is to study the linear dierential equation o the orm A k z) k) A k z) k ) A z) A 0 z) = 0, 2) where A 0 0), A,...,A k are analytic in D. ForF [0, ), the upper lower densities o F are deined by mf [0, r)) dens D F = lim sup r m[0, r)) respectively, where mg) = G dt t mf [0, r)) dens D F = lim in r m[0, r)) or G [0, ). Firstly,wehavetwogeneralresultsasollows. Theorem.2. Let H be a set o complex numbers satisying dens D { z : z H D} > 0, leta 0, A,...,A k be analytic unctions in D such that or some real constants 0 β<α μ > 0 we have A0 z) ) μ } expn 3) z Ai z) ) μ } exp n, i =, 2,...,k, 4) z as z or z H. Then every meromorphic or analytic) solution 0 o Eq. 2) satisies σ n ) = σ n ) μ. Theorem.3. Let H be a set o complex numbers satisying dens D { z : z H D} > 0, leta 0, A,...,A k be analytic unctions in D such that some real constants 0 β<α μ > 0 we have ) μ } T r, A 0 ) exp n 5) z ) μ } T r, A i ) exp n, i =, 2,...,k, 6) z as z or z H. Then every meromorphic or analytic) solution 0 o Eq. 2) satisies σ n ) = σ n ) μ. The other main purpose in this paper is to consider the oscillation o solutions o nonhomogeneous linear dierential equations o the orm k) A k z) k ) A z) A 0 z) = F z). 7) We obtain the ollowing main results. Theorem.4. Let H be a set o complex numbers satisying dens D { z : z H D} > 0,letA 0, A,...,A k be analytic unctions in D such that { max σ M,n A i ): i =, 2,...,k } σ M,n A 0 ) = σ <, 8) or some constants 0 β<α we have, or all ε > 0 suiciently small, A0 z) ) σ ε } expn, 9) z Ai z) ) σ ε } expn, i =, 2,...,k, 0) z as z or z H. Let F 0 be analytic in D.

3 T.-B. Cao / J. Math. Anal. Appl ) i) I σ n F )>σ M,n A 0 ), then all solutions o 7) satisy σ n ) = σ n F ). ii) I σ n F )<σ M,n A 0 ), then all solutions o 7) satisy σ n ) = λ n ) = λ n ) = σ M,n A 0 ) σ n A 0 ), with at most one exception 0 satisying σ n 0 )<σ M,n A 0 ). Theorem.5. Let H be a set o complex numbers satisying dens D { z : z H D} > 0,letA 0, A,...,A k be analytic unctions in D such that max { σ n A i ): i =, 2,...,k } σ n A 0 ) = σ <, ) or some constants 0 β<α we have, or all ε > 0 suiciently small, ) σ ε } T r, A 0 ) exp n, z 2) ) σ ε } T r, A i ) exp n, i =, 2,...,k, 3) z as z or z H. Let F 0 be analytic in D, let α M,n = max{σ M,n A j ): j = 0,,...,k }. i) I σ n F )>α M,n, then all solutions o 7) satisy σ n ) = σ n F ). ii) I σ n F )<α M,n, then all solutions o 7) satisy σ n A 0 ) σ n ) α M,n, with at most one exception 0 satisying σ n 0 )<σ n A 0 ). iii) I σ n F )<σ n A 0 ), then all solutions with σ n ) σ n A 0 ) o 7) satisy σ n A 0 ) σ n ) = λ n ) = λ n ) α M,n. The remainder o the paper is organized as ollows. In Section 2, ater introducing the deinitions o iterated order iterated convergence exponent in D, we shall show the consequences o our main results urther discuss the ixed points o solutions o equations in D. Section 3 is or lemmas, Section 4 or the proos o our results. 2. Consequences urther discussion For n N, the iterated n-order o a meromorphic unction in D is deined by σ n ) = lim sup r logn T r, ) log r), where log x = log x = max{log x, 0}, log n = log log n x.i is analytic in D, then the iterated n-order is deined by σ M,n ) = lim sup r logn Mr, ) log r). I is analytic in D, it is well known that σ M, ) σ ) satisy the inequalities σ ) σ M, ) σ ), which are the best possible in the sense that there are analytic unctions g h such that σ M, g) = σ g) σ M, h) = σ h), see [2]. However, it ollows by Proposition in [23] that σ M,n ) = σ n ) or n 2. For n N a C { }, the iterated n-convergence exponent o the sequence o a-points in D o a meromorphic unction in D is deined by log n λ n a) = Nr, a ) lim sup r log r) ; λ n a), the iterated n-convergence exponent o the sequence o distinct a-points in D o a meromorphic unction in D is deined by

