In this paper, we investigate the iterated order of entire solutions of homogeneous and non-homogeneous linear differential equations with entire coefficients

Belaidi, B.

Electronic journal of qualitative theory of differential equations

Benharrat Belaidi

Crossref

On the iterated order and the fixed points of entire solutions of some complex linear differential equations

Belaidi Benharrat

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2006, No. 9, 1-11; http://www.math.u-szeged.hu/ejqtde/On the iterated order and the fixed points ofentire solutions of some complex lineardifferential equationsBenharrat BELA..IDIDepartment of MathematicsLaboratory of Pure and Applied MathematicsUniversity of MostaganemB. P 227 Mostaganem-(Algeria)belaidi@univ-mosta.dzbelaidi.benharrat@caramail.comAbstract. In this paper, we investigate the iterated order of entire solutionsof homogeneous and non-homogeneous linear differential equations with en-tire coefficients.2000 Mathematics Subject Classification: 34M10, 30D35.Key words: Linear differential equation, growth of entire function, iteratedorder.1 Introduction and statement of results:For the definition of the iterated order of an entire function, we use thesame definition as in [9] , [2, p. 317] , [10, p. 129] . For all r ∈ R, we defineexp1 r := er and expp+1 r := exp(expp r), p ∈ N. We also define for all rsufficiently large log1 r := log r and logp+1 r := log(logp r), p ∈ N. Moreover,we denote by exp0 r := r, log0 r := r, log−1 r := exp1 r and exp−1 r := log1 r.Definition 1.1 Let f be an entire function. Then the iterated p−orderσp (f) of f is defined byσp (f) = limr→+∞logpT (r, f)log r= limr→+∞logp+1M (r, f)log r(p ≥ 1 is an integer) ,(1.1)EJQTDE, 2006 No. 9, p. 1where T (r, f) is the Nevanlinna characteristic function of f and M (r, f) =max|z|=r |f (z)| (see [7] , [13]) . For p = 1, this notation is called order andfor p = 2 hyper-order.Definition 1.2 The finiteness degree of the order of an entire function f isdefined byi (f) =0, for f polynomial,min {j ∈ N : σj (f) < ∞} , for f transcendental for whichsome j ∈ N with σj (f) < ∞ exists,∞, for f with σj (f) = ∞ for all j ∈ N.(1.2)Definition 1.3 Let f be an entire function. Then the iterated convergenceexponent of the sequence of distinct zeros of f (z) is defined byλp (f) = limr→+∞logp N(r, 1f)log r, (1.3)where N(r, 1f)is the counting function of distinct zeros of f (z) in {|z| < r}.Thus λp (f − z) is an indication of oscillation of the fixed points of f (z) .For k ≥ 2, we consider the linear differential equationsf (k) + Ak−1 (z) f(k−1) + ... + A1 (z) f′+ A0 (z) f = 0, (1.4)f (k) + Ak−1 (z) f(k−1) + ... + A1 (z) f′+ A0 (z) f = F (z) , (1.5)where A0 (z) , ..., Ak−1 (z) and F (z) /≡ 0 are entire functions. It is well-knownthat all solutions of equations (1.4) and (1.5) are entire functions.Extensive work in recent years has been concerned with the growth ofsolutions of complex linear differential equations. Many results have beenobtained. Examples of such results are the following two theorems:Theorem A [4]. Let A0 (z) , ..., Ak−1 (z) be entire functions such that thereexists one transcendental As (0 ≤ s ≤ k − 1) satisfying σ (Aj) ≤ σ (As) forall j 6= s. Then equation (1.4) has at least one solution f that satisfiesσ2 (f) = σ (As) .EJQTDE, 2006 No. 9, p. 2Theorem B [4]. Let A0 (z) , ..., Ak−1 (z) satisfy the hypotheses of TheoremA and F (z) /≡ 0 be an entire function with σ (F ) < +∞. Assume that f0is a solution of (1.5) , and g1, ..., gk are a solution base of the correspondinghomogeneous equation (1.4) of (1.5) . Then there exists a gj (1 ≤ j ≤ k) ,say g1, such that all the solutions in the solution subspace {cg1 + f0, c ∈ C}satisfy σ2 (f) = λ2 (f) = σ (As) , with at most one exception.The purpose of this paper is to extend the above two results by consid-ering the iterated order. We will prove the following theorems:Theorem 1.1 Let A0 (z) , ..., Ak−1 (z) be entire functions such that thereexists one transcendental As (0 ≤ s ≤ k − 1) satisfying σp (Aj) ≤ σp (As) <+∞ for all j 6= s. Then equation (1.4) has at least one solution f thatsatisfies i (f) = p + 1 and σp+1 (f) = σp (As) .Theorem 1.2 Let A0 (z) , ..., Ak−1 (z) satisfy the hypotheses of Theorem1.1 and F (z) /≡ 0 be an entire function with i (F ) = q. Assume that f0 isa solution of (1.5) , and g1, ..., gk are a solution base of the correspondinghomogeneous equation (1.4) of (1.5) . If either i (F ) = q < p+1 or q = p+1and σp+1 (F ) < σp (As) < +∞, then there exist a gj (1 ≤ j ≤ k) , say g1,such that all the solutions in the solution subspace {cg1 + f0, c ∈ C} satisfyi (f) = p+1 and σp+1 (f) = λp+1 (f) = σp (As) , with at most one exception.Set g (z) = f (z) − z. Then clearly λp+1 (f − z) = λp+1 (g) and σp+1 (g) =σp+1 (f) . By Theorem 1.1 and Theorem 1.2, we can get the following corol-laries.Corollary 1 Under the hypotheses of Theorem 1.1, if A1 + zA0 /≡ 0, thenequation (1.4) has at least one solution f that satisfies i (f) = p + 1 andλp+1 (f − z) = σp+1 (f) = σp (As) .Corollary 2 Under the hypotheses of Theorem 1.2, if F−A1−zA0 /≡ 0, thenevery solution f of (1.5) with i (f) = p+1 and σp+1 (f) = λp+1 (f) = σp (As)satisfies λp+1 (f − z) = σp (As) .EJQTDE, 2006 No. 9, p. 32 Preliminary LemmasOur proofs depend mainly upon the following lemmas.Lemma 2.1 ([3] , [11]) . Let f (z) =∞∑n=0an zn be an entire with σp+1 (f) = σ,let µ (r) be the maximum term, i.e., µ (r) = max{|an| rn; n = 0, 1, ...} andlet νf (r) be the central index of f , i.e., νf (r) = max{m, µ (r) = |am| rm}.Thenlimr→∞logp+1 νf (r)log r= σ. (2.1)Lemma 2.2 (Wiman-Valiron, [8] , [12]) . Let f (z) be a transcendental en-tire function, and let z be a point with |z| = r at which |f (z)| = M (r, f).Then the estimationf (k) (z)f (z)=(νf (r)z)k(1 + o (1)) (k is an integer), (2.2)holds for all |z| outside a set E2 of r of finite logarithmic measure lm (E2) =∫ +∞1χE2(t)tdt, where χE2is the characteristic function of E2.Lemma 2.3 (See Remark 1.3 of [9]) . If f is a meromorphic function withi (f) = p ≥ 1, then σp (f) = σp(f′).Lemma 2.4 ([5]) . Let f1, ..., fk be linearly independent meromorphic solu-tions of the differential equationf (k) + Ak−1 (z) f(k−1) + ... + A1 (z) f′+ A0 (z) f = 0, (2.3)with meromorphic coefficients A0 (z) , ...Ak−1 (z). Thenm (r, Aj) = O{log(max1≤n≤kT (r, fn))}(j = 0, ..., k − 1) . (2.4)Lemma 2.5 ([9]) . Let f be a meromorphic function for which i (f) = p ≥ 1and σp (f) = σ, and let k ≥ 1 be an integer. Then for any ε > 0,m(r,f (k)f)= O(expp−2{rσ+ε}), (2.5)EJQTDE, 2006 No. 9, p. 4outside of a possible exceptional set E3 of finite linear measure.To avoid some problems caused by the exceptional set we recall thefollowing Lemma.Lemma 2.6 ([1, p. 68] , [9]) . Let g : [0, +∞) → R and h : [0, +∞) → Rbe monotone non-decreasing functions such that g (r) ≤ h (r) outside of anexceptional set E of finite linear measure. Then for any α > 1, there existsr0 > 0 such that g (r) ≤ h (αr) for all r > r0.3 Proof of Theorem 1.1Suppose that f is a solution of (1.4) . We can rewrite (1.4) asf (k)f+ Ak−1 (z)f (k−1)f+ ... + As+1 (z)f (s+1)f+ As (z)f (s)f+As−1 (z)f (s−1)f+ ... + A1 (z)f ′f+ A0 (z) = 0. (3.1)By Lemma 2.2, there exists a set E2 ⊂ (1, +∞) with logarithmic measurelm (E2) < +∞ and we can choose z satisfying |z| = r /∈ [0, 1] ∪ E2 and|f (z)| = M (r, f), such that (2.2) holds. For given small ε > 0 