FIXED POINTS OF MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 3, 2006, 9 2 FIXED POINTS OF MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS Liu Ming-Sheng and Zhang Xiao-Mei South China Normal University, Department of Mathematics Guangzhou 5063, Guangdong, P. R. China; liumsh@scnu.edu.cn Abstract. In this paper we discuss the problems on the fixed points of meromorphic solutions of higher order linear differential equations with meromorphic coefficients and their derivatives. Because of the restriction of differential equations, we obtain that the properties of fixed points of meromorphic solutions of higher order linear differential equations with meromorphic coefficients and their derivatives are more interesting than those of general transcendental meromorphic functions. Some estimates of the exponent of convergence of fixed points of solutions and their derivatives are obtained.. Introduction and main results Many important results have been obtained on the fixed points of general transcendental meromorphic functions for almost four decades see [4]. However, there are few studies on the fixed points of solutions of differential equations. In [3], Chen Zong-Xuan at first studied the problems on the fixed points and hyper-order of solutions of second order linear differential equations with entire coefficients. In [], Wang and Yi studied the problems on the fixed points and hyper order of differential polynomials generated by solutions of second order linear differential equations with meromorphic coefficients. In [9], I. Laine and J. Rieppo had given an extension and improvement of the results in []; they studied the problems on the fixed points and iterated order of differential polynomials generated by solutions of second order linear differential equations with meromorphic coefficients. In [0], Wang and Lü studied the problems on the fixed points and hyper-order of solutions of second order linear differential equations with meromorphic coefficients and their derivatives. The main purpose of this paper is to extend some results in [0] to the case of higher order linear differential equations with meromorphic coefficients. In this paper, we shall assume that the reader is familiar with the fundamental results and the standard notation of R. Nevanlinna s theory of meromorphic functions see [4], [3], [2], [8]. In addition, let σf denote the order of growth 2000 Mathematics Subject Classification: Primary 30D35, 34M05. This wor is supported by the National Natural Science Foundation of China No

2 92 Liu Ming-Sheng and Zhang Xiao-Mei of the meromorphic function fz, λf denote the exponent of convergence of the sequence of distinct zeros of f, v f r denote the central index of fz. In order to give some estimates of fixed points, we recall the following definitions see [3], [7], [9]. Definition.. Let z, z 2,..., z j r j, 0 r r 2, be the sequence of distinct fixed points of a transcendental meromorphic function f. Then τf, the exponent of convergence of the sequence of distinct fixed points of f, is defined by It is evident that { τf inf τ > 0 j z j τ < +. log N r, f z τf lim. r log r Definition.2. Suppose that fz be a meromorphic function of infinite order. Then the hyper-order σ 2 f of fz is defined by σ 2 f lim r log log T r, f. log r Definition.3. Let f be a meromorphic function. Then λ 2 f, the hyperexponent of convergence of the sequence of distinct zeros of f, is defined by log log N r, f λ 2 f lim, r log r and τ 2 f, the hyper-exponent of convergence of the sequence of distinct fixed points of f, is defined by log log N r, f z τ 2 f lim. r log r Definition.4 [2], [6]. Let P z be rational and of the form P z/p 2 z, where P z and P 2 z are two polynomials. We denote by dip, the degree at infinity of P, defined by. dip deg P z deg P 2 z. In [0], Wang and Lü obtained the following results.

3 Fixed points of meromorphic solutions 93 Theorem A. Suppose that P z P z/p 2 z 0 be a rational function with n dip. Then every transcendental meromorphic solution f of the equation.2 f + P zf 0 satisfies that f and f, f all have infinitely many fixed points and τf τf τf σf max { 2 n + 2, 0. Theorem B. Suppose that Az be a transcendental meromorphic function satisfying δ, A > 0, σa σ < +. Then every meromorphic solution f 0 of the equation.3 f + Azf 0 satisfies that f and f, f all have infinitely many fixed points and τf τf τf σf +, τ 2 f τ 2 f τ 2 f σ 2 f σ. Remar.. In [0], Wang and Lü did not give the detailed proof of Theorem A, they only proved that σf max { 2 n + 2, 0, and omit the proof of τf τf τf σf. In fact, when n 2, we cannot prove that τf τf σf by using the similar method as the proof of Theorem B in [0]; please loo at Remar 3. in Section 3. In this paper, we shall prove the following two theorems. Theorem.. Suppose that P z P z/p 2 z 0 be a rational function with n dip, and be an integer with 2. Then: Every transcendental meromorphic solution f of the equation.4 f + P zf 0 satisfies σf max{n + /, 0. 2 If n, then every meromorphic solution f 0 of the equation.4 satisfies that f and f, f,..., f all have infinitely many fixed points and { n + τf τf τf max, 0. 3 If n, then every transcendental meromorphic solution f of the equation.4 satisfies that f and f, f,..., f 2 all have infinitely many fixed points and τf τf τf 2 0.