4 742 T.-B. Cao / J. Math. Anal. Appl ) log n λ n a) = Nr, a ) lim sup r log r). Note that we may replace the integrated counting unction Nr, a ) with the unintegrated counting unction nr, a ) in the deinition o the convergence exponent, since ) n r, log r a 2r N r, a ) N r 0, r r 2 r a nt, ) r = r 0 a ) t nt, While A k z), Eq. 2) can be rewritten as dt N a ) t ) r 2, log r a 2r ) dt n r, a log r r 0. k) A k z) k ) A z) A 0 z) = 0, 4) where the coeicients A 0, A,...,A k are analytic unctions in D, it is well known that all solutions are analytic in D. For Eq. 4), we have the ollowing consequences o Theorems.2.3 respectively, with μ being replaced by σ ε, which are the improvements extensions o [6]. Theorem 2.. Assume that H, A 0, A,...,A k satisy the hypotheses o Theorem.4. Then every solution 0 o 4) satisies σ n ) = σ n ) = σ M,n A 0 ). Theorem 2.2. Assume that H, A 0, A,...,A k satisy the hypotheses o Theorem.5. Then every solution 0 o 4) satisies σ n ) = α M,n σ n ) σ n A 0 ),whereα M,n = max{σ M,n A j ): j = 0,,...,k }. Many important results have been obtained on the ixed points o general transcendental meromorphic unctions or almost our decades see []). However, there are ew studies on the ixed points o solutions o dierential equations. Z.-X. Chen [8] studied the problems on the ixed points hyper-order o solutions o second order linear dierential equations with entire coeicients. Thus it is naturally interesting to consider the ixed points o analytic solutions o dierential equations in the unit disc. Set gz) = z) z, z D. It is obvious that λ n z) = λ n g), σ n ) = σ n g). We obtain the ollowing results on the ixed points o analytic solutions. Theorem 2.3. Under the hypothesis o one o Theorems , ia z) za 0 z) 0, then every solution 0 o 4) satisies λ n z) = σ n ). Theorem 2.4. Under the hypothesis o either Theorem.4ii) or Theorem.5iii), ifz) A z) za 0 z) 0, then every solution with σ n ) = λ n ) o 7) satisies λ n z) = σ n ). In [30], Wang Yi studied the problems on the ixed points hyper-order o dierential polynomials generated by solutions o second order linear dierential equations with meromorphic coeicients. In [24], I. Laine J. Rieppo had given an extension improvement o the results in [30]; they studied the problems on the ixed points iterated order o dierential polynomials generated by solutions o second order linear dierential equations with meromorphic coeicients. In [29], Wang Lü studied the problems on the ixed points hyper-order o solutions o second order linear dierential equations with meromorphic coeicients their derivatives. In [26], Liu Zhang extended some results in [29] to the case o higher order linear dierential equations with meromorphic coeicients. Thus there exists a naturally interesting question: How about the ixed points iterated order o dierential polynomials generated by solutions o linear dierential equations in the unit disc? 3. Some lemmas For the proos o our main results, we need the ollowing lemmas. Lemma 3.. See [6] [28].) Let be a meromorphic unction in the unit disc, let k N.Then m r, k) ) = Sr, ), where Sr, ) = Olog T r, )) Olog r )), possibly outside a set E [0, ) with dr <. I is o inite order namely, E r inite iterated -order) o growth, then

