Nonhomogeneous linear differential polynomials generated by solutions of complex differential equations in the unit disc

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1 ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 20, Number 1, June 2016 Available online at Nonhomogeneous linear differential polynomials generated by solutions of complex differential equations in the unit disc Benharrat Belaïdi Abstract. We consider the complex oscillation of nonhomogeneous linear differential polynomials g k = k j=0 djf j) +b, where d j j =0, 1,..., k) b are meromorphic functions of finite [p,q]-order in the unit disc, generated by meromorphic solutions of linear differential equations with meromorphic coefficients of finite [p,q]-order in. 1. Introduction Throughout this paper, we assume that the reader is familiar with the fundamental results the stard notations of Nevanlinna s value distribution theory on the complex plane in the unit disc = z : z < 1} see [11], [12], [16], [17], [25]). First, let us 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 [6], [15], [16]). 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 = log r log p+1 r = log log p r ), p N. Moreover, we use the notations exp 0 r = r, log 0 r = r, exp 1 r = log 1 r, log 1 r = exp 1 r. Definition 1.1 see [8]). The iterated p-order of a meromorphic function f in is defined by ρ p f) = lim sup + logp T r, f) log 1 p 1). Received March 21, Mathematics Subject Classification. 34M10, 30D35. Key words phrases. Linear differential equations, differential polynomials, meromorphic solutions, [p,q]-order, [p,q]-exponent of convergence of the sequence of distinct zeros, unit disc. 45

2 46 BENHARRAT BELAÏDI For an analytic function f in, we also define ρ M,p f) = lim sup + logp+1 M r, f) log 1 p 1). Remark 1.1. It follows by M. Tsuji [25] that if f is an analytic function in, then ρ 1 f) ρ M,1 f) ρ 1 f) + 1. However, by Proposition in [16], we have ρ M,p f) = ρ p f) p 2). Definition 1.2 see [8]). The growth index of the iterated order of a meromorphic function fz) in is defined by 0, if f is non-admissible, i f) = min ρ j f) < : j N}, if f is admissible, +, if ρ j f) = for all j N. For an analytic function f in, we also define 0, if f is non-admissible, i M f) = min ρ M,j f) < : j N}, if f is admissible, +, if ρ M,j f) = for all j N. Definition 1.3 see [7]). Let f be a meromorphic function. Then the iterated p-convergence exponent of the sequence of zeros of f z) is defined by λ p f) = lim sup + logp N r, 1 f log 1 where N r, 1 f is the integrated counting function of zeros of f z) in z C: z r}. Similarly, the iterated p-convergence exponent of the sequence of distinct zeros of f z) is defined by where N λ p f) = lim sup + logp N r, 1 f log 1 r, 1 f is the integrated counting function of distinct zeros of f z) in z C: z r}. Definition 1.4 see [7]). The growth index of the convergence exponent of the sequence of the zeros of fz) in is defined by 0, if N r, 1 f = O log 1, i λ f) = min λ j f) < : j N}, if λ j f) < for some j N, +, if λ j f) = for all j N.,,

3 NONHOMOGENEOUS LINEAR DIFFERENTIAL POLYNOMIALS 47 Remark 1.2. Similarly, we can define the growth index i λ f) of λ p f). Definition 1.5 see [11]). For a C = C }, the deficiency of a with respect to a meromorphic function f in is defined as 1 1 m r, f a N r, f a δ a, f) = lim inf = 1 lim sup T r, f) T r, f) provided that f has unbounded characteristic. Consider the complex differential equation f k) + A k 1 z) f k 1) + + A 1 z) f + A 0 z) f = 0 1.1) the k th order nonhomogeneous linear differential polynomial g k = d k f k) + d k 1 f k 1) + + d 0 f + b, 1.2) where A j j = 0, 1,..., k 1), d i i = 0, 1,..., k), b are meromorphic functions in. Let L G) denote a differential subfield of the field M G) of meromorphic functions in a domain G C. If G =, we simply write L instead of L ). A special case of such a differential subfield is L p+1,ρ = g meromorphic: ρ p+1 g) < ρ}, where ρ is a positive constant. In [18], Laine Rieppo considered value distribution theory of differential polynomials generated by solutions of linear differential equations in the complex plane. After that, Cao et al. [7] studied the complex oscillation of differential polynomial generated meromorphic solutions of second order linear differential equations with meromorphic coefficient in, obtained the following result. Theorem A see [7]). Let A be an admissible meromorphic function of finite iterated order ρ p A) = ρ > 0 1 p < ) in the unit disc such that let f δ, A) : = lim inf m r, A) T r, A) = δ > 0, be a non-zero meromorphic solution of the differential equation such that δ, f) > 0. Moreover, let f + A z) f = 0 P [f] = k 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 not vanish identically. If ϕ L p+1,ρ

