J. Nonlinear Funct. Anal (2016), Article ID 16 Copyright c 2016 Mathematical Research Press.

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1 J. Nonlinear Funct. Anal , Article ID 16 Copyright c 2016 Mathematical Research Press. ON THE VALUE DISTRIBUTION THEORY OF DIFFERENTIAL POLYNOMIALS IN THE UNIT DISC BENHARRAT BELAÏDI, MOHAMMED AMIN ABDELLAOUI Department of Mathematics, Laboratory of Pure Applied Mathematics, University of Mostaganem UMAB, B. P. 227 Mostaganem, Algeria Abstract. In this paper, we investigate the relationship between small functions non-homogeneous differential polynomials g = d z f + + d 1 z f + d 0 z f + bz, where d 0 z, d 1 z,, d z bz are finite [p,q] order meromorphic functions in the unit disc 2 is an integer, which are not all equal to zero generated by the complex higher order non-homogeneous linear differential equation f + A z f + + A 1 z f + A 0 z f = F, for 2, where A 0 z, A 1 z,, A z are finite [p,q] order meromorphic functions in unit disc. Keywords. Non-homogeneous linear differential equations; Differential polynomials; Analytic solutions; Meromorphic solutions; Unit disc. 1. Introduction The study on value distribution of differential polynomials generated by solutions of a given complex differential equation in the case of complex plane seems to have been started by Ban [2]. Many authors have investigated the growth oscillation of solutions of complex linear differential equations in C, see [2, 6, 7, 19, 20, 22]. In the unit disc, there already exist many results [3,4,5,8,9,10,11,12,15,16,23,24] but the study is more difficult than that in the addresses: benharrat.belaidi@univ-mosta.dz B. Belaïdi, abdellaouiamine13@yahoo.fr M A. Abdellaoui * Corresponding author. Received January 28,

2 2 B. BELAÏDI, M. A. ABDELLAOUI complex plane. In [11] Fenton-Strumia obtained some results of Wiman-Valiron type for power series in the unit disc, Fenton-Rossi [12] obtained an asymptotic equality of Wiman-Valiron type for the derivatives of analytic functions in the unit disc applied to ODEs with analytic coefficients. Throughout this paper, we shall assume that reader is familiar with the fundamental results the stard notations of the Nevanlinna s value distribution theory on the complex plane in the unit disc = {z : z < 1} see [14],[15],[20],[21],[25]. Firstly, we will recall some notations about the finite iterated order the growth index to classify generally meromorphic functions of fast growth in as those in C see [7],[19],[20]. Let us define inductively, for r 0,+, exp 1 r := e r exp p+1 r := exp exp p r, p N. We also define for all r sufficiently large in 0,+, log 1 r := logr log p+1 r := log log p r, p N. Moreover, we denote by exp 0 r := r, log 0 r := r, exp 1 r := log 1 r,log 1 r := exp 1 r. Definition 1.1. [8] The iterated p order of a meromorphic function f in is defined by ρ p f = limsup log+ p T r, f r 1 log 1 1 r p 1 is an integer, where T r, f is the characteristic function of Nevanlinna of f, log + 1 x = log+ x = max{logx,0}, log + p+1 x = log+ log + p x. For p = 1, this notation is called order for p = 2 hyper-order see [15,21]. For an analytic function f in we also define + log p+1 M r, f ρ M,p f = limsup r 1 log 1 r 1 p 1 is an integer, in which M r, f = max f z is the maximum modulus function. z =r Remar 1.1. It follows by M. Tsuji in [25] that if f is an analytic function in, then ρ 1 f ρ M,1 f ρ 1 f +1. However, it follows by Proposition in [20], that we have ρ M,p f = ρ p f, for p 2. Definition 1.2. [9] Let f be a meromorphic function. Then the iterated p convergence exponent of the sequence of zeros of f z is defined by log+ λ p f = limsup r 1 p N r, 1 f log 1 1 r p 1 is an integer,