4 94 Liu Ming-Sheng and Zhang Xiao-Mei Remar.2. Setting 2 in Theorem., we get Theorem of [0] or Theorem A, which contains the related results in []. From Theorem., we now that every transcendental meromorphic solution f of equation.4 has infinitely many fixed points. In addition, if n, there exists some equations of the form.4 that may have solutions with finite fixed points. For example, the equation f 6 zz 2z 5 f 0 has a family of polynomial solutions {f c czz 2z 5; c is a constant, which have at most three fixed points. Theorem.2. Suppose that 2 and Az be a transcendental meromorphic function satisfying δ, A δ > 0, σa σ < +. Then every meromorphic solution f 0 of the equation.5 f + Azf 0 satisfies that f and f, f,..., f all have infinitely many fixed points and.6 τf τf j σf +, τ 2 f τ 2 f j σ 2 f σ, for j,...,. Remar.3. Setting 2 in Theorem.2, we get Theorem 2 of [0] or Theorem B. 2. Some lemmas Lemma 2. []. Suppose that fz gz/dz is a meromorphic function with σf ϱ, where gz is an entire function and dz is a polynomial. Then there exists a sequence {r j, r j, such that for all z satisfying z r j, gz Mr j, g, when j sufficiently large, we have f n n z vg r j + o, n, fz z log v g r j σf lim. j log r j Lemma 2.2. Suppose that 2 and Az is a transcendental meromorphic function with δ, A δ > 0 and σa σ < +. Then every meromorphic solution f 0 of equation.5 satisfies σf + and σ 2 f σ.

5 Fixed points of meromorphic solutions 95 as Proof. Suppose that f 0 is a meromorphic solution of.5. Rewrite.5 2. A f f. By the lemma of logarithmic derivatives, there exists a set E with finite linear measure such that mr, A c log rt r, f for r / E, where c is a positive constant. It follows from the definition of deficiency that for sufficiently large r, we have mr, A 2δT r, A. So when r / E sufficiently large, we have 2.2 T r, A 2c δ log rt r, f. Hence by the definition of hyper-order, we obtain that σf +, and log log T r, f 2.3 σ 2 f lim r + log r σa σ. On the other hand, we now that the poles of f can only occur at the poles of A, so λ/f σa. According to the Hadamard factorization theorem, we can write f as fz gz/dz, where dz and gz are entire functions satisfying λd σd λ/f σ σ 2 g σ 2 f. Substituting fz gz/dz into.5, by Theorem 2. in [7], we get 2.4 σ 2 f σ 2 g σ. Hence σ 2 f σ. This completes the proof. Lemma 2.3. Suppose that be a positive integer and fz be a nonzero solution of equation.5. Let w 0 f z, w f z, w 2 f z. Then w 0, w and w 2 satisfy the following equations: w 0 Aw 0 za, if 2; Aw + A w A 2 w za 2, if 3; A 2 w 2 + 2AA w 2 + AA 2A 2 w 2 2 A 3 w 2 za 3, if 4; Aw + A w A2 w za 2 A, if 2; A 2 w 2 + 2AA w 2 + AA 2A 2 A 3 w 2 za 3 2AA + 2zA 2 zaa, if 2; A 2 w 2 + 2AA w 2 + AA 2A 2 w 2 A3 w 2

