Fixed points of the derivative and k-th power of solutions of complex linear differential equations in the unit disc

Transcription

1 Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 48, -9; Fixed points of the derivative and -th power of solutions of complex linear differential equations in the unit disc Guowei Zhang and Ang Chen Abstract In this paper we consider the question of the existence of fixed points of the derivatives of solutions of complex linear differential equations in the unit disc. This wor improves some very recent results of T.-B. Cao. Keywords: fixed points, solutions, complex linear differential equations, unit disc MR subject classification: 30D35. Introduction and main results In this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna s theory on the complex plane and in the unit disc D = {z C : z < } (see [6, 5, 5]). Many authors have investigated the growth and oscillation of the solutions of complex linear differential equations in C. In the unit disc, there already exist many results [, 7, 8], but the study is more difficult than that in the complex plane, because the efficient tool, Wiman-Valiron theory, in the complex plane doesn t hold in the unit disc. 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, specially in the unit disc. In [3], Z.-X. Chen firstly studied the problem on the fixed points and hyper-order of solutions of second order linear differential equations with entire coefficients. After that, there were some results which improve those of Z.-X. Chen, see [0, 3, 4, 9]. Recently, T.-B. Cao [] firstly investigated the fixed points of solutions of linear complex differential equations in the unit disc. In the present paper, we continue to study the problem in the unit disc. In order to Department of Mathematics, Shandong University, Jinan 25000, P. R. China address: zhirobo@yahoo.com.cn (G. W. Zhang); ang.chen.jr@gmail.com (A. Chen) The second author is the corresponding author. The authors were supported by the NNFC of China (No.0772), the NSF of Shang-dong Province, China (No Z2008A0) EJQTDE, 2009 No. 48, p.

2 be read more clearly, we give some definitions as following. For n N, the iterated n-order of a meromorphic function f in D is defined by σ n (f) = lim sup logn T(r,f) log( r), where log = log = max{log x,0}, log n x = log log n x. If f is analytic in D, then the iterated n-order is defined by σ M,n (f) = lim sup logn M(r,f) If f is analytic in D, it is well nown that σ M, (f) and σ (f) satisfy the inequalities σ (f) σ M, (f) σ (f) which are the best possible in the sense, see [2]. However, it follows by Proposition in [2] that σ M,n (f) = σ n (f) for n 2. For n N and a C { }, the iterated n-convergence exponent of the sequence of a-points in D of a meromorphic function f in D is defined by λ n (f a) = lim sup log n N(r, f a ) Similarly, λ n (f a), the iterated n-convergence exponent of the sequence of distinct a-points in D of a meromorphic function f in D is defined by λ n (f a) = lim sup logn N(r, f a ) For n N, the iterated n-convergence exponent of the sequence of fixed points in D of a meromorphic function f in D is defined by τ n (f) = lim sup logn N(r, f z ) Similarly, λ n (f z), the iterated n-convergence exponent of the sequence of distinct fixed points in D of a meromorphic function f in D is defined by τ n (f) = lim sup logn N(r, f z ) Finally, we give the definition about the degree of small growth order of functions in D as polynomials on the complex plane. Let f be a meromorphic function in D and D(f) = lim sup T(r,f) log( r) = b. If b <, we say that f is non-admissible; if b =, we say that f is admissible. Moreover, for F [0,), the upper and lower densities of F are defined by dens D F = lim sup m(f [0,r)) m([0,r)), dens D F = lim inf m(f [0,r)), m([0,r)) EJQTDE, 2009 No. 48, p. 2