3 ON THE VALUE DISTRIBUTION THEORY 3 where N r, 1 f is the integrated counting function of zeros of f z in {z C : z r}. For p = 1, this notation is called exponent of convergence of the sequence of zeros for p = 2 hyper-exponent of convergence of the sequence of zeros. Similarly, the iterated p convergence exponent of the sequence of distinct zeros of f z is defined by log+ λ p f = limsup r 1 p N r, 1 f log 1 1 r p 1 is an integer, where N r, 1 f is the integrated counting function of distinct zeros of f z in {z C : z r}. For 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. In the following, we will give similar definitions as in [17,18] for analytic meromorphic functions of [p,q]-order, [p,q]-type [p,q]-exponent of convergence of the zero-sequence in the unit disc. Definition 1.3. [4] Let p q 1 be integers, let f be a meromorphic function in, the [p,q]-order of f z is defined by For an analytic function f in, we also define ρ [p,q] f = limsup log+ p T r, f. r 1 1 log q log ρ M,[p,q] f = limsup r 1 1 r + p+1 M r, f. log q 1 1 r Remar 1.2. It is easy to see that 0 ρ [p,q] f + 0 ρ M,[p,q] f +, for any p q 1. By Definition 1.3, we have that ρ [p,1] = ρ p f ρ M,[p,1] = ρ M,p f. In [24], Tu Huang extended Proposition 1.1 in [4] with more details, as follows. Proposition 1.1. [24] Let f be an analytic function of [p,q]-order in. Then the following five statements hold: i If p = q = 1, then ρ f ρ M f ρ f + 1. ii If p = q 2 ρ [p,q] f < 1, then ρ [p,q] f ρ M,[p,q] f 1. iii If p = q 2 ρ [p,q] f 1, or p > q 1, then ρ [p,q] f = ρ M,[p,q] f.

4 4 B. BELAÏDI, M. A. ABDELLAOUI iv If p 1 ρ [p,p+1] f > 1, then D f = limsup D f = 0. T r, f r 1 log 1 r 1 v If p 1 ρ M,[p,p+1] f > 1, then D M f = limsup then D M f = 0. log+ Mr, f r 1 log 1 r 1 = ; if ρ [p,p+1] f < 1, then = ; if ρ M,[p,p+1] f < 1, Definition 1.4. [23] Let p q 1 be integers. The [p,q]-type of a meromorphic function f z in of [p,q]-order ρ [p,q] f 0 < ρ [p,q] f < + is defined by τ [p,q] f = limsup r 1 + log p 1 T r, f logq r ρ[p,q] f. For an analytic function f z in, the [p,q]-type about maximum modulus of f of [p,q]-order ρ M,[p,q] f 0 < ρ M,[p,q] f < + is defined by τ M,[p,q] f = limsup log+ p M r, f r 1 1 ρm,[p,q] f. logq 1 By Definition 1.4, we have that τ [p,1] = τ p f τ M,[p,1] = τ M,p f τ [1,1] = τ f τ M,[1,1] = τ M f. Definition 1.5. [23] Let p q 1 be integers. The [p,q]-exponent of convergence of the zero-sequence of f z in is defined by log+ λ [p,q] f = limsup r 1 1 r p N r, 1 f log q 1 1 r Similarly, the [p, q]-exponent of convergence of the sequence of distinct zeros of f z is defined by log+ λ [p,q] f = limsup r 1 p N r, 1 f log q 1 1 r Consider the complex nonhomogeneous linear differential equation.. f + A 1 z f + A 0 z f = F, 1.1 where A 0 z, A 1 z F are analytic functions in of finite order. In [22], Laine Rieppo considered value distribution theory of differential polynomials generated by solutions of linear differential equations in the complex plane. It is well-nown that all solutions of equation 1.1 are analytic functions in that there are exactly two linearly independent solutions

5 ON THE VALUE DISTRIBUTION THEORY 5 of 1.1, see [15]. In [10], El Farissi, Belaïdi Latreuch studied the complex oscillation of differential polynomial generated by solutions of second order linear differential equation 1.1 with analytic coefficients in obtained the following results. Set α 0 = d 0 d 2 A 0, β 0 = d 2 A 0 A 1 d 2 A 0 d 1 A 0 + d 0, α 1 = d 1 d 2 A 1, β 1 = d 2 A 2 1 d 2 A 1 d 1 A 1 d 2 A 0 + d 0 + d 1, h = α 1 β 0 α 0 β ψ z = α 1 ϕ d 2 F α 1 F β 1 ϕ d 2 F. h Theorem A. [10] Let A 1 z, A 0 z 0 F be analytic functions in of finite order. Let d 0 z, d 1 z, d 2 z be analytic functions in that are not all equal to zero with ρ d j < j = 0,1,2 such that h 0, where h is defined by 1.2. If f is an infinite order solution of 1.1 with ρ 2 f = ρ, then the differential polynomial g f = d 2 f +d 1 f +d 0 f satisfies ρ g f = ρ f = ρ 2 g f = ρ2 f = ρ. Theorem B. [10] Let A 1 z, A 0 z 0 F be analytic functions in of finite order. Let d 0 z, d 1 z, d 2 z be analytic functions in which are not all equal to zero with ρ d j < j = 0,1,2 such that h 0, let ϕ z 0 be an analytic function in of finite order such that ψ z is not a solution of 1.1. If f is an infinite order solution of 1.1 with ρ 2 f = ρ, then the differential polynomial g f = d 2 f + d 1 f + d 0 f satisfies λ g f ϕ = λ g f ϕ = ρ f =, λ 2 g f ϕ = λ 2 g f ϕ = ρ 2 f = ρ. In 2011, Belaïdi studied the growth, the oscillation the relation between small functions some class of differential polynomials generated by non-homogeneous second order linear differential equation 1.1 when the solutions are of finite iterated p order see [3]. Set β 0 = d 0 d 1 A 0, β 1 = d 1 + d 0 d 1 A 1, h = d 1 β 0 d 0 β 1 1.3

