RELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS GENERATED BY MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 147 161. RELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS GENERATED BY MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS BENHARRAT BELAÏDI1 AND ZINELÂABIDINE LATREUCH1 Abstract. This paper is devoted to studying the growth oscillation of higher order differential polynomial with meromorphic coefficients generated by meromorphic solutions of the linear differential equation f (k) + A (z) f = 0 (k 2), where A is a meromorphic function in the complex plane. 1. Introduction main results In this paper, we shall assume that the reader is familiar with the fundamental results the stard notations of the Nevanlinna value distribution theory of meromorphic functions see [10, 17]. For the definition of the iterated order of a meromorphic function, we use the same definition as in [11], ([2], p. 317), ([12], p. 129). For all r R, we define exp 1 r := e r exp p+1 r := exp ( exp p r ), p N. We also define for all r sufficiently large log 1 r := log r log p+1 r := log ( log p r ), p N. Moreover, we denote by exp 0 r := r, log 0 r := r, log 1 r := exp 1 r exp 1 r := log 1 r. Definition 1.1. [11, 12] Let f be a meromorphic function. Then the iterated p order ρ p (f) of f is defined as log p T (r, f) ρ p (f) = lim sup r + log r (p 1 is an integer), Key words phrases. Linear differential equations, Differential polynomials, Meromorphic solutions, Iterated order, Iterated exponent of convergence of the sequence of distinct zeros. 2010 Mathematics Subject Classification. Primary: 34M10. Secondary: 30D35. Received: March 21, 2013 Revised: May 6, 2014. 147

148 B. BELAÏDI AND Z. LATREUCH where T (r, f) is the Nevanlinna characteristic function of f. For p = 1, this notation is called order for p = 2 hyper-order, see [10, 17]. Definition 1.2. [11] The finiteness degree of the order of a meromorphic function f is defined as 0, for f rational, min {j N : ρ i (f) = j (f) < + }, for f transcendental for which some j N with ρ j (f) < + exists, +, for f with ρ j (f) = + for all j N. Definition 1.3. [4, 6] The type of a meromorphic function f of iterated p order ρ (0 < ρ < ) is defined as log p 1 T (r, f) τ p (f) = lim sup (p 1 is an integer). r + r ρ Definition 1.4. [11] Let f be a meromorphic function. Then the iterated exponent of convergence of the sequence of zeros of f (z) is defined as ( ) log p N r, 1 f λ p (f) = lim sup (p 1 is an integer), r + log r where N ( ) r, 1 is the counting function of zeros of f (z) in {z : z r}. For p = 1, f this notation is called exponent of convergence of the sequence of zeros for p = 2 hyper-exponent of convergence of the sequence of zeros, see [9]. Similarly, the iterated exponent of convergence of the sequence of distinct zeros of f (z) is defined as where N log p N λ p (f) = lim sup r + ( r, 1 f ) (p 1 is an integer), log r ( ) r, 1 is the counting function of distinct zeros of f (z) in {z : z r}. For f p = 1, this notation is called exponent of convergence of the sequence of distinct zeros for p = 2 hyper-exponent of convergence of the sequence of distinct zeros, see [9]. Consider for k 2 the complex linear differential equation (1.1) f (k) + A (z) f = 0 the differential polynomial (1.2) g f = d k f (k) + d k 1 f (k 1) + + d 1 f + d 0 f, where A d j (j = 0, 1,..., k) are meromorphic functions in the complex plane. In [9], Chen studied the fixed points hyper-order of solutions of second order linear differential equations with entire coefficients obtained the following results.

RELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS 149 Theorem 1.1. [9] For all non-trivial solutions f of (1.3) f + A (z) f = 0, the following hold: (i) If A is a polynomial with deg A = n 1, then we have λ (f z) = ρ (f) = n + 2. 2 (ii) If A is transcendental ρ (A) <, then we have λ (f z) = ρ (f) = λ 2 (f z) = ρ 2 (f) = ρ (A). After him, in [16] Wang, Yi Cai generalized the precedent theorem for the differential polynomial g f with constant coefficients as follows. Theorem 1.2. [16] For all non-trivial solutions f of (1.3) the following hold: (i) If A is a polynomial with deg A = n 1, then we have λ (g f z) = ρ (f) = n + 2. 2 (ii) If A is transcendental ρ (A) <, then we have λ (g f z) = ρ (f) = λ 2 (g f z) = ρ 2 (f) = ρ (A). Theorem 1.1 has been generalized from entire to meromorphic solutions for higher order differential equations by the author as follows, see [3]. Theorem 1.3. [3] Let k 2 A (z) be a transcendental meromorphic function m(r,a) of finite iterated order ρ p (A) = ρ > 0 such that δ (, A) = lim inf = δ > 0. r + T (r,a) Suppose, moreover, that either: (i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (, f) > 0. If ϕ (z) 0 is a meromorphic function with finite iterated p order ρ p (ϕ) < +, then every meromorphic solution f (z) 0 of (1.1) satisfies λ p (f ϕ) = λ p (f ϕ) = = λ p ( f (k) ϕ ) = ρ p (f) = +, λ p+1 (f ϕ) = λ p+1 (f ϕ) = = λ p+1 ( f (k) ϕ ) = ρ p+1 (f) = ρ.

150 B. BELAÏDI AND Z. LATREUCH Let L (G) denote a differential subfield of the field M (G) of meromorphic functions in a domain G C. If G = C, we simply denote L instead of L (C). Special case of such differential subfield L p+1,ρ = {g meromorphic: ρ p+1 (g) < ρ}, where ρ is a positive constant. In [13], Laine Rieppo have investigated the fixed points iterated order of the second order differential equation (1.3) have obtained the following result. Theorem 1.4. [13] Let A (z) be a transcendental meromorphic function of finite iterated order ρ p (A) = ρ > 0 such that δ (, A) = δ > 0, let f be a transcendental meromorphic solution of equation (1.3). Suppose, moreover, that either: (i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (, f) > 0. Then ρ p+1 (f) = ρ p (A) = ρ. Moreover, let (1.4) P [f] = P ( f, f,..., f (m)) = m p j f (j) j=0 be a linear differential polynomial with coefficients p j L p+1,ρ, assuming that at least one of the coefficients p j does vanish identically. Then for the fixed points of P [f], we have λ p+1 (P [f] z) = ρ, provided that neither P [f] nor P [f] z vanishes identically. Remark 1.1. ([13], p. 904) In Theorem 1.4, in order to study P [f], the authors consider m 1. Indeed, if m 2, we obtain, by repeated differentiation of (1.3), that f (k) = q k,0 f + q k,1 f, q k,0, q k,1 L p+1,ρ for k = 2,..., m. Substitution into (1.4) yields the required reduction. The present article may be understood as an extension improvement of the recent article of the authors [14] from usual order to iterated order. The main purpose of this paper is to study the growth oscillation of the differential polynomial (1.2) generated by meromorphic solutions of equation (1.1). The method used in the proofs of our theorems is simple quite different from the method used in the paper of Laine Rieppo [13]. For some related papers in the unit disc see [7, 8, 15]. Before we state our results, we define the sequence of functions α i,j (j = 0,..., k 1) by { α α i,j = i,j 1 + α i 1,j 1, for all i = 1,..., k 1, α 0,j 1 Aα k 1,j 1, for i = 0 α i,0 = { di, for all i = 1,..., k 1, d 0 d k A, for i = 0.

RELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS 151 We define also α 0,0 α 1,0.. α k 1,0 α 0,1 α 1,1.. α k 1,1 h =.......... α 0,k 1 α 1,k 1.. α k 1,k 1 ψ (z) = C 0 ϕ + C 1 ϕ + + C k 1 ϕ (k 1), where C j (j = 0,..., k 1) are finite iterated p order meromorphic functions depending on α i,j ϕ 0 is a meromorphic function with ρ p (ϕ) <. Theorem 1.5. Let A (z) be a meromorphic function of finite iterated p order. Let d j (z) (j = 0, 1,..., k) be finite iterated p order meromorphic functions that are not all vanishing identically such that h 0. If f (z) is an infinite iterated p order meromorphic solution of (1.1) with ρ p+1 (f) = ρ, then the differential polynomial (1.2) satisfies ρ p (g f ) = ρ p (f) = ρ p+1 (g f ) = ρ p+1 (f) = ρ. Furthermore, if f is a finite iterated p order meromorphic solution of (1.1) such that (1.5) ρ p (f) > max {ρ p (A), ρ p (d j ) (j = 0, 1,..., k)}, then ρ p (g f ) = ρ p (f). Remark 1.2. In Theorem 1.5, if we do not have the condition h 0, then the conclusions of Theorem 1.5 cannot hold. For example, if we take d k = 1, d 0 = A d j 0 (j = 1,..., k 1), then h 0. It follows that g f 0 ρ p (g f ) = 0. So, if f (z) is an infinite iterated p order meromorphic solution of (1.1), then ρ p (g f ) = 0 < ρ p (f) =, if f is a finite iterated p order meromorphic solution of (1.1) such that (1.5) holds, then ρ p (g f ) = 0 < ρ p (f). Corollary 1.1. Let A (z) be a transcendental entire function of finite iterated order ρ p (A) = ρ > 0, let d j (z) (j = 0, 1,..., k) be finite iterated p order entire functions that are not all vanishing identically such that h 0. If f 0 is a solution of (1.1), then the differential polynomial (1.2) satisfies ρ p (g f ) = ρ p (f) = ρ p+1 (g f ) = ρ p+1 (f) = ρ p (A) = ρ. Corollary 1.2. Let k 2 A (z) be a transcendental meromorphic function of finite iterated order ρ p (A) = ρ > 0 such that δ (, A) = δ > 0, let f 0 be a meromorphic solution of equation (1.1). Suppose, moreover, that either:

152 B. BELAÏDI AND Z. LATREUCH (i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (, f) > 0. Let d j (z) (j = 0, 1,..., k) be finite iterated p order meromorphic functions that are not all vanishing identically such that h 0. Then the differential polynomial (1.2) satisfies ρ p (g f ) = ρ p (f) = ρ p+1 (g f ) = ρ p+1 (f) = ρ p (A). Theorem 1.6. Under the hypotheses of Theorem 1.5, let ϕ (z) 0 be a meromorphic function with finite iterated p order such that ψ (z) is not a solution of (1.1). If f (z) is an infinite iterated p order meromorphic solution of (1.1) with ρ p+1 (f) = ρ, then the differential polynomial (1.2) satisfies λ p (g f ϕ) = λ p (g f ϕ) = ρ p (f) = λ p+1 (g f ϕ) = λ p+1 (g f ϕ) = ρ p+1 (f) = ρ. Furthermore, if f is a finite iterated p order meromorphic solution of (1.1) such that (1.6) ρ p (f) > max {ρ p (A), ρ p (ϕ), ρ p (d j ) (j = 0, 1, lcdots, k)}, then λ p (g f ϕ) = λ p (g f ϕ) = ρ p (f). Corollary 1.3. Under the hypotheses of Corollary 1.1, let ϕ (z) 0 be an entire function with finite iterated p order such that ψ (z) 0. Then the differential polynomial (1.2) satisfies λ p (g f ϕ) = λ p (g f ϕ) = ρ p (f) = λ p+1 (g f ϕ) = λ p+1 (g f ϕ) = ρ p+1 (f) = ρ p (A). Corollary 1.4. Under the hypotheses of Corollary 1.2, let ϕ (z) 0 be a meromorphic function with finite iterated p order such that ψ (z) 0. Then the differential polynomial (1.2) satisfies λ p (g f ϕ) = λ p (g f ϕ) = ρ p (f) = λ p+1 (g f ϕ) = λ p+1 (g f ϕ) = ρ p+1 (f) = ρ p (A). In the following we give two applications of the above results without the additional conditions h 0 ψ is not a solution of (1.1). Theorem 1.7. Let A (z) be an entire function of finite iterated p order satisfying 0 < ρ p (A) < 0 < τ p (A) <, let d j (z) (j = 0, 1, 2, 3) be finite iterated p order entire functions that are not all vanishing identically such that max {ρ p (d j ) (j = 0, 1, 2, 3)} < ρ p (A). If f is a nontrivial solution of the equation (1.7) f + A (z) f = 0,

RELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS 153 then the differential polynomial (1.8) g f = d 3 f (3) + d 2 f + d 1 f + d 0 f satisfies ρ p (g f ) = ρ p (f) = ρ p+1 (g f ) = ρ p+1 (f) = ρ p (A). Theorem 1.8. Under the hypotheses of Theorem 1.7, let ϕ (z) 0 be an entire function with finite iterated p order. If f is a nontrivial solution of (1.7), then the differential polynomial g f = d 3 f (3) + d 2 f + d 1 f + d 0 f such that d 3 0 satisfies λ p (g f ϕ) = λ p (g f ϕ) = ρ p (f) = λ p+1 (g f ϕ) = λ p+1 (g f ϕ) = ρ p+1 (f) = ρ p (A). 2. Auxiliary lemmas Here, we give a special case of the result due to T. B. Cao, Z. X. Chen, X. M. Zheng J. Tu in [5]. Lemma 2.1. [3] Let p 1 be an integer let A 0, A 1,..., A k 1, F 0 be finite iterated p order meromorphic functions. If f is a meromorphic solution with ρ p (f) = + ρ p+1 (f) = ρ < + of the differential equation (2.1) f (k) + A k 1 (z) f (k 1) + + A 1 (z) f + A 0 (z) f = F, then λ p (f) = λ p (f) = ρ p (f) = λ p+1 (f) = λ p+1 (f) = ρ p+1 (f) = ρ. Lemma 2.2. [5] Let p 1 be an integer let A 0, A 1,..., A k 1, F 0 be meromorphic functions. If f is a meromorphic solution of equation (2.1) such that (i) max {i (F ), i (A j ) (j = 0,..., k 1)} < i (f) = p or that (ii) max {ρ p (F ), ρ p (A j ) (j = 0,..., k 1)} < ρ p (f) < +, then i λ (f) = i λ (f) = i (f) = p λ p (f) = λ p (f) = ρ p (f). The following lemma is a corollary of Theorem 2.3 in [11]. Lemma 2.3. Assume A is an entire function with i (A) = p, assume 1 p < +. Then, for all non-trivial solutions f of (1.1), we have ρ p (f) = ρ p+1 (f) = ρ p (A). Lemma 2.4. [3] Let k 2 A (z) be a transcendental meromorphic function of finite iterated order ρ p (A) = ρ > 0 such that δ (, A) = δ > 0, let f 0 be a meromorphic solution of equation (1.1). Suppose, moreover, that either: (i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (, f) > 0.

154 B. BELAÏDI AND Z. LATREUCH Then ρ p (f) = + ρ p+1 (f) = ρ p (A) = ρ. Remark 2.1. For k = 2, Lemma 2.4 was obtained by Laine Rieppo in [13]. Lemma 2.5. [4] Let f, g be meromorphic functions with iterated p order 0 < ρ p (f), ρ p (g) < iterated p type 0 < τ p (f), τ p (g) < (1 p < ). Then the following statements hold: (i) If ρ p (g) < ρ p (f), then τ p (f + g) = τ p (fg) = τ p (f). (ii) If ρ p (f) = ρ p (g) τ p (g) τ p (f), then ρ p (f + g) = ρ p (fg) = ρ p (f). Lemma 2.6. [11] Let f be a meromorphic function for which i (f) = p 1 ρ p (f) = ρ, let k 1 be an integer. Then for any ε > 0, m (r, f ) (k) = O ( { exp }) f p 2 r ρ+ε, outside of a possible exceptional set E 1 of finite linear measure. Lemma 2.7. [1] Let g : [0, + ) R h : [0, + ) R be monotone nondecreasing functions such that g (r) h (r) outside of an exceptional set E 2 of finite linear measure. Then for any λ > 1, there exists r 0 > 0 such that g (r) h (λr) for all r > r 0. Lemma 2.8. [14] Assume f 0 is a solution of equation (1.1). Then the differential polynomial g f defined in (1.2) satisfies the system of equations g f = α 0,0 f + α 1,0 f + + α k 1,0 f (k 1), g f = α 0,1 f + α 1,1 f + + α k 1,1 f (k 1), g f = α 0,2 f + α 1,2 f + + α k 1,2 f (k 1), g (k 1) f = α 0,k 1 f + α 1,k 1 f + + α k 1,k 1 f (k 1), where α i,j = { α i,j 1 + α i 1,j 1, for all i = 1,..., k 1, α 0,j 1 Aα k 1,j 1, for i = 0 α i,0 = { di, for all i = 1,..., k 1, d 0 d k A, for i = 0.

RELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS 155 3. Proofs of the Theorems the Corollaries Proof of Theorem 1.5. Suppose that f is an infinite iterated p order meromorphic solution of (1.1) with ρ p+1 (f) = ρ. By Lemma 2.8, g f satisfies the system of equations g f = α 0,0 f + α 1,0 f + + α k 1,0 f (k 1), g f = α 0,1 f + α 1,1 f + + α k 1,1 f (k 1), (3.1) g f = α 0,2 f + α 1,2 f + + α k 1,2 f (k 1), g (k 1) f = α 0,k 1 f + α 1,k 1 f + + α k 1,k 1 f (k 1), where (3.2) α i,j = (3.3) α i,0 = { α i,j 1 + α i 1,j 1, for all i = 1,..., k 1, α 0,j 1 Aα k 1,j 1, for i = 0 { di, for all i = 1,..., k 1, d 0 d k A, for i = 0. By Cramer s rule, since h 0 we have g f α 1,0.. α k 1,0 g f α 1,1.. α k 1,1.......... g (k 1) f α 1,k 1.. α k 1,k 1 (3.4) f =. h Then (3.5) f = C 0 g f + C 1 g f + + C k 1 g (k 1) f, where C j are finite iterated p order meromorphic functions depending on α i,j, where α i,j are defined in (3.2). If ρ p (g f ) < +, then by (3.5) we obtain ρ p (f) < +, this is a contradiction. Hence ρ p (g f ) = ρ p (f) = +. Now, we prove that ρ p+1 (g f ) = ρ p+1 (f) = ρ. By (1.2), we get ρ p+1 (g f ) ρ p+1 (f) by (3.5) we have ρ p+1 (f) ρ p+1 (g f ). This yield ρ p+1 (g f ) = ρ p+1 (f) = ρ. Furthermore, if f is a finite iterated p order meromorphic solution of equation (1.1) such that (3.6) ρ p (f) > max {ρ p (A), ρ p (d j ) (j = 0, 1,..., k)}, then (3.7) ρ p (f) > max {ρ p (α i,j ) : i = 0,..., k 1, j = 0,..., k 1}. By (1.2) (3.6) we have ρ p (g f ) ρ p (f). Now, we prove ρ p (g f ) = ρ p (f). If ρ p (g f ) < ρ p (f), then by (3.5) (3.7) we get ρ p (f) max {ρ p (C j ) (j = 0,..., k 1), ρ p (g f )} < ρ p (f)

156 B. BELAÏDI AND Z. LATREUCH this is a contradiction. Hence ρ p (g f ) = ρ p (f). Remark 3.1. From (3.5), it follows that the condition h 0 is equivalent to the condition g f, g f, g f,..., g(k 1) f are linearly independent over the field of meromorphic functions of finite iterated p order. Proof of Corollary 1.1. Suppose that f 0 is a solution of (1.1). Since A is an entire function with i (A) = p, then by Lemma 2.3, we have ρ p (f) = ρ p+1 (f) = ρ p (A). Thus, by Theorem 1.5 we obtain ρ p (g f ) = ρ p (f) = ρ p+1 (g f ) = ρ p+1 (f) = ρ p (A). Proof of Corollary 1.2. Suppose that f 0 is a meromorphic solution of (1.1) such that (i) all poles of f are uniformly bounded multiplicity or that (ii) δ (, f) > 0. Then by Lemma 2.4, we have ρ p (f) = ρ p+1 (f) = ρ p (A). Now, by using Theorem 1.5, we obtain ρ p (g f ) = ρ p (f) = ρ p+1 (g f ) = ρ p+1 (f) = ρ p (A). Proof of Theorem 1.6. Suppose that f is an infinite iterated p order meromorphic solution of equation (1.1) with ρ p+1 (f) = ρ. Set w (z) = g f ϕ. Since ρ p (ϕ) <, then by Theorem 1.5 we have ρ p (w) = ρ p (g f ) = ρ p+1 (w) = ρ p+1 (g f ) = ρ. To prove λ p (g f ϕ) = λ p (g f ϕ) = λ p+1 (g f ϕ) = λ p+1 (g f ϕ) = ρ we need to prove λ p (w) = λ p (w) = λ p+1 (w) = λ p+1 (w) = ρ. By g f = w + ϕ (3.5), we get (3.8) f = C 0 w + C 1 w + + C k 1 w (k 1) + ψ (z), where Substituting (3.8) into (1.1), we obtain ψ (z) = C 0 ϕ + C 1 ϕ + + C k 1 ϕ (k 1). 