and sufficientlylarge r, we have|Aj (z)| ≤ expp{rσp(As)+ε}(j = 0, 1, ..., k − 1) . (3.2)Substituting (2.2) into (3.1), we obtain by using (3.2)(νf (r)|z|)k|1 + o (1)| ≤ k(νf (r)|z|)k−1|1 + o (1)| expp{rσp(As)+ε}, (3.3)(r /∈ [0, 1] ∪ E2) . By Lemma 2.1, Lemma 2.6 and (3.3) , we obtain that i (f) ≤p + 1 andσp+1 (f) = limr→∞logp+1 νf (r)log r≤ σp (As) + ε. (3.4)Since ε > 0 is arbitrary, then σp+1 (f) ≤ σp (As) .EJQTDE, 2006 No. 9, p. 5Assume that {f1, ..., fk} is a solution base of (1.4). Then by Lemma 2.4m (r, As) ≤ M log(max1≤n≤kT (r, fn)). (3.5)We assert that there exists a set E ⊂ (0, +∞) of infinite linear measure suchthatlimr→∞r∈Elogp m (r, As)log r= σp (As) . (3.6)In fact, there exists a sequence {rn} (rn → ∞) such thatlimrn→∞logp m (rn, As)log rn= σp (As) . (3.7)We take E =∞∪n=1[rn, 2rn]. Then on E, (3.6) holds obviously. Now bysetting En = {r : r ∈ E and m (r, As) ≤ M log T (r, fn) (n = 1, ..., k)}, wehavek∪n=1En =E. It is easy to see that there exists at least one En, say E1,which has an infinite linear measure and on whichlimr→∞r∈E1logp m (r, As)log r= σp (As) , (3.8)andm (r, As) ≤ M log T (r, f1) (r ∈ E1) . (3.9)From (3.8) and (3.9) we have i (f1) ≥ p + 1 and σp+1 (f1) ≥ σp (As) . Thisand the fact that i (f1) ≤ p + 1 and σp+1 (f1) ≤ σp (As) yield i (f1) = p + 1and σp+1 (f1) = σp (As). The proof of Theorem 1.1 is complete.4 Proof of Theorem 1.2Assume that f is a solution of (1.5) and g1, ..., gk are k entire solutions of thecorresponding homogeneous equation (1.4) . Then by the proof of Theorem1.1, we know that i (gj) ≤ p + 1, σp+1 (gj) ≤ σp (As) (j = 1, 2, 3..., k) andEJQTDE, 2006 No. 9, p. 6there exists a gj, say g1, satisfying i (g1) = p + 1, σp+1 (g1) = σp (As) . Thusby variation of parameters, f can be expressed in the formf (z) = B1 (z) g1 (z) + ... + Bk (z) gk (z) , (4.1)where B1 (z) , ..., Bk (z) are determined byB′1 (z) g1 (z) + ... + B′k (z) gk (z) = 0B′1 (z) g′1 (z) + ... + B′k (z) g′k (z) = 0.....................................................B′1 (z) g(k−1)1 (z) + ... + B′k (z) g(k−1)k (z) = F. (4.2)Noting that the Wronskian W (g1, g2, ..., gk) is a differential polynomial ing1, g2, ..., gk with constant coefficients, it follows thatσp+1 (W ) ≤ max {σp+1 (gj) : j = 1, ..., k} ≤ σp (As) .SetWj =∣∣∣∣∣∣∣∣∣g1, ...,(j)0 , ..., gk......g(k−1)1 , ..., F, ..., g(k−1)k∣∣∣∣∣∣∣∣∣= F.Gj (j = 1, ..., k) , (4.3)where Gj (g1, g2, ..., gk) are differential polynomials in g1, g2, ..., gk and of theirderivatives with constant coefficients. Soσp+1 (Gj) ≤ max {σp+1 (gj) : j = 1, ..., k} ≤ σp (As) (j = 1, ..., k) ,B′j =WjW=F.GjW(j = 1, ..., k) . (4.4)Since i (F ) = q < p + 1 or i (F ) = p + 1, σp+1 (F ) < σp (As), then by Lemma2.3, we obtainσp+1 (Bj) = σp+1(B′j)≤ max (σp+1 (F ) , σp (As)) = σp (As) (j = 1, ..., k) .(4.5)EJQTDE, 2006 No. 9, p. 7Then from (4.1) and (4.5) , we get i (f) ≤ p + 1 andσp+1 (f) ≤ max {σp+1 (gj) , σp+1 (Bj) : j = 1, ..., k} ≤ σp (As) . (4.6)Now we setH = {fc = cg1 + f0, c ∈ C} , (4.7)where f0 is a solution of (1.5) . Obviously, every fc in H is a solution of (1.5) .Now we prove that for any two solutions fa and fb (a 6= b) in H , there is atleast one solution, say fa, among fa and fb satisfying i (fa) = p + 1 andσp+1 (fa) = λp+1 (fa) = σp (As) . Since fa = (a − b) g1+ fb, thenT (r, g1) ≤ T (r, fa) + T (r, fb) + O (1) . (4.8)Assume that the set E1 satisfies the condition as required in proof of Theorem1.1. Then there exists at least one of fa and fb, say fa, such that there is asubset E4 of E1 with infinite linear measure andT (r, fb) ≤ T (r, fa) , for r ∈ E4. (4.9)We get from (4.8) and (4.9)T (r, g1) ≤ 2T (r, fa) + O (1) , for r ∈ E4. (4.10)Thus, i (fa) ≥ p + 1 and σp+1 (fa) ≥ σp+1 (g1) = σp (As) and hence i (fa) =p + 1, σp+1 (fa) = σp (As) = σ.Now we prove that σp+1 (fa) = λp+1 (fa) = σ. By (1.5) , it is easy to seethat if fa has a zero at z0 of order α (> k), then F must have a zero at z0 oforder α − k. Hence,n(r,1fa)≤ k n(r,1fa)+ n(r,1F)(4.11)andN(r,1fa)≤ k N(r,1fa)+ N(r,1F). (4.12)Now (1.5) can be rewritten as1fa=1F(f(k)afa+ Ak−1f(k−1)afa+ ... + A1f′afa+ A0). (4.13)EJQTDE, 2006 No. 9, p. 8By (4.13), we havem(r,1fa)≤k∑j=1m(r,f(j)afa)+k∑j=1m (r, Ak−j) + m(r,1F)+ O (1) . (4.14)Applying the Lemma 2.5, we havem(r,f(j)afa)= O(expp−1{rσ+ε})(j = 1, ..., k − 1) , (σp+1 (fa) = σ) ,(4.15)holds for all r outside a set E3 ⊂ (0, +∞) with a linear measure m (E3) =δ < +∞. By (4.12) , (4.14) and (4.15), we getT (r, fa) = T(r,1fa)+ O (1)≤ kN(r,1fa)+k∑j=1T (r, Ak−j)+T (r, F )+O(expp−1{rσ+ε})(|z| = r /∈ E3) .(4.16)For sufficiently large r, we haveT (r, A0) + ... + T (r, Ak−1) ≤ k expp−1{rσ+ε}. (4.17)If i (F ) = q < p + 1, then q − 1 ≤ p − 1 andT (r, F ) ≤ expq−1{rσq(F )+ε}≤ expp−1{rσq(F )+ε}(σq (F ) < ∞) . (4.18)Thus, by (4.16) − (4.18), we haveT (r, fa) ≤ k N(r,1fa)+ k expp−1{rσ+ε}+ expp−1{rσq(F )+ε}+ O(expp−1{rσ+ε})(|z| = r /∈ E3) . (4.19)Hence for any fa with σp+1 (fa) = σ, by (4.19) and Lemma 2.6, we haveσp+1 (fa) ≤ λp+1 (fa). Therefore, λp+1 (fa) = σp+1 (fa) = σ.If i (F ) = p + 1 and σp+1 (F ) < σp (As) = σ, thenT (r, F ) ≤ expp{rσp+1(F )+ε}≤ expp−1{rσ+ε}. (4.20)EJQTDE, 2006 No. 9, p. 9Thus, by (4.16) − (4.17) and (4.20), we haveT (r, fa) ≤ k N(r,1fa)+ k expp−1{rσ+ε}+ expp−1{rσ+ε}+ O(expp−1{rσ+ε})(|z| = r /∈ E3) . (4.21)By using similar reasoning as above, we obtain from (4.21) and Lemma 2.6that λp+1 (fa) = σp+1 (fa) = σ. The proof of Theorem 1.2 is complete.Acknowledgement. The author would like to thank the referee for his/herhelpful remarks and suggestions.References[1] S. Bank, A general theorem concerning the growth of solutions offirst-order algebraic differential equations, Compositio Math. 25 (1972), 61-70.[2] L. G. Bernal, On growth k−order of solutions of a complex homo-geneous linear differential equations, Proc. Amer. Math. Soc. 101 (1987),317-322.[3] T. B. Cao, Z. X. Chen, X. M. Zheng and J. Tu, On the iteratedorder of meromorphic solutions of higher order linear differential equations,Ann. of Diff. Eqs., 21, 2 (2005), 111-122.[4] Z. X. Chen and C. C. Yang, Quantitative estimations on the ze-ros and growths of entire solutions of linear differential equations, ComplexVariables Vol. 42 (2000), 119-133.[5] G. Frank and S. Hellerstein, On the meromorphic solutions of non-homogeneous linear differential equations with polynomial coefficients, Proc.London Math. Soc. (3), 53 (1986), 407-428.[6] G. Gundersen, Estimates for the logarithmic derivative of a mero-morphic function, plus similar estimates, J. London Math. Soc. (2) 37(1988), 88-104.[7] W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford,1964.EJQTDE, 2006 No. 9, p. 10[8] W. K. Hayman, The local growth of power series: a survey of theWiman-Valiron method, Canad. Math. Bull., 17(1974), 317-358.[9] L. Kinnunen, Linear differential equations with solutions of finiteiterated order, Southeast Asian Bull. Math., 22: 4 (1998), 385-405.[10] I. Laine, Nevanlinna Theory and Complex Differential Equations,Walter de Gruyter, Berlin, New York, 1993.[11] J. Tu, Z. X. Chen and X. M. Zheng, Growth of solutions of com-plex differential equations with coefficients of finite iterated order, Electron.J. Diff. Eqns., Vol. 2006 (2006), N◦54, 1-8.[12] G. Valiron, Lectures on the General Theory of Integral Functions,translated by E. F. Collingwood, Chelsea, New York, 1949.[13] H. X. Yi and C. C. Yang, The Uniqueness Theory of MeromorphicFunctions, Science Press, Beijing, 1995 (in Chinese).(Received April 2, 2006)EJQTDE, 2006 No. 9, p. 11