6 6 B. BELAÏDI, M. A. ABDELLAOUI η z = d 1 ϕ b d 1 F β 1 ϕ b. h Theorem C. [3] Let A 1 z, A 0 z 0, F be analytic functions of finite iterated p order in. Let d 0 z, d 1 z, bz be analytic functions of finite iterated p order in such that at least one of d 0, d 1 does not vanish identically that h 0, where h is defined by 1.3. If f is a finite iterated p order solution of 1.1 such that max { ρ p A j, ρp d j j = 0,1, ρp b, ρ p F } < ρ p f, 1.4 then the differential polynomial g f = d 1 f + d 0 f + b satisfies ρ p g f = ρp f = ρ. Theorem D. [3] Assume that the assumptions of Theorem C hold, let ϕ z be an analytic function in with ρ p ϕ < ρ p f such that η z is not a solution of 1.1. If f is a finite iterated p order solution of 1.1 such that 1.4 holds, then the differential polynomial g f = d 1 f + d 0 f + b satisfies λ g f ϕ = λ g f ϕ = ρ p f. The remainder of the paper is organized as follows. In Section 2, we shall show our main results which improve extend many results in the above-mentioned papers. Section 3 is for some lemmas basic theorems. The last section is for the proofs of our main results. 2. Main results In this paper, we continue to consider this subject investigate the complex oscillation theory of differential polynomials generated by meromorphic solutions of non-homogeneous linear differential equations in the unit disc. The main purpose of this paper is to study the controllability of solutions of the non-homogeneous higher order linear differential equation f + A z f + + A 1 z f + A 0 z f = F, the differential polynomial g = d z f + d z f + + d 0 z f + bz, 2.2 where A i z i = 0,1,, 1, F z d 0 z,d 1 z,,d z, bz are meromorphic functions in of finite [p,q] order.

7 ON THE VALUE DISTRIBUTION THEORY 7 There exists a natural question: How about the growth oscillation of the differential polynomial 2.2 with meromorphic coefficients of finite [p, q]-order generated by solutions of equation 2.1 in the unit disc? The main purpose of this paper is to consider the above question. Before we state our results, we define the sequence of meromorphic functions α i, j, β j, i = 0,, 1; j = 0,, 1 in by α i, j 1 α i, j = + α i 1, j 1 A i α, j 1, for all i = 1,, 1, α 0, j 1 A 0α, j 1, for i = 0, 2.3 α i,0 = d i d A i, for i = 0,, we define also h by β j 1 β j = + α, j 1F for all j = 1,, 1, d F + b for j = 0, α 0,0 α 1,0.. α,0 α 0,1 α 1,1.. α,1 h = α 0, α 1,.. α, 2.5 ψ z by ψ z = ϕ β 0 α 1,0.. α,0 ϕ β 1 α 1,1.. α, ϕ β α 1,.. α, h z where h 0 α i, j, β j i = 0,.., 1; j = 0,, 1 are defined in , ϕ z is a meromorphic function in with ρ [p,q] ϕ <. The main results of this paper state as follows.,