6 96 Liu Ming-Sheng and Zhang Xiao-Mei 2. za 3 AA + 2A 2, if 3; A 3 w 3 + 3A2 A w 3 + 3AAA 2A 2 w 3 + A 2 A 6AA A + 6A 3 A 4 w 3 za 4 3AAA 2A 2 za 2 A 6AA A + 6A 3, if 3. Proof. From f w 0 + z, we have that f w 0 for 2. Substituting it into.5, we get f w 0 /A, that is, w 0 + z w 0 /A. From this, we obtain 2.5. According to the equality f w + z, we obtain that f w for 3. Substituting it into.5, we get f w /A, so that f w A + w A /A 2, that is, w + z w A + w A /A 2. From this, we obtain 2.6. By the equality f w 2 + z, we get that f w 2 2 for 4. Substituting it into.5, we get f w 2 2 /A, so that that is, f f 2 A w 2 A + w 2 A 2 A2 w 2 + 2AA w 2 + AA 2A 2 w 2 2 A 3, w 2 + z A2 w 2 + 2AA w 2 + AA 2A 2 w 2 2. A 3 From this, we obtain 2.7. We may prove equations similarly. Hence the proof of Lemma 2.3 is complete. Let i, j are two non-negative integers. Now we define the notation H ij A as follows. For any non-negative integer i, we define H i0 A A i ; 2 For any non-negative integer i, H ii A and H ii A are defined by the following recurrence formula: 2.2 H i+i+ A A [H ii A] i + A H ii A, H i+2i+ A A [H i+i A] i + 2A H i+i A + A H i+i+ A, H 00 A, H 0 A A;

7 Fixed points of meromorphic solutions 97 3 For i 3 and j i 2, H ij A are defined as a sum of a finite number of terms of the type 2.3 B ba l 0 A l A i+ l i+, where b is a constant, l 0, l,..., l i+ are non-negative integers such that l 0 + l + + l i+ i and l + 2l i + l i+ j. It is obvious that H ij A is a differential polynomial of A. Moreover, this notation H ij A, j i 2, may represent a different differential polynomial of A in different occurrences, even within one single formula. By the definition of H ij A, simple computation yields 2.4 H 00 A, H 0 A A, H A A ; H 20 A A 2, H 2 A 2AA, H 22 A AA 2A 2, H 32 A 3A 2 A 6AA 2, H 33 A A 2 A 6AA A + 6A 3 ; A H ij A Hi+j+ A for 0 < j i 2; A H ij A H i+j+ A for 0 < j i 2; A H ij A H i+j A for 0 j i. Lemma 2.4. Suppose that fz is a nonzero solution of equation.5 and 2. Let w i f i z, i 0,,..., 2. Then w i satisfy the following equations: 2.5 i H ij Aw j i A i+ w i za i+, i 0,,..., 2, where H ij A are defined by Proof. When 2 or 3, 4, by Lemma 2.3 and , it is evident that 2.5 hold. In the following, we suppose that 5. We shall use an inductive method to prove it. At first, by Lemma 2.3 and 2.4, we get that 2.5 hold for i 0,, 2. Next, suppose that w i, 2 i 3, satisfy 2.5. Now we verify that w i+ also satisfies 2.5.

8 98 Liu Ming-Sheng and Zhang Xiao-Mei From w i + z f i and w i+ + z f i+, we now that w j i Since w i satisfies 2.5, we have w j i+, j 0,,..., i, i 3. f i w i + z i H ijaw j i A i+, so by 2.4 and 2.2, we obtain f i+ f i i H ijaw j i A i+ i H ijaw j i+ A 2i+ A i+ {[ i Hij A j w i + A i A i+ + i i H ij Aw j i+ ] H ij Aw j i+ A i+ { i 2 H A i+2 i+j+ Aw j i+ + A H ii A i w i+ + A H ii A w i i+ + i + A j i i H ij Aw j i+ AH ij Aw j i+ { i A i++ H i+j Aw j i+ + H i+i Aw i i+ + H i+i+ Aw i i+ i + H i+j Aw j i+ + H i0 A Aw i+ j i i + H i+j Aw j i+ j

9 Fixed points of meromorphic solutions 99 { A i++ A i+ w A i++ i+ i+ + i+ j H i+j Aw j i+ H i+j Aw j i+ w i+ + z. From the above equality, we obtain that w i+ also satisfies 2.5. This completes the proof. Lemma 2.5. Suppose that fz 0 is a solution of equation.5 and 2. Let w f z. Then w satisfies 2.6 H j Aw j where H ij are defined by A w za H A, Proof. When 2 or 3, by 2.8, 2.0 and 2.4, it is evident that 2.6 holds. When 4, from w 2 f 2 z and w f z, we obtain w j 2 w j, j 0,, 2,..., 3, w 2 w +. Since w 2 satisfies 2.5, rewrite it as 2 f 2 w 2 + z H 2jAw j 2, A so by 2.4 and 2.2, we have f f 2 2 H 2jAw j 2 A 3 H 2jAw j + H 2 2 Aw + A {[ 3 H A 2 2j A w j 3 + H 2j Aw j + H 2 2 A w + + H 2 2 Aw [ 3 H 2j Aw j + H 2 2 Aw + ] A 2 A ] A