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Electronic Journal of Qualitative Theory of Differential Equations2006, No. 9, 1-11; http://www.math.u-szeged.hu/ejqtde/On the iterated order and the fixed points ofentire solutions of some complex lineardifferential equationsBenharrat BELA..IDIDepartment of MathematicsLaboratory of Pure and Applied MathematicsUniversity of MostaganemB. P 227 Mostaganem-(Algeria)belaidi@univ-mosta.dzbelaidi.benharrat@caramail.comAbstract. In this paper, we investigate the iterated order of entire solutionsof homogeneous and non-homogeneous linear differential equations with en-tire coefficients.2000 Mathematics Subject Classification: 34M10, 30D35.Key words: Linear differential equation, growth of entire function, iteratedorder.1 Introduction and statement of results:For the definition of the iterated order of an entire function, we use thesame definition as in [9] , [2, p. 317] , [10, p. 129] . For all r ∈ R, we defineexp1 r := er and expp+1 r := exp(expp r), p ∈ N. We also define for all rsufficiently large log1 r := log r and logp+1 r := log(logp r), p ∈ N. Moreover,we denote by exp0 r := r, log0 r := r, log−1 r := exp1 r and exp−1 r := log1 r.Definition 1.1 Let f be an entire function. Then the iterated p−orderσp (f) of f is defined byσp (f) = limr→+∞logpT (r, f)log r= limr→+∞logp+1M (r, f)log r(p ≥ 1 is an integer) ,(1.1)EJQTDE, 2006 No. 9, p. 1where T (r, f) is the Nevanlinna characteristic function of f and M (r, f) =max|z|=r |f (z)| (see [7] , [13]) . For p = 1, this notation is called order andfor p = 2 hyper-order.Definition 1.2 The finiteness degree of the order of an entire function f isdefined byi (f) =0, for f polynomial,min {j ∈ N : σj (f) <∞} , for f transcendental for whichsome j ∈ N with σj (f) <∞ exists,∞, for f with σj (f) =∞ for all j ∈ N.(1.2)Definition 1.3 Let f be an entire function. Then the iterated convergenceexponent of the sequence of distinct zeros of f (z) is defined byλp (f) = limr→+∞logpN(r, 1f)log r, (1.3)where N(r, 1f)is the counting function of distinct zeros of f (z) in {|z| < r}.Thus λp (f − z) is an indication of oscillation of the fixed points of f (z) .For k ≥ 2, we consider the linear differential equationsf (k) + Ak−1 (z) f(k−1) + ...+ A1 (z) f′+ A0 (z) f = 0, (1.4)f (k) + Ak−1 (z) f(k−1) + ...+ A1 (z) f′+ A0 (z) f = F (z) , (1.5)where A0 (z) , ..., Ak−1 (z) and F (z) /≡ 0 are entire functions. It is well-knownthat all solutions of equations (1.4) and (1.5) are entire functions.Extensive work in recent years has been concerned with the growth ofsolutions of complex linear differential equations. Many results have beenobtained. Examples of such results are the following two theorems:Theorem A [4]. Let A0 (z) , ..., Ak−1 (z) be entire functions such that thereexists one transcendental As (0 ≤ s ≤ k − 1) satisfying σ (Aj) ≤ σ (As) forall j 6= s. Then equation (1.4) has at least one solution f that satisfiesσ2 (f) = σ (As) .EJQTDE, 2006 No. 9, p. 2Theorem B [4]. Let A0 (z) , ..., Ak−1 (z) satisfy the hypotheses of TheoremA and F (z) /≡ 0 be an entire function with σ (F ) < +∞. Assume that f0is a solution of (1.5) , and g1, ..., gk are a solution base of the correspondinghomogeneous equation (1.4) of (1.5) . Then there exists a gj (1 ≤ j ≤ k) ,say g1, such that all the solutions in the solution subspace {cg1 + f0, c ∈ C}satisfy σ2 (f) = λ2 (f) = σ (As) , with at most one exception.The purpose of this paper is to extend the above two results by consid-ering the iterated order. We will prove the following theorems:Theorem 1.1 Let A0 (z) , ..., Ak−1 (z) be