4 48 BENHARRAT BELAÏDI is a non-zero meromorphic function in, neither P [f] nor P [f] ϕ vanishes identically, then if p > 1, while if p = 1. i f) = i λ P [f] ϕ) = p + 1 λ p+1 P [f] ϕ) = ρ p+1 f) = ρ p A) = ρ ρ p A) λ p+1 P [f] ϕ) ρ p+1 f) ρ p A) + 1 Recently, the author Latreuch investigated the growth oscillation of higher order differential polynomial with meromorphic coefficients in the unit disc generated by solutions of equation 1.1). They obtained the following results. Theorem B see [20]). Let A i z) i = 0, 1,..., k 1) be meromorphic functions in of finite iterated p-order. Let d j z) j = 0, 1,..., k) be finite iterated p-order meromorphic functions in that are not all vanishing identically such that α 0,0 α 1,0... α k 1,0 α 0,1 α 1,1... α k 1,1 h k = , 1.3). α 0,k 1 α 1,k 1... α k 1,k 1 where the meromorphic functions α i,j i, j = 0,..., k 1) in are defined by α i,j 1 +α i 1,j 1 A i α k 1,j 1, if i, j = 1,..., k 1, α i,j = α 0,j 1 A 0α k 1,j 1, if i = 0, j = 1,..., k 1, 1.4) d i d k A i, if j = 0, i = 0,..., k 1. If f z) is an infinite iterated p-order meromorphic solution in of 1.1) with ρ p+1 f) = ρ, then the differential polynomial g k = k j=0 d jf j) satisfies ρ p g k ) = ρ p f) = ρ p+1 g k ) = ρ p+1 f) = ρ. Furthermore, if f is a finite iterated p-order meromorphic solution in such that ρ p f) > max ρ p A i ), ρ p d j )}, then ρ p g k ) = ρ p f).,1,...,k 1 j=0,1,...,k

5 NONHOMOGENEOUS LINEAR DIFFERENTIAL POLYNOMIALS 49 Theorem C see [20]). Under the hypotheses of Theorem B, let ϕ z) 0 be a meromorphic function in with finite iterated p-order such that ϕ α 1,0... α k 1,0 ϕ α 1,1... α k 1, ϕ k 1) α 1,k 1... α k 1,k 1 ψ k z) =, h k z) is not a solution of 1.1). If f z) is an infinite iterated p-order meromorphic solution in of 1.1) with ρ p+1 f) = ρ, then the differential polynomial g k = k j=0 d jf j) satisfies λ p g k ϕ) = λ p g k ϕ) = ρ p f) = λ p+1 g k ϕ) = λ p+1 g k ϕ) = ρ. Furthermore, if f is a finite iterated p-order meromorphic solution in such that then ρ p f) > max,1,...,k 1 j=0,1,...,k ρ p A i ), ρ p d j ), ρ p ϕ)}, λ p g k ϕ) = λ p g k ϕ) = ρ p f). Juneja et al. [13, 14] investigated some properties of entire functions of [p, q]-order, obtained some results concerning their growth. In 2010, Liu et al. [22] firstly studied the growth of solutions of equation 1.1) with entire coefficients of [p, q]-order in the complex plane. After that, many authors applied the concepts of entire meromorphic) functions in the complex plane analytic functions in the unit disc = z C: z < 1} of [p, q]-order to investigate complex differential equations see [2] [5], [19], [21], [23], [24], [26]). In this paper, we use the concept of [p, q]-order to study the growth zeros of differential polynomial 1.2) generated by meromorphic solutions of [p, q]-order in the unit disc to equation 1.1). In the following, we will give similar definitions as in [13, 14] 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 1.6 see [2]). Let p q 1 be integers, let f be a meromorphic function in. The [p, q]-order of f z) is defined by ρ [p,q] f) = lim sup + log p T r, f) log q 1.