10 200 Liu Ming-Sheng and Zhang Xiao-Mei { 4 A A H 2j A 3 j w + AH 2j Aw j + A H 2 3 A w A H 2 3 Aw + AH 2 2 Aw + A H 2 2 A w + A H 2 2 Aw + 4 A H 2j Aw j { 4 A 3 H j+ Aw j + j H j Aw j + AH 20 Aw + H 2Aw + H Aw + 4 H j+ Aw j A { A w { A + 3 j H j Aw j + H 2 Aw + H Aw + H j Aw j + H A w + z. From the above equality we get 2.6. This completes the proof. Lemma 2.6. Suppose that fz 0 is a solution of equation.5 and 2. Let w f z. Then w satisfies 2.7 H j Aw j + H A A + w za + H A zh A, where H ij are defined by Proof. When 2, by 2.9 and 2.4, it is evident that 2.7 holds. When 3, we obtain from w f z and w f z, w j w j, j 0,,..., 3, w w +, w w + z.

11 Fixed points of meromorphic solutions 20 Since w satisfies 2.6, rewrite it as f w + z so by 2.4 and 2.2, we have H jaw j + H A A, f f H jaw j + H A A 3 H jaw j + H 2 Aw + A + H Aw + z + H A A 3 H jaw j + H 2 Aw + {[ 3 A 2 H j A w j A + H Aw + z A 3 + H j Aw j + H 2 A w + + H 2 Aw + H A w + z + H Aw + ] A [ 3 H j Aw j + H 2 Aw + + H Aw + z ]A A { 3 H A + j+ Aw j 3 + AH 0 Aw + H j Aw j j + A H 2 A w + + AH 2 Aw + A H A w + z

12 202 Liu Ming-Sheng and Zhang Xiao-Mei + AH Aw + A H 2 Aw + 3 A H Aw + z H j+ Aw j { 2 H A + j Aw j j + H 2 Aw + H Aw + z A w + H Aw + { H A + j Aw j + H A + zh A w + z. From the above equality we obtain 2.7. This completes the proof. Lemma 2.7. Suppose that 2 and P z is a rational function with n dip < 2. Then there exist two nonzero constants c and c 2 such that 2.8 H P c + o z n ++n as z, and 2.9 H P + zh P c 2 + o z n+ ++n as z. Proof. At first, we prove 2.8 by means of induction. Since P z is a rational function with n dip < 2, we have 2.20 P z c 0 + o z n as z, where c 0 0 is a constant. From this and 2.4, simple computation yields H P P nc 0 + o z n+ as z, where nc 0 0. Hence 2.8 holds, when 2. Furthermore, if 2.8 is assumed to be valid when m > 2, that is, there exists a constant c 0 such that H m m P c + o z nm +m+n as z,

13 Fixed points of meromorphic solutions 203 it must also hold when m +, since, by 2.2, H mm P P [H m m P ] mp H m m P c [ 0 + o c z n + o ] z nm +m+n m nc 0 + o c z n+ + o z nm +m+n m nc 0c + o z nm+ +m++n c + o z nm+ +m++n as z, where c m nc 0 c 0. Hence 2.8 holds. Next, we prove 2.9 by means of induction. By 2.20 and 2.4, simple computation yields H 2 P + zh 22 P 2P P + zp P 2P 2 n nc2 0 + o z 2n+ c 2 + o z 4n+3 as z, where c 2 n nc Hence 2.9 holds, when 2. Furthermore, if 2.9 is assumed to be valid when m > 2, that is, there exists a constant c 2 0 such that H mm P + zh mm P it must also hold when m +, since, by 2.2, c 2 + o z nm+ +m+n as z, H m+m P + zh m+m+ P P [H mm P ] m + P H mm P where c 2 m nc 0 c 2 + P H mm P + zp [H mm P ] m + zp H mm P P [H mm P + zh mm P ] m + [H mm P + zh mm P ] c [ 0 + o c 2 z n + o ] z nm+ +m+n m + nc 0 + o z n+ m nc 0c 2 + o z nm+2 +m++n c 2 + o z nm+ +m+n c 2 + o z nm+2 +m++n as z, 0. Hence 2.9 holds, and the proof is complete.