3 respectively, where m(g) = G dt t for G [0,). In [], T.-B. Cao investigated the fast growth of the solutions of high order complex differential linear equation with analytic coefficients of n-iterated order in the unit disc. For using the results of T.-B. Cao conveniently, we write them in the following form. T.-B. Cao considered the equation f () A(z)f = 0 (.) where A(z) is analytic function in D, and proved the following theorems; Theorem A. [] Let H be a set of complex numbers satisfying dens D { z : z H D} > 0, and let A(z) be an analytic function in D such that σ M,n (A) = σ < and for constant α we have, for all ε > 0 sufficiently small, A(z) exp n {α( z )σ ε } as z for z H. Then every nontrivial solution f of (.) satisfies σ n (f) = and σ n (f) = σ. Theorem B. [] Let H be a set of complex numbers satisfying dens D { z : z H D} > 0, and let A(z) be an analytic function in D such that σ n (A) = σ < and for constant α we have, for all ε > 0 sufficiently small, T(r,A(z)) exp n {α( z )σ ε } as z for z H. Then every nontrivial solution f of (.) satisfies σ n (f) = and σ M,n (A) σ n (f) σ. In [], T.-B. Cao also investigated the fixed points of the solutions of high order complex differential linear equation with analytic coefficients in the unit disc and proposed that: How about the fixed points and iterated order of differential polynomials generated by solutions of linear differential equations in the unit disc. In the present paper we consider the derivatives of the solutions of the equations and get some theorems as following: Theorem.. Let the assumptions of Theorem A hold and assume also that f is a nontrivial solution of Equation (.). Then τ n (f (i) ) = τ n (f (i) ) = λ n (f (i) z) = λ n (f (i) z) = σ n (f) =, (.2) τ n (f (i) ) = τ n (f (i) ) = λ n (f (i) z) = λ n (f (i) z) = σ n (f) = σ. (.3) Theorem.2. Let the assumptions of Theorem B hold and assume also that f is a nontrivial solution of Equation (.). Then τ n (f (i) ) = τ n (f (i) ) = λ n (f (i) z) = λ n (f (i) z) = σ n (f) =, (.4) EJQTDE, 2009 No. 48, p. 3

4 σ M,n (A) τ n (f (i) ) = τ n (f (i) ) = λ n (f (i) z) = λ n (f (i) z) = σ n (f) σ. (.5) In addition, we study the fixed points of f, here f is a nontrivial solution of equation f A(z)f = 0, (.6) where A(z) is an analytic function in D. We get our theorems as following: Theorem.3. Let the assumptions of Theorem A hold and assume also that f is a nontrivial solution of Equation (.6). Then τ n (f ) = τ n (f ) = λ n (f z) = λ n (f z) = σ n (f) =, (.7) τ n (f ) = τ n (f ) = λ n (f z) = λ n (f z) = σ n (f) = σ. (.8) Theorem.4. Let the assumptions of Theorem B hold and assume also that f is a nontrivial solution of Equation (.6). Then τ n (f ) = τ n (f ) = λ n (f z) = λ n (f z) = σ n (f) =, (.9) σ M,n (A) τ n (f ) = τ n (f ) = λ n (f z) = λ n (f z) = σ n (f) σ. (.0) 2 Preliminary lemmas Lemma 2.. [6] Let f be a meromorphic function in the unit disc, and let N. Then m(r, f() f ) = S(r,f), where S(r,f) = O(log T(r,f))O(log( r )), possibly outside a set E [0,) with E. If f is of finite order (namely, finite iterated -order) of growth, then m(r, f() f ) = O(log( r )). dr r < Lemma 2.2. [0] Suppose that f(z) is a nonzero solution of equation (.) and 2. Let ω i = f (i) z, i = 0,,, 2. Then w i satisfy the following equations i H ij (A)ω ( j) i A i ω i = za i, (2.) EJQTDE, 2009 No. 48, p. 4

5 where H ij (A) (j = 0,,,i) are differential polynomials of A. Lemma 2.3. [0] Suppose that f(z) is a nonzero solution of equation (.) and 2. Let ω = f ( ) z. Then w satisfies the following equation H ( )j (A)ω ( j) A ω = za H ( )( ) (A), (2.2) where H ( )j (A) (j = 0,,, ) are differential polynomials of A. Lemma 2.4. [3] Suppose that f(z) is a nonzero solution of equation (.) and 2. Let ω = f () z. Then w satisfies the following equation H j (A)ω ( j) (H (A) A )ω = za H ( ) (A) zh (A), (2.3) where H j (A) (j = 0,,, ) and H (A) are differential polynomials of A. Lemma 2.5. [3] Suppose that f(z) is a nonzero solution of equation (.6) and 2. Let g = f z. Then g satisfies the following equation g g zg (g ) 2 2 (g ) Ag 2 2Azg = Az 2. 3 Proof of Theorems Proof of Theorem. and.2 Proof. We will prove Theorem. and.2 together. Suppose that f(z) is a nontrivial solution of equation (.). Set ω i = f (i) z, i = 0,,,, then for every i, a point z 0 is a fixed points of f (i) if and only if z 0 is a zero of ω i, and σ n (ω i ) = σ n (f (i) ) = σ n (f), τ n (f (i) ) = λ n (ω i ), σ n (ω i ) = σ n (f (i) ) = σ n (f), τ n (f (i) ) = λ n (ω i ), By Lemma 2.2, Lemma 2.3 and Lemma 2.4, we now that ω i, i = 0,,, satisfy the following equations : i H ij (A)ω ( j) i A i ω i = za i, (i = 0,,, 2). (3.) H ( )j (A)ω ( j) A ω = za H ( )( ) (A). (3.2) EJQTDE, 2009 No. 48, p. 5