6 50 BENHARRAT BELAÏDI For an analytic function f in, we also define ρ M,[p,q] f) = lim sup + logp+1 M r, f). log q 1 Remark 1.3. It is easy to see that 0 ρ [p,q] f) + 0 ρ M,[p,q] f) + ) whenever p q 1. By Definition 1.6, we have that ρ [1,1] = ρ f) ρ M,[1,1] = ρ M f)) ρ [2,1] = ρ 2 f) ρ M,[2,1] = ρ M,2 f) ). In [23], Tu Huang extended Proposition 1.1 in [2] with more details, as follows. Proposition 1.1 see [23]). 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). iv) If p 1 ρ [p,p+1] f) > 1, ρ [p,p+1] f) < 1, then D f) = 0. v) If p 1 then D f) = lim sup ρ M,[p,p+1] f) > 1, then D M f) = lim sup ; if ρ M,[p,p+1] f) < 1, then D M f) = 0. T r,f) log 1 = ; if log+ Mr,f) log 1 Definition 1.7 see [19]). 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) = lim sup λ [p,q] f) = lim sup + logp 1 log q 1 1 T r, f) ) ρ[p,q] f). Definition 1.8 see [19]). Let p q 1 be integers. The [p, q]-exponent of convergence of the zero-sequence of f z) in is defined by ) + logp N r, 1 f log q 1 Similarly, the [p, q]-exponent of convergence of the sequence of distinct zeros of f z) is defined by + logp N r, 1 f λ [p,q] f) = lim sup log q 1 There exists a natural question: how about the growth oscillation of the differential polynomial 1.2) with meromorphic coefficients of finite.. =

7 NONHOMOGENEOUS LINEAR DIFFERENTIAL POLYNOMIALS 51 [p, q]-order generated by solutions of equation 1.1) in the unit disc? The main purpose of this paper is to consider the above question. 2. Main results Before we state our results, assuming that b ϕz) are meromorphic functions in with ρ [p,q] ϕ) <, we define the functions ψ k z) by ϕ b α 1,0... α k 1,0 ϕ b α 1,1... α k 1, ϕ k 1) b k 1) α 1,k 1... α k 1,k 1 ψ k z) =, h k z) where h k 0 α i,j i, j = 0,..., k 1) are determined, respectively, in 1.3) 1.4). The main results state as follows. Theorem 2.1. Let A i z) i = 0, 1,..., k 1) be meromorphic functions in of finite [p, q]-order. Let d j z) j = 0, 1,..., k) b be finite [p, q]- order meromorphic functions in that are not all vanishing identically such that h k 0. If f z) is an infinite [p, q]-order meromorphic solution in of 1.1) with ρ [p+1,q] f) = ρ, then the differential polynomial 1.2) satisfies ρ [p,q] g k ) = ρ [p,q] f) = ρ [p+,q] g k ) = ρ [p+1,q] f) = ρ. Furthermore, if f is a finite [p, q]-order meromorphic solution in such that then ρ [p,q] f) > max,1,...,k 1 j=0,1,...,k ρ[p,q] A i ), ρ [p,q] d j ), ρ [p,q] b) }, 2.1) ρ [p,q] g k ) = ρ [p,q] f). Remark 2.1. In Theorem 2.1, if we do not have the condition h k 0, then the conclusions of Theorem 2.1 cannot hold. For example, if we take d i = d k A i i = 0,..., k 1), then h k 0. It follows that g k b ρ [p,q] g k ) = ρ [p,q] b). So, if f z) is an infinite [p, q]-order meromorphic solution of 1.1), then ρ [p,q] g k ) = ρ [p,q] b) < ρ [p,q] f) =, if f is a finite [p, q]-order meromorphic solution of 1.1) such that 2.1) holds, then ρ [p,q] g k ) = ρ [p,q] b) < ρ [p,q] f).