5 m r, k) ) )) = O log. r T r, ) I is non-admissible namely, D ) = lim sup r log r) < ),then m r, ) log r 2 o) ) log log r. T.-B. Cao / J. Math. Anal. Appl ) Lemma 3.2. See [, Lemma C].) Let g : 0, ) R be monotone increasing unctions such that gr) hr) holds outside o an exceptional set E [0, ) o inite logarithmic measure. Then there exists a d 0, ) such that i sr) = d r) then gr) hsr)) or all r [0, ). Lemma 3.3. See [8, Theorem 5.].) Let be a solution o Eq. 4), where the coeicients A j z) j = 0,...,k ) are analytic unctions in the disc D R ={z C: z < R}, 0 < R. Letθ [0, 2π] ε > 0. Iz θ = νe iθ D R is such that A j z θ ) 0 or some j = 0,...,k, then or all ν < R < R, re iθ ) r C exp n c max A j te iθ ) ) k j dt, j=0,...,k ν where C > 0 is a constant satisying j) z θ ) C ε) max j=0,...,k n c ) j max j=0,...,k A n z θ ) j k n ). Lemma 3.4. See [2, Theorem 3.].) Let k j be integers satisying k > j 0, letε > 0 d 0, ). I is a meromorphic in D such that j) does not vanish identically, then k) ) z) 2ε j) z) max{ log z z, T s z ), ) }) k j, z / E, where E [0, ) dr with inite logarithmic measure E r < s z ) = d z ).Moreover,iσ )<,then k) ) z) k j)σ )2ε) j) z), z / E, z ) while i σ n )< or some n 2,then k) ) z) σn )ε) j) z) exp n, z / E. z ) Lemma 3.5. Let A 0, A,...,A k F 0) be analytic unctions in D let z) be a solution o Eq. 7) such that max{σ n F ), σ n A j )j = 0,,...,k )} < σ n ).Thenλ n ) = λ n ) = σ n ). Proo. J. Heittokangas [6, Theorem 7.] obtained that all solution o 7) are analytic in D when A 0,...,A k F 0) are analytic in D. From Eq. 7) we get that = k) k ) A k A 0 ). 5) F I has a zero at z 0 D o order α > k), then we get rom 5) that F has zero at z 0 o order at least α k. Hencewe have N r, ) k N r, ) N r, ). F 6) It ollows rom Lemma 3. 5) that m r, ) m r, ) k ) mr, A j ) O log T r, ) log F r j=0 7) holds or all z =r / E, where E is a subset o [0, ) with E dr <. By 6) 7), we get that or all z with z =r / E, r

6 744 T.-B. Cao / J. Math. Anal. Appl ) T r, ) = T kn r, ) O) r, ) k T r, F ) j=0 { )} O log r T r, ). Set σ n ) := σ. Then there exists {r n } r n ) such that logn lim T r n, ) = σ. r n log r n T r, A j ) 9) dr Set E r := log δ<. Since r n δ dr r n r = logδ ), then there exists r n [r n, r n ] E [0, ) such that δ log n T r n, ) log n T r n, ) log n = T r n, ) log r n log δ r n ) log r n logδ ). Hence, we have logn T r n, ) log lim r r n logn T r n, ) lim in r log r n logδ ) = σ. It yields logn lim T r n, ) = σ. r log r n Set max{σ n F ), σ n A j )j = 0,...,k )}:=b < σ, then or any given ε 0 < 2ε < σ b) or all n large enough, we have ) σ ε T r n, ) exp n, r n 8) 20) ) bε T r n, F ) exp n r n ) bε T r n, A j ) exp n. r n So we get { T rn, F ) max T r n, ), T r } n A j ) exp n r n ) bε T r n, ) exp n r n ) σ ε 0 r n ). Hence, or r n we obtain T r n, F ) k 3 T r n, ), T r n, A j ) k 3 T r n, ) j = 0,...,k ). 22) Furthermore, since σ n )>0, then by Lemma 3. we get that { )} O log T r n, ) = o T r n, ) ). r n Thus or r n,wehave { )} O log T r n, ) r n k 3 T r n, ). 23) Now we can get rom 8) 23) that T r n, ) kk 3)Nr n, ). It gives immediately that λ n ) = λ n ) = σ n ). 2)