2k 2 C k 1 w (2k 1) + φ i w (i) = ( ψ (k) + A (z) ψ ) = H, i=0 where φ i (i = 0,..., 2k 2) are meromorphic functions with finite iterated p order. Since ψ (z) is not a solution of (1.1), it follows that H 0. Then by Lemma 2.1, we obtain λ p (w) = λ p (w) = λ p+1 (w) = λ p+1 (w) = ρ, i. e., λ p (g f ϕ) = λ p (g f ϕ) = λ p+1 (g f ϕ) = λ p+1 (g f ϕ) = ρ. Suppose that f is a finite iterated p order meromorphic solution of equation (1.1) such that (1.6) holds. Set w (z) = g f ϕ. Since ρ p (ϕ) < ρ p (f), then by Theorem 1.5 we have ρ p (w) = ρ p (g f ) = ρ p (f). To prove λ p (g f ϕ) = λ p (g f ϕ) = ρ p (f) we need to prove λ p (w) = λ p (w) = ρ p (f). Using the same reasoning as above, we get 2k 2 C k 1 w (2k 1) + φ i w (i) = ( ψ (k) + A (z) ψ ) = F, i=0

RELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS 157 where C k 1, φ i (i = 0,..., 2k 2) are meromorphic functions with finite iterated p order ρ p (C k 1 ) < ρ p (f) = ρ p (w), ρ p (φ i ) < ρ p (f) = ρ p (w) (i = 0,..., 2k 2) ψ (z) = C 0 ϕ + C 1 ϕ + + C k 1 ϕ (k 1), ρ p (F ) < ρ p (w). Since ψ (z) is not a solution of (1.1), it follows that F 0. Then by Lemma 2.2, we obtain λ p (w) = λ p (w) = ρ p (f), i. e., λ p (g f ϕ) = λ p (g f ϕ) = ρ p (f). Proof of Corollary 1.3. Suppose that f 0 is a solution of (1.1). Then by Lemma 2.3, we have ρ p (f) = ρ p+1 (f) = ρ p (A). Since ψ 0 ρ p (ψ) <, then ψ cannot be a solution of equation (1.1). Thus, by Theorem 1.6 we obtain λ p (g f ϕ) = λ p (g f ϕ) = ρ p (f) = λ p+1 (g f ϕ) = λ p+1 (g f ϕ) = ρ p+1 (f) = ρ p (A). Proof of Corollary 1.4. Suppose that f 0 is a meromorphic solution of (1.1). Then by Lemma 2.4, we have ρ p (f) = ρ p+1 (f) = ρ p (A). Since ψ 0 ρ p (ψ) <, then ψ cannot be a solution of equation (1.1). Now, by using Theorem 1.6, we obtain λ p (g f ϕ) = λ p (g f ϕ) = ρ p (f) = λ p+1 (g f ϕ) = λ p+1 (g f ϕ) = ρ p+1 (f) = ρ p (A). Proof of Theorem 1.7. Suppose that f is a nontrivial solution of (1.7). Lemma 2.3, we have ρ p (f) =, ρ p+1 (f) = ρ p (A). We have by Lemma 2.8 (3.9) By (3.3) we obtain (3.10) α i,0 = Now, by (3.2) we get g f = α 0,0 f + α 1,0 f + α 2,0 f, g f = α 0,1 f + α 1,1 f + α 2,1 f, g f = α 0,2 f + α 1,2 f + α 2,2 f. { di, for all i = 1, 2, d 0 d 3 A, for i = 0. Then by α i,1 = { α i,0 + α i 1,0, for all i = 1, 2, α 0,0 Aα 2,0, for i = 0.