8 52 BENHARRAT BELAÏDI 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 k z) is not a solution of 1.1). If f z) is an infinite [p, q]-order meromorphic solution in of 1.1) with ρ [p+1,q] f) = ρ, then the differential polynomial 1.2) satisfies λ [p,q] g k ϕ) = λ [p,q] g k ϕ) = ρ [p,q] f) = λ [p+1,q] g k ϕ) = λ [p+1,q] g k ϕ) = ρ [p+1,q] f) = ρ. Furthermore, if f is a finite [p, q]-order meromorphic solution in such that ρ [p,q] f) > ρ[p,q] A i ), ρ [p,q] d j ), ρ [p,q] b), ρ [p,q] ϕ) }, 2.2) then max,1,...,k 1 j=0,1,...,k λ p g k ϕ) = λ p g k ϕ) = ρ [p,q] f). Remark 2.2. Obviously, Theorems are generalizations of Theorems A, B C. From Theorems , Lemmas below, we easily obtain the following corollaries. Corollary 2.1. Let p q 1 be integers. Let H be a set of complex numbers with dens z : z H } > 0, let A 0 z),..., A k 1 z) be analytic functions in the unit disc satisfying max ρ [p,q] A i ) ρ [p,q] A 0 ) = ρ.,...,k 1 Suppose that there exists a real number µ satisfying 0 µ < ρ such that for any given ε 0 < ε < ρ µ) sufficiently small, we have } 1 T r, A 0 ) exp p ρ ε) log q 1 z } 1 T r, A i ) exp p µ log q i = 1,..., k 1) 1 z as z 1 for z H. Let d j z) j = 0, 1,..., k) b be finite [p, q]- order analytic functions in that are not all vanishing identically such that h k 0. If f 0 is a solution of 1.1), then the differential polynomial 1.2) satisfies ρ [p,q] g k ) = ρ [p,q] f) = ρ M,[p,q] f) = ρ [p,q] A 0 ) ρ [p+1,q] g k ) = ρ [p+1,q] f) = ρ M,[p+1,q] f) Furthermore, if p > q, then maxρ M,[p,q] A i ): i = 0, 1,..., k 1}. ρ [p+1,q] g k ) = ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ [p,q] A 0 ).

9 NONHOMOGENEOUS LINEAR DIFFERENTIAL POLYNOMIALS 53 Corollary 2.2. Let p, q H be as in Corollary 2.1, let A 0 z),..., A k 1 z) be analytic functions in the unit disc satisfying max ρ M,[p,q] A i ) ρ M,[p,q] A 0 ) = ρ.,...,k 1 Suppose that there exists a real number µ satisfying 0 µ < ρ such that for any given ε 0 < ε < ρ µ) sufficiently small, we have } 1 A 0 z) exp p+1 ρ ε) log q 1 z } 1 A i z) exp p+1 µ log q 1 z i = 1,..., k 1) as z 1 for z H. Let d j z) j = 0, 1,..., k) b be finite [p, q]- order analytic functions in that are not all vanishing identically such that h k 0. If f 0 is a solution of 1.1), then the differential polynomial 1.2) satisfies ρ [p,q] g k ) = ρ [p,q] f) = ρ M,[p,q] f) = ρ [p+1,q] g k ) = ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ M,[p,q] A 0 ) = ρ. Corollary 2.3. Under the hypotheses of Corollary 2.1, let ϕ z) be an analytic function in with finite [p, q]-order such that ψ k z) is not a solution of 1.1). If f 0 is a solution of 1.1), then the differential polynomial 1.2) satisfies λ [p,q] g k ϕ) = λ [p,q] g k ϕ) = ρ [p,q] f) = ρ M,[p,q] f) = ρ [p,q] A 0 ) λ [p+1,q] g k ϕ) = λ [p+1,q] g k ϕ) Furthermore, if p > q, then = ρ [p+1,q] f) = ρ M,[p+1,q] f) maxρ M,[p,q] A j ) : j = 0, 1,..., k 1}. λ [p+1,q] g k ϕ) = λ [p+1,q] g k ϕ) = ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ [p,q] A 0 ). Corollary 2.4. Under the hypotheses of Corollary 2.2, let ϕ z) be an analytic function in with finite [p, q]-order such that ψ k z) is not a solution of 1.1). If f 0 is a solution of 1.1), then the differential polynomial 1.2) satisfies λ [p,q] g k ϕ) = λ [p,q] g k ϕ) = ρ [p,q] f) = ρ M,[p,q] f) = λ [p+1,q] g k ϕ) = λ [p+1,q] g k ϕ) = ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ M,[p,q] A 0 ) = ρ.