8 8 B. BELAÏDI, M. A. ABDELLAOUI Theorem 2.1. Let A i z i = 0,1,, 1, F z be meromorphic functions in of finite [p,q] order. Let d j z j = 0,1,,, bz be finite [p,q] order meromorphic functions in that are not all vanishing identically such that h 0. If f z is an infinite [p,q] order meromorphic solution in of 2.1 with ρ [p+1,q] f = ρ, then the differential polynomial 2.2 satisfies ρ [p,q] g = ρ [p,q] f = ρ [p+1,q] g = ρ [p+1,q] f = ρ. Furthermore, if f is a finite [p,q] order meromorphic solution of 2.1 in such that { } ρ [p,q] f > max max ρ [p,q] A i, max ρ [p,q] d j,ρ[p,q] F, ρ [p,q] b, i 0 j then ρ [p,q] g = ρ [p,q] f. Remar 2.1. In Theorem 2.1, if we do not have the condition h 0, then the conclusions of Theorem 2.1 cannot hold. For example, if we tae d i = d A i i = 0,, 1, then h 0. It follows that g d F +b ρ [p,q] g = ρ [p,q] d F + b. So, if f z is an infinite [p,q] order meromorphic solution of 2.1, then ρ [p,q] g = ρ [p,q] d F + b < ρ [p,q] f =, if f is a finite [p,q] order meromorphic solution of 2.1 such that 2.6 holds, then ρ [p,q] g = ρ [p,q] d F + b max { ρ [p,q] d,ρ [p,q] F, ρ [p,q] b } < ρ [p,q] f. Theorem 2.2. Under the hypotheses of Theorem 2.1, let ϕ z be a meromorphic function in with finite [p,q] order such that ψ z is not a solution of 2.1. If f z is an infinite [p,q] order meromorphic solution in of 2.1 with ρ [p+1,q] f = ρ, then the differential polynomial 2.2 satisfies λ [p,q] g ϕ = λ [p,q] g ϕ = ρ [p,q] f = λ [p+1,q] g ϕ = λ [p+1,q] g ϕ = ρ [p+1,q] f = ρ.

9 ON THE VALUE DISTRIBUTION THEORY 9 Furthermore, if f is a finite [p,q] order meromorphic solution of 2.1 in such that { } ρ [p,q] f > max max ρ [p,q] A i, max ρ [p,q] d j,ρ[p,q] F, ρ [p,q] b,ρ [p,q] ϕ, 0 i 0 j 2.7 then λ [p,q] g ϕ = λ [p,q] g ϕ = ρ [p,q] f. Remar 2.2. Obviously, Theorems are a generalization of Theorems A, B, C D. Remar 2.3. By setting bz 0, ϕ z 0 = 2 in Theorem 2.2 we obtain Theorem 1.1 in [13]. We consider now the differential equation f + A 1 z f + A 0 z f = 0, 2.8 where A 0 z, A 1 z are finite [p,q] order analytic functions in the unit disc. In the following we will give sufficient conditions on A 0, A 1 which satisfied the results of Theorem 2.1 Theorem 2.2 without the conditions h 0 ψ z is not a solution of 2.8 where = 2. Corollary 2.1. Let p q 1 be integers, let A 0 z, A 1 z be analytic functions in. Assume that ρ [p,q] A 1 < ρ [p,q] A 0 = ρ 0 < ρ < + τ [p,q] A 0 = τ 0 < τ < +. Let d 0 z, d 1 z, d 2 z bz be analytic functions in that are not all vanishing identically such that max { } ρ [p,q] b, ρ [p,q] d j : j = 0,1,2 < ρ[p,q] A 0. If f z 0 is a solution of the differential equation 2.8, then the differential polynomial g 2 = d 2 f + d 1 f + d 0 f + b satisfies ρ [p,q] g 2 = ρ [p,q] f = ρ [p,q] A 0 ρ [p+1,q] g 2 = ρ [p+1,q] f max { ρ M,[p,q] A i i = 0,1 }. Furthermore, if p > q then ρ [p+1,q] g 2 = ρ [p+1,q] f = ρ [p,q] A 0.

