International Journal of Analysis and Applications ISSN 2291-8639 Volume 2, Number 2 (2013), 111-123 http://www.etamaths.com COMPLEX OSCILLATION OF SOLUTIONS AND THEIR DERIVATIVES OF NON-HOMOGENOUS LINEAR DIFFERENTIAL EQUATIONS IN THE UNIT DISC ZINELAÂBIDINE LATREUCH AND BENHARRAT BELAÏDI∗ Abstract. In this paper, we study the complex oscillation of solutions and their derivatives of the differential equation f′′ + A (z) f′ + B (z) f = F (z) , where A (z) , B (z) ( 6≡ 0) and F (z) (6≡ 0) are meromorphic functions of finite iterated p-order in the unit disc ∆ = {z : |z| < 1}. 1. Introduction and main results Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna’s value distribution theory on the complex plane and in the unit disc ∆ = {z : |z| < 1} (see [11] , [12] , [15] , [16] , [18]). We need to give some definitions and discussions. Firstly, let us give two defini- tions about the degree of small growth order of functions in ∆ as polynomials on the complex plane C. There are many types of definitions of small growth order of functions in ∆ (see [9, 10]) . Definition 1.1 [9, 10] Let f be a meromorphic function in ∆, and D (f) = lim sup r→1− T (r,f) log 1 1−r = b. 2010 Mathematics Subject Classification. 34M10, 30D35. Key words and phrases. Linear differential equations, Meromorphic functions, Iterated p−exponent of convergence of the sequence of zeros, Unit disc. c©2013 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 111 112 LATREUCH AND BELAÏDI If b < ∞, then we say that f is of finite b degree (or is non-admissible). If b = ∞, then we say that f is of infinite degree (or is admissible), both defined by characteristic function T(r,f). Definition 1.2 [9, 10] Let f be an analytic function in ∆, and DM (f) = lim sup r→1− log+ M (r,f) log 1 1−r = a < ∞ (or a = ∞) . Then we say that f is a function of finite a degree (or of infinite degree) defined by maximum modulus function M(r,f) = max |z|=r |f (z)| . Moreover, for F ⊂ [0, 1) , the upper and lower densities of F are defined by dens∆F = lim sup r→1− m (F ∩ [0,r)) m ([0,r)) , dens∆F = lim inf r→1− m (F ∩ [0,r)) m ([0,r)) respectively, where m (G) = ∫ G dt 1−t for G ⊂ [0, 1) . Now we give the definitions of iterated order and growth index to classify gen- erally the functions of fast growth in ∆ as those in C, see [4] , [14] , [15] . Let us define inductively, for r ∈ [0, 1) , exp1 r = er and expp+1 r = exp(expp r), p ∈ N. We also define for all r sufficiently large in (0, 1) , log1 r = log r and logp+1 r = log(logp r),p ∈ N. Moreover, we denote by exp0 r = r, log0 r = r, exp−1 r = log1 r, log−1 r = exp1 r. Definition 1.3 [5] The iterated p-order of a meromorphic function f in ∆ is defined by ρp (f) = lim sup r→1− log+p T (r,f) log 1 1−r (p ≥ 1) . For an analytic function f in ∆, we also define ρM,p (f) = lim sup r→1− log+p+1 M (r,f) log 1 1−r (p ≥ 1) . COMPLEX OSCILLATION 113 Remark 1.1 It follows by M. Tsuji in [18] that if f is an analytic function in ∆, then ρ1 (f) ≤ ρM,1 (f) ≤ ρ1 (f) + 1. However, it follows by Proposition 2.2.2 in [15] that ρM,p (f) = ρp (f) (p ≥ 