10 54 BENHARRAT BELAÏDI We now consider the differential equation f + A z) f = 0, 2.3) where A z) is a meromorphic function of finite [p, q]-order in the unit disc. In the following, we will give sufficient conditions on A which satisfied the results of Theorem 2.1 Theorem 2.2 without the conditions h k 0 ψ k z) is not a solution of 1.1), where k = 2. Corollary 2.5. Let p q 1 be integers, let Az) be a meromorphic function in with 0 < ρ [p,q] A) = ρ < such that δ, A) > 0. Let d 0, d 1, d 2, b be meromorphic functions in that are not all vanishing identically such that max ρ[p,q] d j ), ρ [p,q] b) } < ρ [p,q] A). j=0,1,2 If f 0 is a meromorphic solution of 2.3) such that δ, f) > 0, 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) ρ [p+1,q] g 2 ) = ρ [p+1,q] f) = ρ M,[p+1,q] f) ρ [p,q] A) + 1. Furthermore, if p > q, then ρ [p+1,q] g 2 ) = ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ [p,q] A). Corollary 2.6. Under the hypotheses of Corollary 2.5, suppose that 0 < τ [p,q] A) < +, let ϕ be a meromorphic function in such that ϕ b 0 with ρ [p,q] ϕ) <. If f 0 is a meromorphic solution of 2.3) such that δ, f) > 0, then the differential polynomial g 2 = d 2 f + d 1 f + d 0 f + b with d 2 0 satisfies Furthermore, if p > q, then λ [p,q] g 2 ϕ) = λ [p,q] g 2 ϕ) = ρ [p,q] f) = ρ [p,q] A) λ [p+1,q] g 2 ϕ) = λ [p+1,q] g 2 ϕ) = ρ [p+1,q] f) = ρ M,[p+1,q] f) ρ [p,q] A) + 1. λ [p+1,q] g 2 ϕ) = λ [p+1,q] g 2 ϕ) = ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ [p,q] A).

11 NONHOMOGENEOUS LINEAR DIFFERENTIAL POLYNOMIALS Auxiliary lemmas Lemma 3.1 see [3]). 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 3.2 see [3]). Let p q 1 be integers, let f g be non-constant meromorphic functions of [p, q]-order in. Then ρ [p,q] f + g) max ρ [p,q] f), ρ [p,q] g) } ρ [p,q] fg) max ρ [p,q] f), ρ [p,q] g) }. Furthermore, if ρ [p,q] f) > ρ [p,q] g), then ρ [p,q] f + g) = ρ [p,q] fg) = ρ [p,q] f). By using similar proof of Lemma 2.6 in [3] or Lemma 2.6 in [19], we easily obtain the following lemma. Lemma 3.3 see [3], [19]). Let p q 1 be integers. Let A i i = 0,..., k 1) F 0 be meromorphic functions in, let f z) be a solution of the differential equation satisfying Then f k) + A k 1 z) f k 1) + + A 1 z) f + A 0 z) f = F max ρ[p,q] A i ), ρ [p,q] F ) } < ρ [p,q] f) = ρ +.,...,k 1 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 see [4]). Let p q 1 be integers. Let H be a set of complex numbers satisfying dens z : z H } > 0, let A 0 z),..., A k 1 z) be analytic functions in the unit disc satisfying maxρ [p,q] A i ): i = 1,..., k 1} ρ [p,q] A 0 ) = ρ. Suppose that there exists a real number µ satisfying 0 µ < ρ such that for any given ε 0 < ε < ρ µ) sufficiently small, we have } 1 T r, A 0 ) exp p ρ ε) log q 1 z } 1 T r, A i ) exp p µ log q 1 z i = 1,..., k 1)