10 10 B. BELAÏDI, M. A. ABDELLAOUI Corollary 2.2. Let p q 1 be integers, let A 0 z, A 1 z be analytic functions in. Assume that ρ [p,q] A 1 < ρ [p,q] A 0 = ρ 0 < ρ < + τ [p,q] A 0 = τ 0 < τ < +. Let d 0 z, d 1 z, d 2 z bz be analytic functions in that are not all vanishing identically such that max { } ρ [p,q] b, ρ [p,q] d j : j = 0,1,2 < ρ[p,q] A 1. Let ϕ z be an analytic function in of finite [p,q] order such that ϕ b 0. If f z 0 is a solution of the differential equation 2.8, then the differential polynomial g 2 = d 2 f + d 1 f + d 0 f + b satisfies λ [p,q] g 2 ϕ = λ [p,q] g 2 ϕ = ρ [p,q] f = ρ [p,q] A 0 λ [p+1,q] g 2 ϕ = λ [p+1,q] g 2 ϕ = ρ [p+1,q] g 2 = ρ [p+1,q] f max { ρ M,[p,q] A 0, ρ M,[p,q] A 1 }. Furthermore, if p > q then λ [p+1,q] g 2 ϕ = λ [p+1,q] g 2 ϕ = ρ [p+1,q] f = ρ [p,q] A 0. Remar 2.4 By setting bz 0 in Corollaries we obtain Theorems in [23]. 3. Auxiliary lemmas Lemma 3.1. [4] Let p q 1 be integers. Let f be a meromorphic function in the unit disc such that ρ [p,q] f = ρ <, let 1 be an integer. Then for any ε > 0, m r, f { } 1 = O exp f p 1 ρ + εlog q 1 r holds for all r outside a set E 1 [0,1 with E 1 dr 1 r <. Lemma 3.2. [1, 15] Let g : 0, 1 R h : 0, 1 R be monotone increasing functions such that gr hr holds outside of an exceptional set E 2 [0,1 for which E 2 dr 1 r <. Then there exists a constant d 0,1 such that if sr = 1 d 1 r, then gr hsr for all r [0,1.

11 ON THE VALUE DISTRIBUTION THEORY 11 By using similar proof of Lemma 3.5 in [16], we easily obtain the following lemma when ρ [p,q] f = +. Lemma 3.3. Let p q 1 be integers. Let A i z i = 0,, 1, F 0 be meromorphic functions in, let f z be a solution of the differential equation f + A z f + + A 1 z f + A 0 z f = F satisfying max { ρ [p,q] A i i = 0,, 1,ρ [p,q] F } < ρ [p,q] f = ρ +. Then we have λ [p,q] f = λ [p,q] f = ρ [p,q] f λ [p+1,q] f = λ [p+1,q] f = ρ [p+1,q] f. Lemma 3.4. [23] Let p q 1 be integers, let A i z i = 0,, 1 be analytic functions in satisfying max { ρ [p,q] A i i = 1,, 1 } < ρ [p,q] A 0. If f z 0 is a solution of the differential equation f + A z f + + A 1 z f + A 0 z f = 0, then ρ [p,q] f = + ρ [p,q] A 0 ρ [p+1,q] f max { ρ M,[p,q] A i i = 0,, 1 }. Furthermore, if p > q then ρ [p+1,q] f = ρ [p,q] A 0. Lemma 3.5. [5] Let p q 1 be integers, let f be a meromorphic function of [p,q] order in. Then ρ [p,q] f = ρ [p,q] f. Lemma 3.6. [5] Let p q 1 be integers, let f g be non-constant meromorphic functions of [p,q]-order in. Then we have ρ [p,q] f + g max { ρ [p,q] f,ρ [p,q] g }

12 12 B. BELAÏDI, M. A. ABDELLAOUI ρ [p,q] f g max { ρ [p,q] f,ρ [p,q] g }. Furthermore, if ρ [p,q] f > ρ [p,q] g, then we obtain ρ [p,q] f + g = ρ [p,q] f g = ρ [p,q] f. Lemma 3.7. [23] Let p q 1 be integers, let f g be meromorphic functions of [p,q]-order in such that 0 < ρ [p,q] f,ρ [p,q] g < 0 < τ [p,q] f,τ [p,q] g <. We have i If ρ [p,q] f > ρ [p,q] g, then τ [p,q] f + g = τ [p,q] f g = τ [p,q] f. ii If ρ [p,q] f = ρ [p,q] g τ [p,q] f τ [p,q] g, then ρ [p,q] f + g = ρ [p,q] f g = ρ [p,q] f = ρ [p,q] g. Lemma 3.8. Assume that f z is a solution of equation 2.1. Then the differential polynomial g defined in 2.2 satisfies the system of equations g β 0 = α 0,0 f + α 1,0 f + + α,0 f, g β 1 = α 0,1 f + α 1,1 f + + α,1 f, g β 2 = α 0,2 f + α 1,2 f + + α,2 f, g β = α 0, f + α 1, f + + α, f, where α i, j = α i, j 1 + α i 1, j 1 A i α, j 1, for all i = 1,, 1, α 0, j 1 A 0α, j 1, for i = 0, α i,0 = d i d A i, for i = 0,, 1 β j 1 β j = + α, j 1F, for all j = 1,2,, 1, d F + b, for j = 0.