2) . Definition 1.4 [5] The growth index of the iterated order of a meromorphic func- tion f(z) in ∆ is defined by i (f) =   0, if f is non-admissible, min{p ∈ N,ρp (f) < ∞} , if f is admissible, ∞, if ρp (f) = ∞ for all p ∈ N. For an analytic functionf in ∆, we also define iM (f) =   0, if f is non-admissible, min{p ∈ N,ρM,p (f) < ∞} , if f is admissible, ∞, if ρM,p (f) = ∞ for all p ∈ N. Definition 1.5 [6, 7] Let f be a meromorphic function in ∆. Then the iterated p−exponent of convergence of the sequence of zeros of f (z) is defined by λp (f) = lim sup r→1− log+p N ( r, 1 f ) log 1 1−r , where N ( r, 1 f ) is the counting function of zeros of f (z) in {z ∈ C : |z| < r}. Similarly, the iterated p-exponent of convergence of the sequence of distinct zeros of f (z) is defined by λp (f) = lim sup r→1− log+p N ( r, 1 f ) log 1 1−r , where N ( r, 1 f ) is the counting function of distinct zeros of f (z) in {z ∈ C : |z| < r}. 114 LATREUCH AND BELAÏDI Definition 1.6 [8] The growth index of the iterated convergence exponent of the sequence of zeros of f(z) in ∆ is defined by iλ (f) =   0, if N ( r, 1 f ) = O ( log 1 1−r ) , min{p ∈ N,λp (f) < ∞} , if some p ∈ N with λp (f) < ∞, ∞, if λp (f) = ∞ for all p ∈ N. The complex oscillation theory of solutions of linear differential equations in the complex plane C was started by Bank and Laine in 1982 ([1]). After their well- known work, many important results have been obtained on the growth and the complex oscillation theory of solutions of linear differential equations in the unit disc ∆ = {z : |z| < 1} , (see [2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 16, 20]) . Recently, the sec- ond author (see, [2]) extended some results of [6, 20] to the case of higher order linear differential equations with analytic coefficients. He investigated the relation between solutions and their derivatives of the differential equation (1.1) f(k) + A (z) f = 0 and analytic functions of finite iterated p-order, and obtained the following results: Theorem A [2] Let H be a set of complex numbers satisfying dens∆ {|z| : z ∈ H ⊂ ∆} > 0, and let A (z) 6≡ 0 be an analytic function in ∆ such that ρM,p (A) = σ < ∞ and for real number α > 0, we have for all ε > 0 sufficiently small, |A (z)| ≥ expp { α ( 1 1 −|z| )σ−ε} as |z|→ 1− for z ∈ H. If ϕ (z) is an analytic function in ∆ such that ϕ(k−j) (z) 6≡ 0 (j = 0, · · · ,k) with finite iterated p−order ρp (ϕ) < ∞, then every solution f 6≡ 0 of (1.1) , satisfies λp ( f(j) −ϕ ) = λp ( f(j) −ϕ ) = ρp (f) = ∞ (j = 0, · · · ,k) , λp+1 ( f(j) −ϕ ) = λp+1 ( f(j) −ϕ ) = ρp+1 (f) = ρM,p (A) (j = 0, · · · ,k) . COMPLEX OSCILLATION 115 Theorem B [2] Let H be a set of complex numbers satisfying dens∆ {|z| : z ∈ H ⊂ ∆} > 0, and let A (z) 6≡ 0 be an analytic function in ∆ such that ρp (A) = σ < ∞ and for real number α > 0, we have for all ε > 0 sufficiently small, |A (z)| ≥ expp−1 { α ( 1 1 −|z| )σ−ε} as |z|→ 1− for z ∈ H. If ϕ (z) is an analytic function in ∆ such that ϕ(k−j) (z) 6≡ 0 (j = 0, · · · ,k) with finite iterated p−order ρp (ϕ) < ∞, then every solution f 6≡ 0 of (1.1) , satisfies λp ( f(j) −ϕ ) = λp ( f(j) −ϕ ) = ρp (f) = ∞ (j = 0, · · · ,k) , σ ≤ λp+1 ( f(j) −ϕ ) = λp+1 ( f(j) −ϕ ) = ρp+1 (f) ≤ ρM,p (A) (j = 0, · · · ,k) . In this paper we consider the oscillation problem of solutions and their deriva- tives of second order non-homogenous linear differential equation (1.2) f′′ + A (z) f′ + B (z) f = F (z) , where A (z) , B (z) 6≡ 0 and F (z) 6≡ 0 are meromorphic functions of finite iterated p-order in ∆. It is a natural to ask what about the exponent of convergence of zeros of f(j) (z) (j = 0, 1, 2, · · ·) , where f is a solution of (1.2) . For some related papers in the complex plane on the usual order see, [17, 19] . The main purpose of this paper is give an answer to this question. Before we state our results we need to define the following notations (1.3) Aj (z) = Aj−1 (z) − B′j−1 (z) Bj−1 (z) for j = 1, 2, 3, · · · , (1.4) Bj (z) = A ′ j−1 (z) −Aj−1 (z) B′j−1 (z) Bj−1 (z) + Bj−1 (z) for j = 1, 2, 3, · · · and (1.5) Fj (z) = F ′ j−1 (z) −Fj−1 (z) B′j−1 (z) Bj−1 (z) for j = 1, 2, 3, · · · , where A0 (z) = A (z) , B0 (z) = B (z) and F0 (z) = F (z) . We obtain the following results. 116 LATREUCH AND BELAÏDI Theorem 1.1 Let A (z) , B (z) 6≡ 0 and F (z) 6≡ 0 be meromorphic functions of finite iterated p−order in ∆ such that Bj (z) 6≡ 0 and Fj (z) 6≡ 0 (j = 1, 2, 3, · · ·) . If f is a meromorphic solution in ∆ of (1.2) with ρp (f) = ∞ and ρp+1 (f) = ρ, then f satisfies λp ( f(j) ) = λp ( f(j) ) = ρp (f) = ∞ (j = 0, 1, 2, · · ·) and λp+1 ( f(j) ) = λp+1 ( f(j) ) = ρp+1 (f) = ρ (j = 0, 1, 2, · · ·) . Theorem 1.2 Let A (z) , B (z) 6≡ 0 and F (z) 6≡ 0 be meromorphic functions in ∆ with finite iterated p−order such that Bj (z) 6≡ 0 and Fj (z) 6≡ 0 (j = 1, 2, 3, · · ·) . If f is a meromorphic solution in ∆ of (1.2) with ρp (f) > max{ρp (A) ,ρp (B) ,ρp (F)} , then λp ( f(j) ) = λp ( f(j) ) = ρp (f) (j = 0, 1, 2, · · ·) . Remark 1.2 In Theorems 1.1, 1.2, the conditions Bj (z) 6≡ 0 and Fj (z) 6≡ 0 (j = 1, 2, 3, · · ·) are necessary. For example f (z) = exp ( 1 1−z )2 − 1 satisfies (1.2) where A (z) = −3 1−z , B (z) = − 4 (1−z)6 , F (z) = 4 (1−z)6 and ρ1 (f) = 1 > max{ρ1 (A) ,ρ1 (B) ,ρ1 (F)} = 0. On the other hand, we have A1 = A− B′ B = − 9 1 −z , B1 = A ′ −A B′ B + B = 15 (1 −z)2 − 4 (1 −z)6 , F1 = F ′ −F B′ B ≡ 0, and λ1 (f) = 1 > λ1 (f ′) = 0. Here, we give some sufficient conditions on the coefficients which guarantee Bj (z) 6≡ 0 and Fj (z) 6≡ 0 (j = 1, 2, 3, · · ·) , and we obtain: Theorem 1.3 Let A (z) , B (z) 6≡ 0 and F (z) 6≡ 0 be analytic functions in ∆ with finite iterated p−order such that β = ρp (B) > max{ρp (A) ,ρp (F)} . Then all nontrivial solutions of (1.2) satisfy ρp (B) ≤ λp+1 ( f(j) ) = λp+1 ( f(j) ) = ρp+1 (f) ≤ ρM,p (B) (j = 0, 1, 2, · · ·) COMPLEX OSCILLATION 117 with at most one possible exceptional solution f0 such that ρp+1 (f0) < ρp (B) . In the next, we set σp (f) = lim sup r→1− logp m (r,f) log 1 1−r . Theorem 1.4 Let A (z) , B (z) 6≡ 0 and F (z) 6≡ 0 be meromorphic functions in ∆ with finite iterated p-order such that σp (B) > max{σp (A) ,σp (F)} . If f is a meromorphic solution in ∆ of (1.2) with ρp (f) = ∞ and ρp+1 (f) = ρ, then f satisfies λp ( f(j) ) = λp ( f(j) ) = ρp (f) = ∞ (j = 0, 1, 2, · · ·) and λp+1 ( f(j) ) = λp+1 ( f(j) ) = ρp+1 (f) = ρ (j = 0, 1, 2, · · ·) . 