12 56 BENHARRAT BELAÏDI as z 1 for z H. Then every solution f 0 of 1.1) satisfies ρ [p,q] f) = ρ M,[p,q] f) = ρ [p,q] A 0 ) ρ [p+1,q] f) = ρ M,[p+1,q] f) Furthermore, if p > q, then ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ [p,q] A 0 ). max ρ M,[p,q] A i ).,1,...,k 1 Lemma 3.5 see [3]). Let p q 1 be integers. Let H be a set of complex numbers with dens z : z H } > 0, let A 0 z),..., A k 1 z) be analytic functions in the unit disc satisfying maxρ M,[p,q] A i ): i = 1,..., k 1} ρ M,[p,q] A 0 ) = ρ. Suppose that there exists a real number µ satisfying 0 µ < ρ such that for any given ε 0 < ε < ρ µ) sufficiently small, we have } 1 A 0 z) exp p+1 ρ ε) log q 1 z } 1 A i z) exp p+1 µ log q 1 z i = 1,..., k 1) as z 1 for z H. Then every solution f 0 of 1.1) satisfies ρ [p,q] f) = ρ M,[p,q] f) = ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ M,[p,q] A 0 ) = ρ. Lemma 3.6 see [11], [12], [25]). Let f be a meromorphic function in the unit disc let k N. Then m r, f k) = S r, f), f where ) 1 S r, f) = O log + T r, f) + log 1 r possibly outside a set E 1 [0, 1) with E 1 dr <. Lemma 3.7 see [2]). Let p q 1 be integers. Let f be a meromorphic function in the unit disc such that ρ [p,q] f) = ρ <, let k 1 be an integer. Then for any ε > 0, m r, f k) f ) = O }) 1 exp p 1 ρ + ε) log q 1 r holds for all r outside a set E 2 [0, 1) with E 2 dr <.

13 NONHOMOGENEOUS LINEAR DIFFERENTIAL POLYNOMIALS 57 Lemma 3.8 see [1], [12]). Let g : 0, 1) R h: 0, 1) R be monotone increasing functions such that g r) h r) holds outside of an exceptional set E 3 [0, 1) for which dr E 3 <. Then there exists a constant d 0, 1) such that if s r) = 1 d 1 r), then g r) h s r)) for all r [0, 1). Lemma 3.9 see [10], Corollary 2.5). Suppose that 0 < ρ < r < t < R < the path Γ = Γθ 0, ρ, t) is given by the segment followed by the circle Γ 1 : z = τe iθ 0, ρ τ t < 1 3r + R), 4 Γ 2 : z = te iθ, θ 0 θ θ 0 + 2π. We suppose that f is a meromorphic solution of the equation f k) + a k 1 z) f k 1) + + a 1 z) f + a 0 z) f = 0, where the coefficients a 0 z), a 1 z),..., a k 1 z) are meromorphic in the disc z R. We also define [ 20R k 1 ) ] R C = C a n, ρ, r, R) = k +2) exp T R, a n ) + p n log, R r ρ where p n is the multiplicity of the pole of a n at the origin if a n 0) =, p n = 0 otherwise. If δ = δ, f) > 0 0 ε < δ, then 1 T r, f) 2π + 1) RC, r 1 ε) < r < R. δ ε Lemma Let p q 1 be integers, let Az) be a meromorphic function with 0 < ρ [p,q] A) = ρ < such that δ, A) > 0. If f 0 is a meromorphic solution of n=0 such that δ, f) > 0, then ρ [p,q] f) = n=0 f k) + A z) f = 0 3.1) ρ [p,q] A) ρ [p+1,q] f) = ρ M,[p+1,q] f) ρ [p,q] A) + 1. Furthermore, if p > q, then ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ [p,q] A). Proof. First, we prove that ρ [p,q] f) =. We suppose that ρ [p,q] f) = β < + then we obtain a contradiction. It follows from the definition of deficiency see Theorem A) δ, A) that, for r 1, we have m r, A) δ T r, A). 2