13 ON THE VALUE DISTRIBUTION THEORY 13 Proof. Suppose that f is a solution of 2.1. We can rewrite 2.1 as which implies f = F A i f i, 3.1 g = d f + d f + + d 1 f + d 0 f + b = We can rewrite 3.2 as g β 0 = d i d A i f i + d F + b. 3.2 α i,0 f i, 3.3 where α i,0 are defined in 2.4 β 0 = d F + b. Differentiating both sides of equation 3.3 replacing f with f = F A i f i, we obtain g β 0 = α i,0 f i + = α 0,0 f + α i,0 f i + = α 0,0 f + α i,0 f i + α i,0 f i+1 = α i 1,0 f i α i,0 f i + α i 1,0 f i + α,0 f α i 1,0 f i α,0 A i f i + α,0 F = α 0,0 α,0 A 0 f + α i,0 + α i 1,0 α,0 A i f i + α,0 F. 3.4 We can rewrite 3.4 as where g β 1 = α i,1 f i, 3.5 α i,0 α i,1 = + α i 1,0 A i α,0, for all i = 1,, 1, α 0,0 A 0α,0, for i = 0 β 1 = β 0 + α,0 F. 3.6 Differentiating both sides of equation 3.5 replacing f with f = F A i f i, we obtain g β 1 = α i,1 f i + = α 0,1 f + α i,1 f i + α i,1 f i+1 = α i,1 f i + α i 1,1 f i + α,1 f α i 1,1 f i

14 14 B. BELAÏDI, M. A. ABDELLAOUI = α 0,1 f + α i,1 f i + α i 1,1 f i A i α,1 f i + α,1 F = α 0,1 α,1 A 0 f + α i,1 + α i 1,1 A i α,1 f i + α,1 F, 3.7 which implies that where g β 2 = α i,2 f i, 3.8 α i,1 α i,2 = + α i 1,1 A i α,1, for all i = 1,, 1, α 0,1 A 0α,1, for i = 0 β 2 = β 1 + α,1 F. By using the same method as above we can easily deduce that 3.9 where g j β j = α i, j f i, j = 0,1,, 1, 3.10 α i, j 1 α i, j = + α i 1, j 1 A i α, j 1, for all i = 1,, 1, α 0, j 1 A 0α, j 1, for i = 0, α i,0 = d i d A i, for all i = 0,1,, β j 1 β j = + α, j 1F, for all j = 1,2,, 1, d F + b, for j = 0. By we obtain the system of equations g β 0 = α 0,0 f + α 1,0 f + + α,0 f, g β 1 = α 0,1 f + α 1,1 f + + α,1 f, g β 2 = α 0,2 f + α 1,2 f + + α,2 f, g β = α 0, f + α 1, f + + α, f. This completes the proof of Lemma Proof of the theorems corollaries

15 ON THE VALUE DISTRIBUTION THEORY 15 Proof of Theorem 2.1. Suppose that f z is an infinite [p, q] order meromorphic solution of 2.1 with ρ [p+1,q] f = ρ. By Lemma 3.8, g satisfies the system of equations g β 0 = α 0,0 f + α 1,0 f + + α,0 f, where g β 1 = α 0,1 f + α 1,1 f + + α,1 f, g β 2 = α 0,2 f + α 1,2 f + + α,2 f, g β = α 0, f + α 1, f + + α, f, α i, j 1 α i, j = + α i 1, j 1 A i α, j 1, for all i = 1,, 1, α 0, j 1 A 0α, j 1, for i = 0, α i,0 = d i d A i, for all i = 0,1,, β j 1 β j = + α, j 1F, for all j = 1,2,, 1, d F + b, for j = 0. By Cramer s rule, since h 0, then we have g β 0 α 1,0.. α,0 g β 1 α 1,1.. α, g β α 1,.. α, f =. h 4.4 It follows that f = C 0 g β 0 +C 1 g β 1 + +C g β = C 0 g +C 1 g + +C g =0 C j β j, 4.5 where C j are finite [p,q] order meromorphic functions in depending on α i, j, where α i, j are defined in 4.2, 4.3 β j are defined in 4.4. If ρ [p,q] g < +, then by 4.5 we obtain ρ [p,q] f < +, which is a contradiction. Hence ρ [p,q] g = ρ [p,q] f = +.