entire functions such that thereexists one transcendental As (0 ≤ s ≤ k − 1) satisfying σp (Aj) ≤ σp (As) <+∞ for all j 6= s. Then equation (1.4) has at least one solution f thatsatisfies i (f) = p+ 1 and σp+1 (f) = σp (As) .Theorem 1.2 Let A0 (z) , ..., Ak−1 (z) satisfy the hypotheses of Theorem1.1 and F (z) /≡ 0 be an entire function with i (F ) = q. Assume that f0 isa solution of (1.5) , and g1, ..., gk are a solution base of the correspondinghomogeneous equation (1.4) of (1.5) . If either i (F ) = q < p+1 or q = p+1and σp+1 (F ) < σp (As) < +∞, then there exist a gj (1 ≤ j ≤ k) , say g1,such that all the solutions in the solution subspace {cg1 + f0, c ∈ C} satisfyi (f) = p+1 and σp+1 (f) = λp+1 (f) = σp (As) , with at most one exception.Set g (z) = f (z) − z. Then clearly λp+1 (f − z) = λp+1 (g) and σp+1 (g) =σp+1 (f) . By Theorem 1.1 and Theorem 1.2, we can get the following corol-laries.Corollary 1 Under the hypotheses of Theorem 1.1, if A1 + zA0 /≡ 0, thenequation (1.4) has at least one solution f that satisfies i (f) = p + 1 andλp+1 (f − z) = σp+1 (f) = σp (As) .Corollary 2 Under the hypotheses of Theorem 1.2, if F−A1−zA0 /≡ 0, thenevery solution f of (1.5) with i (f) = p+1 and σp+1 (f) = λp+1 (f) = σp (As)satisfies λp+1 (f − z) = σp (As) .EJQTDE, 2006 No. 9, p. 32 Preliminary LemmasOur proofs depend mainly upon the following lemmas.Lemma 2.1 ([3] , [11]) . Let f (z) =∞∑n=0an zn be an entire with σp+1 (f) = σ,let µ (r) be the maximum term, i.e., µ (r) = max{|an| rn; n = 0, 1, ...} andlet νf (r) be the central index of f , i.e., νf (r) = max{m, µ (r) = |am| rm}.Thenlimr→∞logp+1 νf (r)log r= σ. (2.1)Lemma 2.2 (Wiman-Valiron, [8] , [12]) . Let f (z) be a transcendental en-tire function, and let z be a point with |z| = r at which |f (z)| = M (r, f).Then the estimationf (k) (z)f (z)=(νf (r)z)k(1 + o (1)) (k is an integer), (2.2)holds for all |z| outside a set E2 of r of finite logarithmic measure lm (E2) =∫ +∞1χE2(t)tdt, where χE2is the characteristic function of E2.Lemma 2.3 (See Remark 1.3 of [9]) . If f is a meromorphic function withi (f) = p ≥ 1, then σp (f) = σp(f′).Lemma 2.4 ([5]) . Let f1, ..., fk be linearly independent meromorphic solu-tions of the differential equationf (k) + Ak−1 (z) f(k−1) + ...+ A1 (z) f′+ A0 (z) f = 0, (2.3)with meromorphic coefficients A0 (z) , ...Ak−1 (z). Thenm (r, Aj) = O{log(max1≤n≤kT (r, fn))}(j = 0, ..., k − 1) . (2.4)Lemma 2.5 ([9]) . Let f be a meromorphic function for which i (f) = p ≥ 1and σp (f) = σ, and let k ≥ 1 be an integer. Then for any ε > 0,m(r,f (k)f)= O(expp−2{rσ+ε}), (2.5)EJQTDE, 2006 No. 9, p. 4outside of a possible exceptional set E3 of finite linear measure.To avoid some problems caused by the exceptional set we recall thefollowing Lemma.Lemma 2.6 ([1, p. 68] , [9]) . Let g : [0,+∞)→ R and h : [0,+∞)→ Rbe monotone non-decreasing functions such that g (r) ≤ h (r) outside of anexceptional set E of finite linear measure. Then for any α > 1, there existsr0 > 0 such that g (r) ≤ h (αr) for all r > r0.3 Proof of Theorem 1.1Suppose that f is a solution of (1.4) . We can rewrite (1.4) asf (k)f+ Ak−1 (z)f (k−1)f+ ...+ As+1 (z)f (s+1)f+ As (z)f (s)f+As−1 (z)f (s−1)f+ ...