6 H j (A)ω ( j) (H (A) A )ω = za H ( ) (A) zh (A). (3.3) where H ij (A) are differential polynomials of A. Since A is admissible analytic functions in D, we have za i 0, i = 0,,, 2. We claim that and za H ( )( ) (A) 0, za H ( ) (A) zh (A) 0. In fact, if za H ( )( ) (A) 0, rewrite it as A = H ( )( )(A) za. Since A is analytic in D, we have T(r, A) = m(r, A) = S(r, A) in D, which is impossible. If za H ( ) (A) zh (A) 0, rewrite it as A = H ( )(A) za H (A) A. Similarly, we obtain that T(r, A) = m(r, A) = S(r, A) in D. Hence our claim holds. Rewrite the above equations (3.)-(3.3) as ω i = ω = za i( i H ij (A) ω( j) A iω i ), (i = 0,,, 2). (3.4) i za H ( )( ) (A) ( H ( )j (A) ω( j) A ), (3.5) ω ω = za H ( ) (A) zh (A) ( H j (A) ω( j) H (A) A ω ). (3.6) By Lemma 2., there exists E [0,) with E m(r, ω i ) O(m(r, A )) C(log T(r,ω i ) log dr r <, such that for r E, we have, ), (3.7) r here i = 0,,,. By the assumption, we now that A and H ij (A) are analytic in D. Now we split three cases to discuss the zeros of ω i. Case : for i = 0,,, 2, if ω i has a zero at z 0 D of order m(> ), then from (3.4) we now z 0 is a zero of za i of order at least m. Hence we have N(r, ω i ) N(r, ω i ) N(r, zai). (3.8) EJQTDE, 2009 No. 48, p. 6

7 Case 2: for i =, if ω i has a zero at z 0 D of order m(> ), then from (3.5) we see z 0 is a zero of za H ( )( ) (A) of order at least m. Hence we have N(r, ω i ) N(r, ω i ) N(r, za ). (3.9) H ( )( ) (A) Case 3: for i =, if ω i has a zero at z 0 D of order m(> ), then we get from (3.6) that za H ( ) (A) zh (A) has zeros at z 0 of order at least m. Hence we have N(r, ω i ) N(r, ω i ) N(r, za ). (3.0) H ( ) (A) zh (A) Note that for r, C(log T(r,ω i ) log r ) 2 T(r,ω i), where C is a constant. Thus we have T(r,ω i ) 2N(r, ω i ) O(T(r,A)), (i = 0,,,.) (3.) Under the hypotheses of Theorem., we now that σ n (f) = by Theorem A. Then, for any given sufficiently large positive number N > σ, there exists {r n}(r n ) such that σ n (f) = lim sup r n Set dr E r = log δ <. Since r n ] \ E [0,), such that r n δ Hence, we have It yields r n δ log n T(r n,f) log( r n ) log n T(r n,f) log δ = r n lim inf r n log n T(r n,f) lim sup log( r n ) r n logn T(r n,f) log( r n) N. dr r = log(δ ), then there exists r n [r n, log n T(r n,f) log r log(δ ). n log log r n log n lim T(r n,f) r n log( r n ) N. n T(r n,f) N. log(δ ) Since σ n (f) σ M,n (f) = σ, for any given ε, we have T(r n,a) exp n ( r n ) σε. So, for r n [r n, r n δ ] \ E [0,), we get T(r n,a) T(r n,f) exp n( r n ) σε exp n ( r n ) 0. N So we have T(r,A) = o(t(r,f)). Since σ n (ω i ) = σ n (f), we have T(r,A) = o(t(r,ω i )). From (3.), we have λ n (ω i ) = λ n (ω i ) = σ n (ω i ), (3.2) EJQTDE, 2009 No. 48, p. 7