7 T.-B. Cao / J. Math. Anal. Appl ) Proos 4.. Proo o Theorem.2 Suppose that 0 is a meromorphic solution o Eq. 2) with σ n )<. Then rom 2), we have A k k) A k A 0 A 0 k ) A = 0. 24) A 0 By Lemma 3.4, there exists a set E 0 D with a inite logarithmic measure such that or all z satisying z / [0, ] E 0 or j =, 2,...,k, wehave j) ) z) jσ )2ε) z), z / E 0, 25) z ) i σ )<, while i σ n )< or some n 2, we have j) ) z) σn )ε) z) exp n, z / E 0. 26) z ) On the other h, rom the conditions o Theorem.2, there is a set H o complex numbers with dens D { z : z H D} > 0 such that z H, we have 3) 4) as z. We note that since E 0 has inite logarithmic measure, the density o E 0 is zero. Thereore we get rom 25), 26), 3) 4) that or all z satisying z H, z / [0, ] E 0, A j j) ) μ } ) jσ )2ε) {β exp α), j =, 2,...,k, z z or A 0 A j A 0 Hence either k or j) exp n z )μ } exp n z )μ } exp n j= k j= A j A 0 A j A 0 j) {β k exp α) z z ) j) exp n k z )μ } exp n z )μ } exp n ) μ } ) σn )ε), j =, 2,...,k. z z ) ) kσ )2ε), ) σn )ε) holds or all z satisying z H, z / [0, ] E 0. One can deduce that or all z H, z / [0, ] E 0, both the limits o the right h o the above inequalities are zero as z. Thus we get a contradiction. This proves that every nonzero meromorphic solution o Eq. 2) has ininite iterated n-order. Now let be a nonzero meromorphic solution o Eq. 2) with σ n ) =.AgainromEq.2)wehave A0 z) k A j j= j). By Lemma 3.4, there exist s z ) = d z ) dr a set with inite logarithmic measure E < such that z / E, r E [0, ) we have j) ) z) 2ε z) max{ log z z, T s z ), ) }) j, j =, 2,...,k. 28) Again rom the conditions o Theorem.2, there is a set H o complex numbers with dens D { z : z H D} > 0 such that or all z H, 3) 4) hold as z.henceorallz satisying z H, z / [0, ] E, we get rom 27), 28), 3) 4) that ) μ } exp n A0 z) { ) μ } ) 2ε k expn β max{ log z z z Noting that α >β 0, i ollows rom the above inequality that z, T s z ), ) }) k. 27)

8 746 T.-B. Cao / J. Math. Anal. Appl ) exp { { ) μ }} α γ ) exp n o) ) T s z ), ) k, z holds or all z satisying z H, z / [0, ] E, where γ 0 < γ < ) is a real number. Hence by Lemma 3.2, we obtain σ n ) = lim sup r Theorem.2 is thus proved. logn T r, ) μ. log r) 4.2. Proo o Theorem.3 Suppose that 0 is a meromorphic unction o Eq. 2). From the conditions o Theorem.3, there is a set H o complex numbers with dens D { z : z H D} > 0 such that z H, we have 5) 6) as z. It ollow rom 2), 5), 6) Lemma 3. that ) μ } exp n T r, A 0 ) = mr, A 0 ) z k k mr, A j ) m r, j) ) O) j= j= k exp n z ) μ } exp n z ) μ } O log T r, ) ) O log O log T r, ) log r r ) holds or all z satisying z H, z / [0, ] E 2, where E 2 is a set in [0, ) dr with inite logarithmic measure E 2 r <. Hence { { ) μ }} exp α γ ) exp n 2 O log T r, ) log ) 29) z r holds or all z satisying z H z / [0, ] E 2, where γ 0 < γ < ) is a real number. I σ n )<, then one can obtain a contradiction by 29) whether n = orn 2. Thereore, the iterated n-order o is ininity. Again by 29) by Lemma 3.2 we have σ n ) = lim sup r Theorem.3 is thus proved. logn T r, ) μ. log r) ) 4.3. Proo o Theorem 2. We assume that 0 be a solution o Eq. 4), then or any given ε > 0, by the result o Theorem.2, we have σ n ) = σ n ) σ ε. Sinceε is arbitrary, we get σ n ) σ = σ M,n A 0 ). By Lemma 3.3 we deduce that σ n ) = σ M,n ) max { σ M,n A j ): j = 0,,...,k } = σ M,n A 0 ) = σ. Thereore, we obtain σ n ) = σ M,n A 0 ) = σ Proo o Theorem 2.2 We also assume that 0 be a solution o Eq. 4), then or any given ε > 0, by the result o Theorem.3, we have σ n ) = σ n ) σ ε. Sinceε is arbitrary, we get σ n ) σ = σ n A 0 ). By Lemma 3.3 we also have σ n ) = σ M,n ) max { σ M,n A j ): j = 0,,...,k } = α M,n. Thereore, we obtain α M,n σ n ) = σ n A 0 ) = σ.