16 16 B. BELAÏDI, M. A. ABDELLAOUI Now, we prove that ρ [p+1,q] g = ρ [p+1,q] f = ρ. By 2.2, we get ρ [p+1,q] g ρ [p+1,q] f by 4.5 we have ρ [p+1,q] f ρ [p+1,q] g. This yield ρ [p+1,q] g = ρ [p+1,q] f = ρ. Furthermore, if f z is a finite [p, q] order meromorphic solution of equation 2.1 such that { ρ [p,q] f > max max ρ [p,q] A i, 0 i max 0 j ρ [p,q] } d j,ρ[p,q] F, ρ [p,q] b, 4.6 then ρ [p,q] f > max { ρ [p,q] αi, j,ρ[p,q] Cj β j : i = 0,, 1; j = 0,, 1 }. 4.7 By we have ρ [p,q] g ρ [p,q] f. Now, we prove ρ [p,q] g f = ρ[p,q] f. If ρ [p,q] g < ρ [p,q] f, then by we get ρ [p,q] f max { ρ [p,q] Cj β j j = 0,, 1,ρ[p,q] g } < ρ [p,q] f which is a contradiction. Hence ρ [p,q] g = ρ [p,q] f. Remar 4.1. From 4.5, it follows that the condition h 0 is equivalent to the condition g β 0,g β 1,,g β are linearly independent over the field of meromorphic functions of finite [p,q] order. As it was noted in the paper by Laine Rieppo [22], one may assume that d 0. Note that the linear dependence of g β 0,g β 1,,g β implies that f satisfies a linear differential equation of order smaller than with appropriate coefficients, vise versa e.g. Theorem 2.3 in the paper of Laine Rieppo [22]. Proof of Theorem 2.2. Suppose that f z is an infinite [p, q] order meromorphic solution of equation 2.1 with ρ [p+1,q] f = ρ. Set wz = g ϕ. Since ρ [p,q] ϕ <, then by Lemma 3.6 Theorem 2.1 we have ρ [p,q] w = ρ [p,q] g = ρ [p+1,q] w = ρ [p+1,q] g = ρ. To prove λ [p,q] g ϕ = λ [p,q] g ϕ = λ [p+1,q] g ϕ = λ [p+1,q] g ϕ = ρ we need to prove λ [p,q] w = λ [p,q] w = λ [p+1,q] w = λ [p+1,q] w = ρ. By g = w + ϕ, using 4.5, we get f = C 0 w +C 1 w + +C w + ψ z, 4.8 where ψ z = C 0 ϕ β 0 +C 1 ϕ β 1 + +C ϕ β.

17 Substituting 4.8 into 2.1, we obtain C w j=0 ON THE VALUE DISTRIBUTION THEORY 17 φ j w j = F ψ + A zψ + + A 0 zψ = H, where φ j j = 0,,2 2 are meromorphic functions of finite [p,q]-order. Since ψ z is not a solution of 2.1, it follows that H 0. Then by Lemma 3.3, we obtain λ [p,q] w = λ [p,q] w = λ [p+1,q] w = λ [p+1,q] w = ρ, i. e., λ [p,q] g ϕ = λ [p,q] g ϕ = λ [p+1,q] g ϕ = λ [p+1,q] g ϕ = ρ. Suppose that f z is a finite [p, q] order meromorphic solution of equation 2.1 such that 2.7 holds. Set wz = g ϕ. Since ρ [p,q] ϕ < ρ [p,q] f, then by Lemma 3.6 Theorem 2.1 we have ρ [p,q] w = ρ [p,q] g = ρ [p,q] f. To prove λ [p,q] g ϕ = λ [p,q] g ϕ = ρ [p,q] f we need to prove λ [p,q] w = λ [p,q] w = ρ [p,q] f. Using the same reasoning as above, we get C w j=0 φ j w j = F ψ + A zψ + + A 0 zψ = H, where φ j j = 0,,2 2 are meromorphic functions in with [p,q] order such that ρ [p,q] φ j < ρ [p,q] f j = 0,,2 2 ψ z = C 0 ϕ β 0 +C 1 ϕ β 1 + +C ϕ β, ρ [p,q] H < ρ [p,q] f. Since ψ z is not a solution of 2.1, it follows that H 0. Then by Lemma 3.3, we obtain λ [p,q] w = λ [p,q] w = ρ [p,q] f, i. e., λ [p,q] g ϕ = λ [p,q] g ϕ = ρ [p,q] f. Proof of Corollary 2.1. Suppose that f 0 is a solution of 2.8. Then by Lemma 3.4, we have ρ [p,q] f = ρ [p,q] A 0 ρ [p+1,q] f = ρ M,[p+1,q] f max { ρ M,[p,q] A i i = 0,1 }.