158 B. BELAÏDI AND Z. LATREUCH Hence (3.11) α 0,1 = α 0,0 Aα 2,0 = (d 0 d 3 A) Ad 2 = d 0 (d 2 + d 3) A d 3 A, α 1,1 = α 1,0 + α 0,0 = d 0 + d 1 d 3 A, α 2,1 = α 2,0 + α 1,0 = d 1 + d 2. Finally, by (3.2) we have α i,2 = { α i,1 + α i 1,1, for all i = 1, 2, α 0,1 Aα 2,1, for i = 0. So, we obtain α 0,2 = α 0,1 Aα 2,1 = d 0 (d 1 + 2d 2 + d 3) A (d 2 + 2d 3) A d 3 A, (3.12) α 1,2 α 2,2 = α 1,1 + α 0,1 = 2d 0 + d 1 (d 2 + 2d 3) A 2d 3 A, = α 2,1 + α 1,1 = d 0 + 2d 1 + d 2 d 3 A. Hence (3.13) g f = (d 0 d 3 A) f + d 1 f + d 2 f, g f = (d 0 (d 2 + d 3) A d 3 A ) f + (d 0 + d 1 d 3 A) f + (d 1 + d 2) f, g f = (d 0 (d 1 + 2d 2 + d 3) A (d 2 + 2d 3) A d 3 A ) f + (2d 0 + d 1 (d 2 + 2d 3) A 2d 3 A ) f + (d 0 + 2d 1 + d 2 d 3 A) f. First, we suppose that d 3 0. By (3.13), we have h = H 0 H 1 H 2 H 3 H 4 H 5 H 6 H 7 H 8 where H 0 = d 0 d 3 A, H 1 = d 1, H 2 = d 2, H 3 = d 0 (d 2 + d 3) A d 3 A, H 4 = d 0 + d 1 d 3 A, H 5 = d 1 + d 2, H 6 = d 0 (d 1 + 2d 2 + d 3) A (d 2 + 2d 3) A d 3 A, H 7 = 2d 0 + d 1 (d 2 + 2d 3) A 2d 3 A, H 8 = d 0 + 2d 1 + d 2 d 3 A. Then h = (3d 0 d 1 d 2 + 3d 0 d 1 d 3 + 3d 0 d 2 d 2 6d 0 d 3 d 1 + 3d 1 d 2 d 1 + 3d 1 d 3 d 0 +d 0 d 2 d 3 2d 0 d 3 d 2 + d 1 d 2 d 2 + d 1 d 3 d 1 + d 2 d 3 d 0 + 2d 0 d 2d 3 + 2d 1 d 1d 3 4d 2 d 0d 3 +2d 2 d 1d 2 + 2d 3 d 0d 2 d 1 d 2d 3 + d 1 d 3d 2 + d 2 d 1d 3 d 2 d 1d 3 d 3 d 1d 2 +d 3 d 2d 1 d 3 1 3d 2 0d 3 2d 1 (d 2) 2 3d 2 1d 2 2d 3 (d 1) 2 d 2 2d 1 d 2 1d 3 3d 2 2d 0) A + (2d 0 d 2 d 3 + 2d 0 d 3 d 2 d 1 d 2 d 2 + 2d 1 d 3 d 1 4d 2 d 3 d 0 + d 1 d 3 d 2 ) d 2 d 3 d 1 2d 1 d 2d 3 + 2d 2 d 1d 3 + 3d 0 d 1 d 3 + d 0 d 2 2 d 2 1d 2 + d 2 2d 1 2d 2 1d 3 A + (d 2 d 3 d 1 + d 0 d 2 d 3 d 1 d 3 d 2 d 2 1d 3 )A + (2d 2 d 3 d 3 3d 1 d 2 3 + 2d 2 2d 3 2d 2 3d 2)AA,

RELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS 159 + ( d 3 2 3d 1 d 2 d 3 3d 1 d 3 d 3 3d 2 d 3 d 2 d 2 d 3 d 3 2d 3 d 2d 3 ) +3d 0 d 2 3 + 3d 2 3d 1 + 2d 2 (d 3) 2 + 3d 2 2d 3 + d 2 3d 2 A 2 d 3 3A 3 + 2d 2 d 2 3(A ) 2 d 2 d 2 3AA 3d 0 d 1 d 0 d 0 d 1 d 1 d 0 d 2 d 0 2d 0 d 0d 2 + d 1 d 0d 2 + d 2 d 0d 1 d 2 d 1d 0 + d 3 0 + 2d 0 (d 1) 2 + 3d 2 0d 1 + 2d 2 (d 0) 2 + d 2 1d 0 + d 2 0d 2 2d 1 d 0d 1 + d 0 d 1d 2 d 0 d 2d 1 d 1 d 0d 2. By d 3 0, A 0, 0 < ρ p (A) <, 0 < τ p (A) < Lemma 2.5, we have ρ p (h) = ρ p (A), hence h 0. For the cases (i) d 3 0, d 2 0; (ii) d 3 0, d 2 0 d 1 0 by using a similar reasoning as above we get h 0. Finally, if d 3 0, d 2 0, d 1 0 d 0 0, we have h = d 3 0 0. Hence h 0. By h 0, we obtain f = 1 g f d 1 d 2 h g f d 0 + d 1 d 3 A d 1 + d 2 g f 2d 0 + d 1 (d 2 + 2d 3) A 2d 3 A d 0 + 2d 1 + d 2 d 3 A, which we can write (3.14) f = 1 h where ( D0 g f + D 1 g f + D 2 g f), D 0 = (d 1 d 2 2d 0 d 3 + 2d 1 d 3 + d 2 d 2 3d 3 d 1 d 3 d 2 + 2d 2d 3) A + (2d 1 d 3 + 2d 3 d 2) A + A 2 d 2 3 + 3d 0 d 1 2d 1 d 0 + d 0 d 2 d 1 d 1 2d 0d 2 + d 1d 2 d 2d 1 + d 2 0 + 2(d 1) 2, D 1 = ( ) d 1 d 3 