14 204 Liu Ming-Sheng and Zhang Xiao-Mei 3. Proofs of Theorems. and.2 Proof of Theorem.2. Suppose that f 0 is a meromorphic solution of.5. Then, by Lemma 2.2, we obtain that σf, σ 2 f σ. Set w i f i z, i 0,,...,. Then for every i, a point z 0 points of f i if and only if z 0 is a zero of w i, and is a fixed 3. σw i σf i σf +, τf i λw i, and 3.2 σ 2 w i σ 2 f i σ 2 f σ, τ 2 f i λ 2 w i. By Lemmas , we now that w i, i 0,,...,, satisfy the following equations: i H ij Aw j i A i+ w i za i+, i 0,,..., 2; H j Aw j H j Aw j A w za H A; + H A A + w za + H A zh A. Since A is a transcendental meromorphic function, A i+ z 0, i 0,,..., 2. We claim that za H A 0 and za + H A zh A 0. In fact, if za H A 0, rewrite it as A H A za. From this, we obtain that mr, A Sr, A, which contradicts δ, A > 0. If za + H A zh A 0, rewrite it as A H A za + H A A.

15 Fixed points of meromorphic solutions 205 Similarly, we may obtain that mr, A Sr, A, which contradicts δ, A > 0; hence the claims hold. Rewrite 3.3, 3.4 and 3.5 as i H za i+ ij A w j A iw i+, i 0,,..., 2, i w i za H A za + H A zh A H j A w j w H j A w j + H A A w + A, w. w Hence there exists a set E with finite linear measure such that for r / E, we have 3.9 m r, wi O m r, + C log rt r, w i, i 0,,...,. A On the other hand, since H ij A are differential polynomials of A, according to their definition, it is easy to get that if z 0 is a pole of order l of A, then z 0 is a pole of order li + j of H ij A. In the following, we split it into three cases to discuss the poles of w i : Case : i 0,,..., 2. Since the coefficients of w j i are H ij A, j 0,,..., i, and the right-hand sides of the equations 3.3 are A i+ z, by 3.3, all zeros except for z 0 of w i, whose order is larger than, are zeros of Az each is not the pole of A, if not, it will lead to a contradiction as follows. Hence 3.0 N r, wi N r, w i + N r,. za i+ Case 2: i. Suppose that w has a zero of order 2 +2 at point z, and z is also the pole of order l of Az. We claim that 0 l. In fact, if l, then z is a pole of order l of za H A, and z is also a pole of order l, l l < l, of the left-hand side in equality 3.4; this is a contradiction; hence 0 l. At this time, z is a pole of order l +j of H j A for j 0,,...,. Therefore, by 3.4, direct computation yields that z is a zero of order l++l of the left-hand side of

16 206 Liu Ming-Sheng and Zhang Xiao-Mei equality 3.4, so z is also a zero of order l++l of za H A. Especially, l + + l 2 + for l. Hence 3. N r, 2 +2N r, +N r,. za H A w w Case 3: i. Suppose that w has a zero of order at point z 2, and z 2 is also the pole of order t of Az. We claim that 0 t. In fact, if t +, then z 2 is a pole of order + t of za + H A zh A, and z 2 is also a pole of order + t 2 of the left-hand side in equality 3.5; this is a contradiction. Hence 0 t. At this time, z 2 is a pole of order t + j of H j A for j 0,,...,. Therefore, by 3.5, direct computation yields that z 2 is a zero of order 2 t + of the left-hand side of equality 3.5, so z 2 is also a zero of order 2 t + of za + H A zh A. Especially, t for t. Hence 3.2 N r, w N r, w + N r,. za + H A zh A Note that for sufficiently large r, C log rt r, w i 2 T r, w i, i 0,,...,. Because w i, i 0,,...,, are transcendental meromorphic functions and σa σ < +, so for any given ε > 0, it follows from and 3.9 that 3.3 T r, w i N r, + Or σ+ε, i 0,,...,, for r / E 2 sufficiently large, where E 2 has finite linear measure. By Definition.2, 3., 3.2 and 3.3, we obtain that for i 0,,...,, we have 3.4 τf i λw i σw i +, τ 2 f i λ 2 w i σ 2 w i σ. That is, each meromorphic solution f 0 of.5 and its f,..., f have infinitely many fixed points and satisfy.6. This completes the proof. Proof of Theorem.. Suppose that fz is a transcendental meromorphic solution of.4. Then we now that the poles of f can only occur at the poles of P z, so f has only finite poles. According to the Hadamard factorization theorem, we can write f as fz gz/dz, where gz is an entire function satisfying σg σf and dz is a polynomial. By Lemma 2., for any ε > 0, there exists a sequence {r j, r j, such that for all z satisfying z r j, gz Mr j, g, when j is large enough, we have f z vg r j o, fz z log v g r j 3.6 σf lim. j log r j w i all