+ A1 (z)f ′f+ A0 (z) = 0. (3.1)By Lemma 2.2, there exists a set E2 ⊂ (1,+∞) with logarithmic measurelm (E2) < +∞ and we can choose z satisfying |z| = r /∈ [0, 1] ∪ E2 and|f (z)| =M (r, f), such that (2.2) holds. For given small ε > 0 and sufficientlylarge r, we have|Aj (z)| ≤ expp{rσp(As)+ε}(j = 0, 1, ..., k − 1) . (3.2)Substituting (2.2) into (3.1), we obtain by using (3.2)(νf (r)|z|)k|1 + o (1)| ≤ k(νf (r)|z|)k−1|1 + o (1)| expp{rσp(As)+ε}, (3.3)(r /∈ [0, 1] ∪E2) . By Lemma 2.1, Lemma 2.6 and (3.3) , we obtain that i (f) ≤p+ 1 andσp+1 (f) = limr→∞logp+1 νf (r)log r≤ σp (As) + ε. (3.4)Since ε > 0 is arbitrary, then σp+1 (f) ≤ σp (As) .EJQTDE, 2006 No. 9, p. 5Assume that {f1, ..., fk} is a solution base of (1.4). Then by Lemma 2.4m (r, As) ≤M log(max1≤n≤kT (r, fn)). (3.5)We assert that there exists a set E ⊂ (0,+∞) of infinite linear measure suchthatlimr→∞r∈Elogpm (r, As)log r= σp (As) . (3.6)In fact, there exists a sequence {rn} (rn →∞) such thatlimrn→∞logpm (rn, As)log rn= σp (As) . (3.7)We take E =∞∪n=1[rn, 2rn]. Then on E, (3.6) holds obviously. Now bysetting En = {r : r ∈ E and m (r, As) ≤M log T (r, fn) (n = 1, ..., k)}, wehavek∪n=1En =E. It is easy to see that there exists at least one En, say E1,which has an infinite linear measure and on whichlimr→∞r∈E1logpm (r, As)log r= σp (As) , (3.8)andm (r, As) ≤M log T (r, f1) (r ∈ E1) . (3.9)From (3.8) and (3.9) we have i (f1) ≥ p + 1 and σp+1 (f1) ≥ σp (As) . Thisand the fact that i (f1) ≤ p+ 1 and σp+1 (f1) ≤ σp (As) yield i (f1) = p+ 1and σp+1 (f1) = σp (As). The proof of Theorem 1.1 is complete.4 Proof of Theorem 1.2Assume that f is a solution of (1.5) and g1, ..., gk are k entire solutions of thecorresponding homogeneous equation (1.4) . Then by the proof of Theorem1.1, we know that i (gj) ≤ p + 1, σp+1 (gj) ≤ σp (As) (j = 1, 2, 3..., k) andEJQTDE, 2006 No. 9, p. 6there exists a gj, say g1, satisfying i (g1) = p+ 1, σp+1 (g1) = σp (As) . Thusby variation of parameters, f can be expressed in the formf (z) = B1 (z) g1 (z) + ... +Bk (z) gk (z) , (4.1)where B1 (z) , ..., Bk (z) are determined byB′1 (z) g1 (z) + ... +B′k (z) gk (z) = 0B′1 (z) g′1 (z) + ... +B′k (z) g′k (z) = 0.....................................................B′1 (z) g(k−1)1 (z) + ... +B′k (z) g(k−1)k (z) = F. (4.2)Noting that the Wronskian W (g1, g2, ..., gk) is a differential polynomial ing1, g2, ..., gk with constant coefficients, it follows thatσp+1 (W ) ≤ max {σp+1 (gj) : j = 1, ..., k} ≤ σp (As) .SetWj =∣∣∣∣∣∣∣∣∣g1, ...,(j)0 , ..., gk......g(k−1)1 , ..., F, ..., g(k−1)k∣∣∣∣∣∣∣∣∣= F.Gj (j = 1, ..., k) , (4.3)where Gj (g1, g2, ..., gk) are differential polynomials in g1, g2, ..., gk and of theirderivatives with constant coefficients. Soσp+1 (Gj) ≤ max {σp+1 (gj) : j = 1, ..., k} ≤ σp (As) (j = 1, ..., k) ,B′j =WjW=F.GjW(j = 1, ..., k) . (4.4)Since i (F ) = q < p+1 or i (F ) = p+1, σp+1 (F ) < σp (As), then by Lemma2.3, we obtainσp+1 (Bj) = σp+1(B′j)≤ max (σp+1 (F ) , σp (As)) = σp (As) (j = 1, ..., k) .(4.5)EJQTDE, 2006 No. 9, p. 7Then from (4.1) and (4.5) , we get i (f) ≤ p+ 1 andσp+1 (f) ≤ max {σp+1 (gj) , σp+1 (Bj) : j = 1, ..., k} ≤ σp (As) . (4.6)Now we setH = {fc = cg1 + f0, c ∈ C} , (4.7)where f0 is a solution of (1.5) . Obviously, every fc in H is a solution of (1.5) .Now we prove that for any two solutions fa and fb (a 6= b) in H , there is atleast one solution, say fa, among fa and fb satisfying i (fa) = p + 1 andσp+1 (fa) = λp+1 (fa) = σp (As) . Since fa = (a− b) g1+ fb, thenT (r, g1) ≤ T (r, fa) + T (r, fb) +O (1) . (4.8)Assume that the set E1 satisfies the condition as required in proof of Theorem1.1. Then there exists at least one of fa and fb, say fa, such that there is asubset E4 of E1 with infinite linear measure andT (r, fb) ≤ T (r, fa) , for r ∈ E4. (4.9)We get from (4.8) and (4.9)T (r, g1) ≤ 2T (r, fa) +O (1) , for r ∈ E4. (4.10)Thus, i (fa) ≥ p + 1 and σp+1 (fa) ≥ σp+1 (g1) = σp (As) and hence i (fa) =p+ 1, σp+1 (fa) = σp (As) = σ.Now we prove that σp+1 (fa) = λp+1 (fa) = σ. By (1.5) , it is easy to seethat if fa has a zero at z0 of order α (> k), then F must have a zero at z0 oforder α− k. Hence,n(r,1fa)≤ k n(r,1fa)+ n(r,1F)(4.11)andN(r,1fa)≤ k N(r,1fa)+N(r,1F). (4.12)Now (1.5) can be rewritten as1fa=1F(f(k)afa+ Ak−1f(k−1)afa+ ... + A1f′afa+ A0). (4.13)EJQTDE, 2006 No. 9, p. 8By (4.13), we havem(r,1fa)≤k∑j=1m(r,f(j)afa)+k∑j=1m (r, Ak−j) +m(r,1F)+O (1) . (4.14)Applying the Lemma 2.5, we havem(r,f(j)afa)= O(expp−1{rσ+ε})(j = 1, ..., k − 1) , (σp+1 (fa) = σ) ,(4.15)holds for all r outside a set E3 ⊂ (0,+∞) with a linear measure m (E3) =δ < +∞. By (4.12) , (4.14) and (4.15), we getT (r, fa) = T(r,1fa)+O (1)≤ kN(r,1fa)+k∑j=1T (r, Ak−j)+T (r, F )+O(expp−1{rσ+ε})(|z| = r /∈ E3) .