9 T.-B. Cao / J. Math. Anal. Appl ) Proo o Theorem.4 Recall that every solution o non-homogeneous linear dierential equation 7) is analytic in D. Thus we can assume that {, 2,..., k } is a solution base o Eq. 4). By Theorem 2., we know that σ n j ) = σ M,n A 0 ) σ n A 0 ) j =, 2,...,k ). Then by the elementary o dierential equations, any solution o 7) can be represented in the orm = C C 2 2 C k k, where C,...,C k are given by the system o equations C C 2 2 C k k = 0, C C 2 2 C k k = 0,... C k 2) C 2 k 2) 2 C k k 2) = 0, k C k ) C 2 k ) 2 C k k ) = F. k Since the Wronskian o,..., k satisies W,..., k ) = exp A k dz), weobtain ) C j = F G j,..., k ) exp A k dz j =,...,k ), 32) where G j,..., k ) is a dierential polynomial o,..., k o their derivative, with constant coeicients. Hence we obtain 30) 3) σ n ) max { σ n F ), σ M,n A 0 ) }. 33) i) I σ n F )>σ M,n A 0 ), it ollows rom 33) Eq. 7) that σ n ) = σ n F ). ii) I σ n F )<σ M,n A 0 ), then all solutions z) o 7) satisy σ n ) σ M,n A 0 ). Now we assert that all solutions o 7) satisy σ n ) = σ M,n A 0 ) with at most one exception. In act, i there exists two distinct solutions g g 2 o 7) satisy that σ n g i )<σ M,n A 0 ) or i =, 2. Then g = g g 2 satisies that σ n g) = σ n g g 2 )<σ M,n A 0 ).Butg = g g 2 is a nonzero solution o 4) satisying σ n g) = σ n g g 2 ) = σ M,n A 0 ) by Theorem 2.. This is a contradiction. By Lemma 3.5, we know that all solutions o 7) with σ n ) = σ M,n A 0 ) satisy σ n ) = λ n ) = λ n ). Thereore, Theorem.4 is proved Proo o Theorem.5 Assume that {, 2,..., k } is a solution base o 4), then by Theorem 2.2, we know that α M,n σ n j ) σ n A 0 ). Thus we also have 30) 32) so σ n ) max { σ n F ), α M,n }. 34) i) I σ n F )>α M,n, it ollows rom 34) Eq. 7) that σ n ) = σ n F ). ii) I σ n F )<α M,n, then all solutions z) o 7) satisy σ n ) σ M,n A 0 ). Now we assert that all solutions o 7) satisy σ n ) σ n A 0 ) with at most one exception. In act, i there exists two distinct solutions g g 2 o 7) satisy that σ n g i )<σ n A 0 ) or i =, 2. Then g = g g 2 satisies that σ n g) = σ n g g 2 )<σ n A 0 ). But g = g g 2 is a nonzero solution o 4) satisying σ n g) = σ n g g 2 ) σ A 0 ) by Theorem 2.2. This is a contradiction. iii) I σ n F )<σ n A 0 ), then by Lemma 3.5, we know that all solutions with σ n ) σ n A 0 ) o 7) satisy σ n ) = λ n ) = λ n ). Thereore, Theorem.5 ollows Proo o Theorem 2.3 Set gz) = z) z, z D. It obvious that λ n z) = λ n g), σ n ) = σ n g). Eq. 4) becomes g k) A k z)g k ) A 0 z)g = A z) za 0 z) ). Assume that A z) za 0 z) 0. By Theorems 2. or 2.2 we have σ n g) = σ n )>max { σ n A j ), σ n A za 0 ) } j = 0,,...,k ). Hence, we deduce by Lemma 3.5 that λ n g) = σ n g). Thereore, we obtain λ n z) = λ n g) = σ n g) = σ n ).