17 Substituting 3.5 into.4, we have Fixed points of meromorphic solutions vg r j r j + o r n j a + o, where a is a nonzero constant. From 3.6 and 3.7, we obtain that σf max{n + /, 0. 2: dip n. Suppose that f 0 is a meromorphic solution of.4. Then f must be a transcendental meromorphic function. In fact, if not, then f is a rational function. We conclude that f z/fz b + o /z as z refer to p. 62 in [4]. On the other hand, since dip n and P z 0, we have that P z b 0 + o/z n as z, where b 0 0; this contradicts f z/fz P z and n. Hence fz is a transcendental meromorphic function. From part, we have { n σf max, D. Set w i f i z, i 0,,...,. Then for every i, a point z 0 is a fixed point of f i if and only if z 0 is a zero of w i, and { n σw i σf i σf max, D, τf i λw i. According to Lemmas , we now that w i, i 0,,...,, satisfy the following equations: 3.20 i H ij P w j i P i+ w i zp i+, i 0,,..., 2; 3.2 H j P w j P w zp H P, if 3; 3.22 H j P w j + H P P + w zp + H P zh P. Since P 0, we conclude that P i+ z 0 for i 0,,..., 2. We claim that zp H P 0 and zp + H P zh P 0. We discuss three subcases below.

18 208 Liu Ming-Sheng and Zhang Xiao-Mei Subcase I: dip n 0. If zp H P 0, rewrite it as P H P /P z. Notice that P z is a rational function. We have that P j z/p z 0 as z. From this we obtain that H P /zp 0 as z, which contradicts that P z or a nonzero constant. If zp + H P zh P 0, rewrite it as Similarly, we may obtain that P H P zp + H P P. H P zp + H P P 0 as z, which contradicts that P z or a nonzero constant; hence the claims hold. Subcase II: When < dip n, we have P z c 0 + o /z n as z, where c 0 is a nonzero constant. From this, we have 3.23 zp c 0 + o z n, zp + c+ 0 + o as z. z n+ On the other hand, by , using a similar argument as in the proof of Lemma 2.7, we have 3.24 H P d + o z n ++n as z, and 3.25 H P + zh P d 2 + o z n+ ++n as z, where d, d 2 are two constants note that d or d 2 may be zero. Hence it follows from and + n > 0 that zp H P 0 and zp + H P zh P 0, that is, the claims hold. Subcase III: n dip <. constants c, c 2 such that By Lemma 2.7, there exist two nonzero 3.26 H P c + o z n ++n as z,

19 Fixed points of meromorphic solutions 209 and 3.27 H P + zh P c 2 + o z n+ ++n as z. Hence it follows from 3.23, 3.26, 3.27 and + n < 0 that zp H P 0 and zp + H P zh P 0, that is, the claims hold. Rewrite 3.20, 3.2 and 3.22 as 3.28 i H zp i+ ij P w j P iw i+, i 0,,..., 2, i w i 3.29 H zp j P w j H P w P, w 3.30 zp + H P zh P H j P w j + H P P w +. w Hence there exists a set E with finite linear measure such that for r / E, and we have 3.3 m r, O m r, + C log rt r, w i, i 0,,...,. P w i Applying similar arguments as in the proof of Theorem.2, we obtain that 3.32 N r, 3.33 N r, w w i N r, w i 2 +2N 3.34 N r, N w r, w + N r, zp i+, i 0,,..., 2; r, w +N r, zp ; H P + N r,. zp + H P zh P