18 18 B. BELAÏDI, M. A. ABDELLAOUI Furthermore, if p > q, then ρ [p+1,q] f = ρ M,[p+1,q] f = ρ [p,q] A 0. On the other h, we have g 2 = d 2 f + d 1 f + d 0 f + b. 4.9 It follows by Lemma 3.8 that g 2 β 0 = α 0,0 f + α 1,0 f, g 2 β 1 = α 0,1 f + α 1,1 f By 2.4, we obtain Now, by 2.3, we get by 2.5 we get d 1 d 2 A 1, for i = 1, α i,0 = d 0 d 2 A 0, for i = 0. α 1,0 α i,1 = + α 0,0 A 1 α 1,0, for i = 1, α 0,0 A 0α 1,0, for i = 0, β 0 = d 2 F + b = b, β 1 = β 0 + α 1,0 F = b F Hence α 0,1 = d 2 A 0 A 1 d 2 A 0 d 1 A 0 + d 0, α 1,1 = d 2 A 2 1 d 2A 1 d 1 A 1 d 2 A 0 + d 0 + d 1 α h 2 = 0,0 α 1,0 α 0,1 α 1,1 = d2 2A d 0 d 2 A 2 1 d 2d 1 + d 1d 2 + 2d 0 d 2 d1 2 d 2d 0 d 2 d 0 + d 0 d 1 A1 d 1 d 2 A 1 A 0 + d 1 d 2 A 0 d 0 d 2 A 1 A d 2 2A 0A 1 + d 2 2A 0 A 1 d 0d 1 + d 0 d 1 + d First we suppose that d 2 0. By d 2 0, A 0 0 Lemmas we have ρ [p,q] h = ρ [p,q] A 0 > 0. Hence h 0. Now suppose d 2 0, d 1 0 or d 2 0, d 1 0 d 0 0. Then, by using a similar reasoning as above we get h 2 0. By h , we obtain f = α 1,1 g 2 β 0 α 1,0 g 2 β 1 h 2 = α 1,1 g 2 b α 1,0 g 2 b h

19 ON THE VALUE DISTRIBUTION THEORY 19 By 4.9, Lemma 3.5 Lemma 3.6, we have ρ [p,q] g 2 ρ [p,q] f ρ [p+1,q] g 2 ρ [p+1,q] f by 4.14 we have ρ [p,q] f ρ [p,q] g 2 ρ [p+1,q] f ρ [p+1,q] g 2. Hence ρ [p,q] g 2 = ρ [p,q] f ρ [p+1,q] g 2 = ρ [p+1,q] f. Proof of Corollary 2.2. Set wz = d 2 f + d 1 f + d 0 f + b ϕ. Then, by ρ [p,q] ϕ <, we have ρ [p,q] w = ρ [p,q] g 2 = ρ [p,q] f ρ [p+1,q] w = ρ [p+1,q] g 2 = ρ [p+1,q] f. In order to prove λ [p,q] g 2 ϕ = λ [p,q] g 2 ϕ = ρ [p,q] f λ [p+1,q] g 2 ϕ = λ [p+1,q] g 2 ϕ = ρ [p+1,q] f, we need to prove only λ [p,q] w = λ [p,q] w = ρ [p,q] f λ [p+1,q] w = λ [p+1,q] w = ρ [p+1,q] f. Using g 2 = w + ϕ, we get from 4.14 where Substituting 4.15 into equation 2.8, we obtain f = α 1,0w + α 1,1 w h 2 + ψ 2, 4.15 ψ 2 z = α 1,1 ϕ b α 1,0 ϕ b h α 1,0 h 2 w + φ 2 w + φ 1 w + φ 0 w = ψ 2 + A 1 zψ 2 + A 0 zψ 2 = G, 4.17 where φ j j = 0,1,2 are meromorphic functions in with ρ [p,q] φ j < j = 0,1,2. First, we prove that ψ 2 0. Suppose that ψ 2 0. By ϕ b we obtain α 1,1 = α 1,0 ϕ b ϕ b Since ρ [p,q] ϕ b max { ρ [p,q] ϕ,ρ [p,q] b } = α <, then it follows that by using Lemma 3.1, we have { } 1 mr,α 1,1 mr,α 1,0 + O exp p 1 α + ε log q, 1 r holds for all r outside a set E 1 [0,1 with E 1 dr 1 r <, that is, mr,d 2 A 2 1 d 2 A 1 d 1 A 1 d 2 A 0 + d 0 + d 1 mr,d 1 d 2 A 1 { } 1 +O exp p 1 α + ε log q, r / E r i If d 2 0, then by Lemma we obtain ρ [p,q] A 0 ρ [p,q] A 1,

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