8 and λ n (ω i ) = λ n ω i ) = σ n (ω i ). (3.3) Combining (3.2), (3.3) with Theorem A, we can get (.2) and (.3). Thus we complete the proof of Theorem.. Combining (3.2), (3.3) with Theorem B, we can get (.4) and (.5), which are the results of Theorem.2. Proof of Theorem.3 and.4 Proof. Suppose that f(z) is a nonzero solution of equation (.6) and 2. Let g = f z. By Lemma 2.5, g satisfies the following equation g g zg (g ) 2 2 (g ) Ag 2 2Azg = Az 2. (3.4) Obviously, we now that Az 2 0. For z satisfying g(z) <, we have g(z) >. From (3.4), we have From (3.5) we have Az 2 g ( z ) g g m(r, g ) O( 2 i= ( g g )2 2 g g A 2 Az. (3.5) m(r, g(i) )) O(m(r,A)). (3.6) g From (3.4), we now that if z 0 is a zero of g with multiplicity m, then z 0 is a zero of Az 2 with multiplicity at least m 2. So we have From (3.6) and (3.7), we have From (3.8), we have and Since N(r, g ) 2N(r, g ) N(r, Az2). (3.7) T(r,g) 2N(r, ) O(T(r,A)) S(r,g). (3.8) g σ n (g) = λ n (g) = λ n (g) = τ n (f ) = τ n (f ). (3.9) σ n (g) = λ n (g) = λ n (g) = τ n (f ) = τ n (f ). (3.20) σ n (g) = σ n (f ) = σ n (f) (3.2) From (3.9)- (3.2), and Theorem A (Theorem B), we can get Theorem.3 (Theorem.4). Acnowledgement The authors would lie to express their sincere thans to the referee for helpful comments and suggestion. EJQTDE, 2009 No. 48, p. 8

9 References [] T.-B. Cao, The growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc, J. Math. Anal. Appl 2009, doi:0.06/j.jmaa [2] T.-B. Cao, H.-X. Yi, The growth of solutions of linear differential equations with coefficients of iterated order in the unit disc, J. Math. Anal. Appl 39 (2006), [3] Z.-X. Chen, The fixed points and hyper-order of solutions of sencond order complex differential equations, Acta Math. Sci. Ser. A. Chin. Ed. 20(3) (2000), (in Chinese). [4] C.-T. Chuang, C.-C. Yang, The fixed points and factorization theory of meromorphic functions, Beijing University Press, Beijing, 988 (in Chinese). [5] W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 964. [6] J. Heittoangas, On complex linear differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss. 22 (2000), -54. [7] J. Heittoangas, R. Korhonen, J. Rättyä, Fast growing solutions of linear differential equations in the unit disc, Results Math. 49 (2006), [8] K.-H. Kwon, On the growth of entire functions satisfying second order linear differential equations, Bull. Korean Math. Soc. 33(3) (996), [9] I. Laine, J. Rieppo, Differential polynomials generated by linear differential equations, Complex Var. Elliptic Equ. 49() (2004), [0] M.-S. Liu, X.-M. Zhang, Fixed points of meromorphic solutions of higher order linear differential equations, Ann. Acad. Sci. Fenn. Math. 3 (2006), 9-2. [] D. Shea, L. Sons, Value distribution theory for meromorphic functions of slow growth in the disc, Houston J. Math. 2(2) (986), [2] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, New Yor, (975), reprint of the 959 edition. [3] J. Wang, H.-X. Yi, The fixed points of -th power and differential polynomials generated by solutions of sencond order differential equations, J. Sys. Sci. and Math. Scis. 24(2) (2004), [4] J. Wang, H.-X. Yi, Fixed points and hyper-order of differential polynomials generated by solutions of differential equations, Complex Var. Elliptic Equ. 48() (2003), [5] L. Yang, Value Distribution Theory, Springer-Verlag, Berlin, 993. (Received February 26, 2009) EJQTDE, 2009 No. 48, p. 9