20 20 Liu Ming-Sheng and Zhang Xiao-Mei Note that for sufficiently large r, C log rt r, w i 2 T r, w i, i 0,,...,. Because w i, i 0,,...,, are transcendental meromorphic functions and P is a rational function, so for any given ε > 0, from , we have T r, w i N r, wi + Olog r, i 0,,...,, for r / E 3 sufficiently large, where E 3 has finite linear measure. By 3.9, we obtain that for i 0,,...,, f i z all have infinitely many fixed points and { 3.35 τf i λw n + i σw i max, 0. That is, each meromorphic solution f 0 of.4 and its f,..., f have infinitely many fixed points and satisfy Case 3: dip n. Suppose that fz is a transcendental meromorphic solution of.4. Set w i z f i z z, i 0,,..., 2, since P i+ z 0 for i 0,,..., 2. Using similar arguments as in part 2, we may prove that for any given ε > 0, 3.36 T r, w i 2N r, wi + Olog r, i 0,,..., 2, for r / E 4 sufficiently large, where E 4 has finite linear measure. Note that w i, i 0,,..., 2, are transcendental meromorphic functions. It follows from 3.36 that f and f, f,..., f 2 all have infinitely many fixed points and { n + τf τf τf 2 σw i max, 0 0. Hence the proof of Theorem. is complete. Remar 3.. When n in Theorem., we cannot prove that τf τf σf by using similar arguments as in part 2 of the proof of Theorem.. Since we cannot prove that zp H P 0 and zp + H P zh P 0 in 3.2 and 3.22, respectively. In fact, if n, there exist some equations of the form.4 such that zp H P 0 or zp + H P zh P 0. For example, when 2, by , simple computation yields and zp 2 H P zp 2 P 0 for P z 2 z 2, all

21 Fixed points of meromorphic solutions 2 zp 3 H 2 P zh 22 P zp 3 2P P + 2zP 2 zp P 0 for P z 6 z 2. When 3, simple computation yields zp 3 H 22 P zp 3 P P +2P 2 0 for P z 6/z 3, and for P z 24/z 3, we have zp 4 H 32 P zh 33 P zp 4 3P P P 2P 2 + 6zP P P zp 2 P 6zP 3 0. Acnowledgements. The authors than the referees for their helpful comments and suggestions to improve our manuscript. References [] Ban, S., and I. Laine: On the oscillation theory of f + Af 0 where A is entire. - Trans. Amer. Math. Soc. 273, 982, [2] Ban, S., and I. Laine: On the zeros of meromorphic solutions of second-order linear differential equations. - Comment. Math. Helv. 58, 983, [3] Chen, Z. X.: The fixed points and hyper-order of solutions of second order linear differential equations. - Acta Math. Scientia 20, 2000, Chinese. [4] Hayman, W.: Meromorphic Functions. - Clarendon Press, Oxford,964. [5] He, Y. Z., and X. Z. Xiao: Algebroid Functions and Ordinary Differential Equations. - Science Press, Beijing, 988 Chinese. [6] Hellerstein, S., and J. Rossi: Zeros of meromorphic solutions of second order linear differential equations. - Math. Z. 92, 986, [7] Kinnunen, L.: Linear differential equations with solutions of finite iterated order. - Southeast Asian Bull. Math. 22, 998, [8] Laine, I.: Nevanlinna Theory and Complex Differential Equations. - W. de Gruyter, Berlin, 993. [9] Laine, I., and J. Rieppo: Differential polynomials generated by linear differential equations. - Complex Variables 49, 2004, [0] Wang, J., and W. R. Lü: The fixed points and hyper-order of solutions of second order linear differential equations with meromorphic coefficients. - Acta Math. Appl. Sinica 27, 2004, Chinese. [] Wang, J., and H. X. Yi: Fixed points and hyper-order of differential polynomials generated by solutions of differential equation. - Complex Variables 48, 2003, [2] Yang, C. C., and H. X. Yi: Uniqueness Theory of Meromorphic Functions. - Science Press/ Kluwer Academic Publishers, Beijing New Yor, [3] Yang, L.: Value Distribution Theory and Its New Research. - Science Press, Beijing, 982 Chinese. [4] Zhang, Q. T. and C. C. Yang: The Fixed Points and Resolution Theory of Meromorphic Functions. - Beijing University Press, Beijing, 988 Chinese. Received 30 December 2004