10 748 T.-B. Cao / J. Math. Anal. Appl ) Proo o Theorem 2.4 Set gz) = z) z, z D. It obvious that λ n z) = λ n g), σ n ) = σ n g). Eq. 7) becomes g k) A k z)g k ) A 0 z)g = F A z) za 0 z) ). Assume that F z) A z) za 0 z)) 0. Then by Theorem.4ii) or Theorem.5iii), or any solution with σ n ) = λ n ), wehave σ n g) = σ n )>max { σ n A j ), σ n F A za 0 ) } j = 0,,...,k ). Hence, we deduce by Lemma 3.5 that λ n g) = σ n g). Thereore, we obtain λ n z) = λ n g) = σ n g) = σ n ). Acknowledgment The author would like to thank the reeree or making valuable suggestions comments to improve the present paper. Reerences [] S. Bank, A general theorem concerning the growth o solutions o irst-order algebraic dierential equations, Compos. Math ) [2] S. Bank, I. Laine, On the oscillation theory o A = 0whereA is entire, Trans. Amer. Math. Soc ) [3] S. Bank, I. Laine, On the zeros o meromorphic solutions o second order linear dierential equations, Comment. Math. Helv ) [4] D. Benbourenane, L.R. Sons, On global solutions o complex dierential equations in the unit disc, Complex Var. Elliptic Equ. 49 3) 2004) [5] T.-B. Cao, H.-X. Yi, The growth o solutions o linear dierential equations with coeicients o iterated order in the unit disc, J. Math. Anal. Appl ) [6] T.-B. Cao, H.-X. Yi, On the complex oscillation o second order linear dierential equations with analytic coeicients in the unit disc, Chinese Ann. Math. Ser. A 28 5) 2007) in Chinese). [7] T.-B. Cao, H.-X. Yi, On the complex oscillation o higher order linear dierential equations with meromorphic coeicients, J. Syst. Sci. Complex. 20) 2007) [8] Z.-X. Chen, The ixed points hyper order o solutions o secon order complex dierential equations, Acta Math. Sci. Ser. A Chin. Ed. 20 3) 2000) in Chinese). [9] Z.-X. Chen, K.-H. Shon, The growth o solutions o dierential equations with coeicients o small growth in the disc, J. Math. Anal. Appl ) [0] Z.-X. Chen, C.-C. Yang, Some uther results on the zeros growths o entire solutions o second order linear dierential equations, Kodai Math. J ) [] C.-T. Chuang, C.-C. Yang, The Fixed Points Factorization Theory o Meromorphic Functions, Beijing University Press, Beijing, 988 in Chinese). [2] I. Chyzhykov, G. Gundersen, J. Heittokangas, Linear dierential equations logarithmic derivative estimates, Proc. London Math. Soc ) [3] G.G. Gundersen, Finite order solutions o second order linear dierential equations, Trans. Amer. Math. Soc ) [4] G.G. Gundersen, Finite order solutions o nonhomogeneous linear dierential equations, Ann. Acad. Sci. Fenn. Math ) [5] W. Hayman, Meromorphic Functions, Clarendon Press, Oxord, 964. [6]J.Heittokangas,Oncomplexdierentialequationsintheunitdisc,Ann.Acad.Sci.Fenn.Math.Diss ) 54. [7] J. Heittokangas, Blaschke-oscillatory equations o the orm Az) = 0, J. Math. Anal. Appl ) [8] J. Heittokangas, R. Korhonen, J. Rättyä, Growth estimates or solutions o linear complex dierential equations, Ann. Acad. Sci. Fenn. Math ) [9] J. Heittokangas, R. Korhonen, J. Rättyä, Linear dierential equations with coeicients in weighted Bergman Hardy spaces, Trans. Amer. Math. Soc ) 2007) [20] J. Heittokangas, R. Korhonen, J. Rättyä, Linear dierential equations with solutions in the dirichlet type subspace o the Hardy space, Nagoya Math. J ) 9 3. [2] J. Heittokangas, R. Korhonen, J. Rättyä, Fast growing solutions o linear dierential equations in the unit disc, Results Math ) [22] K.-H. Kwon, On the growth o entire unctions satisying second order linear dierential equations, Bull. Korean Math. Soc. 33 3) 996) [23] I. Laine, Nevanlinna Theory Complex Dierential Equations, W. de Gruyter, Berlin, 993. [24] I. Laine, J. Rieppo, Dierential polynomials generated by linear dierential equations, Complex Var. Elliptic Equ ) [25] I. Laine, R. Yang, Finite order solutions o complex linear dierential equations, Electron. J. Dierential Equations ) 8. [26] M.-S. Liu, X.-M. Zhang, Fixed points o meromorphic solutions o higher order linear dierential equations, Ann. Acad. Sci. Fenn. Math ) 9 2. [27] C. Pommenrenke, On the mean growth o solutions o complex linear dierential equations in the disk, Complex Var. Elliptic Equ. ) 982) [28] D. Shea, L. Sons, Value distribution theory or meromorphic unctions o slow growth in the disk, Houston J. Math. 2 2) 986) [29] J. Wang, W.-R. Lü, The ixed points hyper-order o solutions o second order linear dierential equations with meromorphic coeicients, Acta Math. Appl. Sin ) in Chinese). [30] J. Wang, H.-X. Yi, Fixed points hyper order o dierential polynomials generated by solutions o dierential equation, Complex Var. Elliptic Equ. 48 ) 2003) [3] L. Yang, Value Distribution Theory, Springer-Verlag/Science Press, Berlin/Beijing, 993. [32] L.-Z. Yang, Growth o linear dierential equations their applications, Israel J. Math )