(4.16)For sufficiently large r, we haveT (r, A0) + ... + T (r, Ak−1) ≤ k expp−1{rσ+ε}. (4.17)If i (F ) = q < p+ 1, then q − 1 ≤ p− 1 andT (r, F ) ≤ expq−1{rσq(F )+ε}≤ expp−1{rσq(F )+ε}(σq (F ) <∞) . (4.18)Thus, by (4.16)− (4.18), we haveT (r, fa) ≤ k N(r,1fa)+ k expp−1{rσ+ε}+expp−1{rσq(F )+ε}+O(expp−1{rσ+ε})(|z| = r /∈ E3) . (4.19)Hence for any fa with σp+1 (fa) = σ, by (4.19) and Lemma 2.6, we haveσp+1 (fa) ≤ λp+1 (fa). Therefore, λp+1 (fa) = σp+1 (fa) = σ.If i (F ) = p+ 1 and σp+1 (F ) < σp (As) = σ, thenT (r, F ) ≤ expp{rσp+1(F )+ε}≤ expp−1{rσ+ε}. (4.20)EJQTDE, 2006 No. 9, p. 9Thus, by (4.16)− (4.17) and (4.20), we haveT (r, fa) ≤ k N(r,1fa)+ k expp−1{rσ+ε}+expp−1{rσ+ε}+O(expp−1{rσ+ε})(|z| = r /∈ E3) . (4.21)By using similar reasoning as above, we obtain from (4.21) and Lemma 2.6that λp+1 (fa) = σp+1 (fa) = σ. The proof of Theorem 1.2 is complete.Acknowledgement. The author would like to thank the referee for his/herhelpful remarks and suggestions.References[1] S. Bank, A general theorem concerning the growth of solutions offirst-order algebraic differential equations, Compositio Math. 25 (1972), 61-70.[2] L. G. Bernal, On growth k−order of solutions of a complex homo-geneous linear differential equations, Proc. Amer. Math. Soc. 101 (1987),317-322.[3] T. B. Cao, Z. X. Chen, X. M. Zheng and J. Tu, On the iteratedorder of meromorphic solutions of higher order linear differential equations,Ann. of Diff. Eqs., 21, 2 (2005), 111-122.[4] Z. X. Chen and C. C. Yang, Quantitative estimations on the ze-ros and growths of entire solutions of linear differential equations, ComplexVariables Vol. 42 (2000), 119-133.[5] G. Frank and S. Hellerstein, On the meromorphic solutions of non-homogeneous linear differential equations with polynomial coefficients, Proc.London Math. Soc. (3), 53 (1986), 407-428.[6] G. Gundersen, Estimates for the logarithmic derivative of a mero-morphic function, plus similar estimates, J. London Math. Soc. (2) 37(1988), 88-104.[7] W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford,1964.EJQTDE, 2006 No. 9, p. 10[8] W. K. Hayman, The local growth of power series: a survey of theWiman-Valiron method, Canad. Math. Bull., 17(1974), 317-358.[9] L. Kinnunen, Linear differential equations with solutions of finiteiterated order, Southeast Asian Bull. Math., 22: 4 (1998), 385-405.[10] I. Laine, Nevanlinna Theory and Complex Differential Equations,Walter de Gruyter, Berlin, New York, 1993.[11] J. Tu, Z. X. Chen and X. M. Zheng, Growth of solutions of com-plex differential equations with coefficients of finite iterated order, Electron.J. Diff. Eqns., Vol. 2006 (2006), N◦54, 1-8.[12] G. Valiron, Lectures on the General Theory of Integral Functions,translated by E. F. Collingwood, Chelsea, New York, 1949.[13] H. X. Yi and C. C. Yang, The Uniqueness Theory of MeromorphicFunctions, Science Press, Beijing, 1995 (in Chinese).(Received April 2, 2006)EJQTDE, 2006 No. 9, p. 11

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On the iterated order and the fixed points of entire solutions of some complex linear differential equations

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