2d 2 d 3 d 2 2 A + d2 d 1 d 0 d 1 2d 1 d 1 + 2d 2 d 0 d 1 d 2, D 2 = d 2 d 3 A + d 2 1 d 2 d 1 + d 1 d 2 d 0 d 2. If ρ p (g f ) < +, then by (3.14) we obtain ρ p (f) < +, this is a contradiction. Hence ρ p (g f ) = ρ p (f) = +. Now, we prove that ρ p+1 (g f ) = ρ p+1 (f) = ρ p (A). By (1.8), we get ρ p+1 (g f ) ρ p+1 (f) by (3.14) we have ρ p+1 (f) ρ p+1 (g f ). This yield ρ p+1 (g f ) = ρ p+1 (f) = ρ p (A). Proof of Theorem 1.8. Suppose that f is a nontrivial solution of (1.7). By setting w = g f ϕ in (3.14), we have (3.15) f = 1 h (D 0w + D 1 w + D 2 w ) + ψ, where (3.16) ψ = D 2ϕ + D 1 ϕ + D 0 ϕ. h

160 B. BELAÏDI AND Z. LATREUCH Since d 3 0, then h 0. It follows from Theorem 1.7 that g f is of infinite iterated p order ρ p+1 (g f ) = ρ p (A). Substituting (3.15) into (1.7), we obtain D 2 4 h w(5) + φ i w (i) = ( ψ (3) + A (z) ψ ), i=0 where φ i (i = 0,..., 4) are meromorphic functions with finite iterated p order. First, we prove that ψ 0. Suppose that ψ 0, then by (3.16) we obtain (3.17) D 2 ϕ + D 1 ϕ + D 0 ϕ = 0 by Lemma 2.5, we have (3.18) ρ p (D 0 ) > max {ρ p (D 1 ), ρ p (D 2 )}. By (3.17) we can write ϕ D 0 = (D 2 ϕ + D 1 ϕ Since ρ p (ϕ) = β <, then by Lemma 2.6 we obtain ϕ ). T (r, D 0 ) T (r, D 1 ) + T (r, D 2 ) + O ( exp p 2 { r β+ε }), r / E, where E is a set of finite linear measure. Then, by Lemma 2.7 we have ρ p (D 0 ) max {ρ p (D 1 ), ρ p (D 2 )}, which is a contradiction with (3.18). Hence ψ 0. It is clear now that ψ 0 cannot be a solution of (1.7) because ρ p (ψ) <. Then, by Lemma 2.1 we obtain λ p (w) = λ p (w) = λ p (g f ϕ) = λ p (g f ϕ) = ρ p (f) = λ p+1 (w) = λ p+1 (w) = λ p+1 (g f ϕ) = λ p+1 (g f ϕ) = ρ p+1 (f) = ρ p (A). Acknowledgments: The authors would like to thank the referees for their valuable comments suggestions to improve the paper. References [1] S. Bank, A general theorem concerning the growth of solutions of first-order algebraic differential equations, Compositio Math. 25 (1972), 61-70. [2] L. G. Bernal, On growth k order of solutions of a complex homogeneous linear differential equations, Proc. Amer. Math. Soc. 101 (1987), 317-322. [3] B. Belaïdi, Oscillation of fixed points of solutions of some linear differential equations, Acta Math. Univ. Comenian. (N. S.) 77(2) (2008), 263 269. [4] B. Belaïdi, Complex oscillation of differential polynomials generated by meromorphic solutions of linear differential equations, Publ. Inst. Math. (Beograd) (N. S.) 90(104) (2011), 125 133. [5] T. B. Cao, Z. X. Chen, X. M. Zheng J. Tu, On the iterated order of meromorphic solutions of higher order linear differential equations, Ann. Differential Equations 21(2) (2005), 111 122. [6] T. B. Cao, J. F. Xu Z. X. Chen, On the meromorphic solutions of linear differential equations on the complex plane, J. Math. Anal. Appl. 364(1) (2010), 130 142.

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