2. Some lemmas Lemma 2.1 [2] Let f be a meromorphic function in the unit disc for which i (f) = p ≥ 1 and ρp (f) = β < ∞, and let k ∈ N. Then for any ε > 0, m ( r, f(k) f ) = O ( expp−2 ( 1 1 −r )β+ε) for all r outside a set E1 ⊂ [0, 1) with ∫ E1 dr 1−r < ∞. Lemma 2.2 [7] Let A0,A1, · · · ,Ak−1,F 6≡ 0 be meromorphic functions in ∆, and let f be a meromorphic solution of the differential equation (2.1) f(k) + Ak−1 (z) f (k−1) + · · · + A0 (z) f = F (z) such that i (f) = p (0 < p < ∞) . If either max{i (Aj) (j = 0, 1, · · · ,k − 1) , i (F)} < p or max{ρp (Aj) (j = 0, 1, · · · ,k − 1) ,ρp (F)} < ρp (f) , 118 LATREUCH AND BELAÏDI then iλ (f) = iλ (f) = i (f) = p and λp (f) = λp (f) = ρp (f) . Using the same arguments as in the proof of Lemma 2.2 (see, the proof of Lemma 2.5 in [7]), we easily obtain the following lemma. Lemma 2.3 Let A0, A1, · · · , Ak−1, F 6≡ 0 be finite iterated p−order meromorphic functions in the unit disc ∆. If f is a meromorphic solution with ρp (f) = ∞ and ρp+1 (f) = ρ < ∞ of equation (2.1) , then λp (f) = λp (f) = ρp (f) = ∞ and λp+1 (f) = λp+1 (f) = ρp+1 (f) = ρ. Lemma 2.4 [7] Let p ∈ N, and assume that the coefficients A0, · · · ,Ak−1 and F 6≡ 0 are analytic in ∆ and ρp (Aj) < ρp (A0) for all j = 1, · · · ,k − 1. Let αM := max{ρM,p (Aj) : j = 0, · · · ,k − 1} . (i) If ρM,p+1 (F) > αM, then all solutions f of (2.1) satisfy ρM,p+1 (f) = ρM,p+1 (F) . (ii) If ρM,p+1 (F) < αM, then all solutions f of (2.1) satisfy ρp (A0) ≤ ρM,p+1 (f) ≤ αM, with at most one exeption f0 satisfying ρM,p+1 (f0) < ρp (A0) . (iii) If ρM,p+1 (F) < ρp (A0) , then all solutions f of (2.1) satisfy ρp (A0) ≤ λp+1 (f) = λp+1 (f) = ρM,p+1 (f) ≤ αM, with at most one exception f0 satisfy- ing ρM,p+1 (f0) < ρp (A0) . 3. Proof of Theorems Proof of Theorem 1.1. For the proof, we use the principle of mathematical induction. Since B (z) 6≡ 0 and F (z) 6≡ 0, then by using Lemma 2.3 we have λp (f) = λp (f) = ρp (f) = ∞ and λp+1 (f) = λp+1 (f) = ρp+1(f) = ρ. COMPLEX OSCILLATION 119 Dividing both sides of (1.2) by B, we obtain (3.1) 1 B f′′ + A B f′ + f = F B . Differentiating both sides of equation (3.1) , we have (3.2) 1 B f(3) + (( 1 B )′ + A B ) f′′ + (( A B )′ + 1 ) f′ = ( F B )′ . Multiplying now (3.2) by B, we get (3.3) f(3) + A1f ′′ + B1f ′ = F1, where A1 = A− B′ B , B1 = A ′ −A B′ B + B and F1 = F ′ −F B′ B . Since B1 6≡ 0 and F1 6≡ 0 are meromorphic functions with finite iterated p-order, then by using Lemma 2.3 we obtain λp (f ′) = λp (f ′) = ρp (f) = ∞ and λp+1 (f ′) = λp+1 (f ′) = ρp+1(f) = ρ. Dividing now both sides of (3.3) by B1, we obtain (3.4) 1 B1 f(3) + A1 B1 f′′ + f′ = F1 B1 . Differentiating both sides of equation (3.4) and multplying by B1, we get (3.5) f(4) + A2f (3) + B2f ′′ = F2, where A2,B2 6≡ 0 and F2 6≡ 0 are meromorphic functions defined in (1.3) − (1.5) . By using Lemma 2.3 we obtain λp (f ′′) = λp (f ′′) = ρp (f) = ∞ and λp+1 (f ′′) = λp+1 (f ′′) = ρp+1 (f) = ρ. 120 LATREUCH AND BELAÏDI Suppose now that (3.6) λp ( f(k) ) = λp ( f(k) ) = ρp (f) = ∞, λp+1 ( f(k) ) = λp+1 ( f(k) ) = ρp+1 (f) = ρ for all k = 0, 1, 2, · · · ,j − 1, and we prove that (3.6) is true for k = j. By the same procedure as before, we can obtain f(j+2) + Ajf (j+1) + Bjf (j) = Fj, where Aj,Bj 6≡ 0 and Fj 6≡ 0 are meromorphic functions defined in (1.3) − (1.5) . By using Lemma 2.3 we obtain λp ( f(j) ) = λp ( f(j) ) = ρp (f) = ∞ and λp+1 ( f(j) ) = λp+1 ( f(j) ) = ρp+1 (f) = ρ. The proof of Theorem 1.1 is complete. Proof of Theorem 1.2. By a similar reasoning as Theorem 1.1 and by using Lemma 2.2, we obtain λp ( f(j) ) = λp ( f(j) ) = ρp (f) (j = 0, 1, 2, · · ·) . Proof of Theorem 1.3. By Lemma 2.4 (iii), all nontrivial solutions of (1.2) satisfy ρp (B) ≤ λp+1 (f) = λp+1 (f) = ρp+1 (f) ≤ ρM,p (B) with at most one exceptional solution f0 such that ρp (B) > ρp+1 (f0). By using (1.3) and Lemma 2.1 we have m (r,Aj) ≤ m (r,Aj−1) + O ( expp−2 ( 1 1 −r )β+ε) (β = ρp (Bj−1)) outside a set E1 ⊂ [0, 1) with ∫ E1 dr 1−r < ∞, for all j = 1, 2, 3, · · · , which we can write as (3.7) m (r,Aj) ≤ m (r,A) + O ( expp−2 ( 1 1 −r )β+ε) (j = 1, 2, 3, · · ·) . On the other hand, we have from (1.4) COMPLEX OSCILLATION 121 Bj = Aj−1 ( A′j−1 Aj−1 − B′j−1 Bj−1 ) + Bj−1 = Aj−1 ( A′j−1 Aj−1 − B′j−1 Bj−1 ) + Aj−2 ( A′j−2 Aj−2 − B′j−2 Bj−2 ) + Bj−2 (3.8) = j−1∑ k=0 Ak ( A′k Ak − B′k Bk ) + B. Now we prove that Bj 6≡ 0 for all j = 1, 2, 3, · · · . For that we suppose there exists j ∈ N such that Bj = 0. By (3.7) and (3.8) we have T (r,B) = m (r,B) ≤ j−1∑ k=0 m (r,Ak) + O ( expp−2 ( 1 1 −r )β+ε) ≤ jm (r,A) + O ( expp−2 ( 1 1 −r )β+ε) (3.9) = jT (r,A) + O ( expp−2 ( 1 1 −r )β+ε) which implies the contradiction ρp (B) ≤ ρp (A) . Hence Bj 6≡ 0 for all j = 1, 2, 3, · · · . Suppose now there exists j ∈ N such that Fj = 0. Then, from (1.5) F ′j−1 (z) −Fj−1 (z) B′j−1 (z) Bj−1 (z) = 0 which implies (3.10) Fj−1 (z) = cBj−1 (z) , where c ∈ C∗. By (3.8) and (3.10) we have (3.11) 1 c Fj−1 = j−2∑ k=0 Ak ( A′k Ak − B′k Bk ) + B. On the other hand, we have from (1.5) (3.12) m (r,Fj) ≤ m (r,F) + O ( expp−2 ( 1 1 −r )β+ε) (j = 1, 2, 3, · · ·) . 122 LATREUCH AND BELAÏDI By (3.11) , (3.12) and Lemma 2.1, we have T (r,B) = m (r,B) ≤ j−2∑ k=0 m (r,Ak) + m (r,Fj−1) + O ( expp−2 ( 1 1 −r )β+ε) ≤ (j − 1) m (r,A) + m (r,F) + O ( expp−2 ( 1 1 −r )β+ε) = (j − 1) T (r,A) + T (r,F) + O ( expp−2 ( 1 1 −r )β+ε) which implies the contradiction ρp (B) ≤ max{ρp (A) ,ρp (F)} . Since Bj 6≡ 0, Fj 6≡ 0 (j = 1, 2, 3, · · ·) , then by applying Theorem 1.1 and Lemma 2.4 (iii) we have ρp (B) ≤ λp+1 ( f(j) ) = λp+1 ( f(j) ) = ρp+1 (f) ≤ ρM,p (B) (j = 0, 1, 2, · · ·) with at most one exceptional solution f0 such that ρp (B) > ρp+1 (f0) . 4. Proof of Theorem 1.4 Using the same reasoning as Theorem 1.1, we obtain Theorem 1.4. References [1] S. Bank and I. Laine, On the oscillation theory of f′′ + A (z) f = 0. where A is entire, Trans. Amer. Math. Soc. 273 (1982), no. 1, 351–363. [2] B. 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