14 58 BENHARRAT BELAÏDI So, when r 1, we get by 3.1) Lemma 3.7 that T r, A) 2 δ m r, A) = 2 δ m r, f k) }) 1 = O exp f p 1 β log q 1 r holds for all r outside a set E 2 [0, 1) with E 2 <. Therefore, by Lemma 3.8 we obtain ρ [p 1,q] A) < which is a contradiction since A is a meromorphic function with ρ [p,q] A) = ρ > 0. Hence ρ [p,q] f) =. Now, we prove that ρ [p,q] A) ρ [p+1,q] f) ρ [p,q] A) + 1 ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ [p,q] A) if p > q. Since ρ [p,q] f) =, by 3.1) Lemma 3.6 it follows from the definition of deficiency that, for r 1, we have T r, A) 2 δ m r, A) = 2 δ m r, f k) f ) 1 = O log + T r, f) + log. 1 r By Lemma 3.8, there exists a constant d 0, 1) such that if s r) = 1 d 1 r), then ) T r, A) O log + 1 T 1 d 1 r), f) + log d 1 r) for r 1. Hence, by the definition of [p, q]-order, ρ [p+1,q] f) = lim sup dr + logp+1 T r, f) 1 ρ [p,q] A) = ρ. log q On the other h, if δ, f) > 0, then by Lemma 3.9, for any fixed ε, 0 ε < δ 1 := δ, f), r 1 ε) < r < t < R := 1+r 2 < 1, 1 T r, f) 2π + 1) RC 3.2) δ 1 ε holds on the path Γ = Γθ 0, ρ, t) chosen in accordance with Lemma 3.9, where [ ] 20R R C = k + 2) exp R r T R, A) + p 0 log, ρ p 0 is the multiplicity of the pole of A at the origin if A0) =, p 0 = 0 otherwise. By 3.2), we immediately get that ρ [p+1,q] f) = ρ M,[p+1,q] f) ρ [p,q] A) + 1

15 NONHOMOGENEOUS LINEAR DIFFERENTIAL POLYNOMIALS 59 ρ [p+1,q] f) = ρ M,[p+1,q] f) ρ [p,q] A) if p > q. Therefore, ρ [p,q] A) ρ [p+1,q] f) = ρ M,[p+1,q] f) ρ [p,q] A) + 1 ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ [p,q] A) if p > q. Remark 3.1. Lemma 3.10 was proved for q = 1 by Cao et al. [9]. Lemma 3.11 see [19]). 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) <. The following statements hold. i) If ρ [p,q] f) > ρ [p,q] g), then τ [p,q] f + g) = τ [p,q] fg) = 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] fg) = ρ [p,q] f) = ρ [p,q] g). 4. Proofs of main results Proof of Theorem 2.1. Suppose that f is an infinite [p, q]-order meromorphic solution of 1.1). By 1.1) we have which implies f k) = A i f i) 4.1) g k b = d k f k) + d k 1 f k 1) + + d 0 f = d i d k A i ) f i). 4.2) We can rewrite 4.2) as g k b = α i,0 f i), 4.3)

16 60 BENHARRAT BELAÏDI where α i,0 is defined in 1.4). Differentiating both sides of equation 4.3) using 4.1), we obtain g k k b = α i,0f i) + α i,0 f i+1) = α i,0f i) + α i 1,0 f i) = α 0,0f + α i,0f i) + α i 1,0 f i) + α k 1,0 f k) = α 0,0f + α i,0f i) + α i 1,0 f i) α k 1,0 A i f i) = α 0,0 ) α k 1,0 A 0 f + α i,0 + α i 1,0 α k 1,0 A i f i). We can rewrite 4.4) as where α i,1 = 4.4) g k b = α i,1 f i), 4.5) α i,0 + α i 1,0 α k 1,0 A i, if i = 1,..., k 1, α 0,0 A 0α k 1,0, if i = 0. Differentiating both sides of equation 4.5) using 4.1), we obtain g k k b = α i,1f i) + α i,1 f i+1) = α i,1f i) + α i 1,1 f i) This implies where = α 0,1f + α i,1f i) + α i 1,1 f i) + α k 1,1 f k) = α 0,1f + α i,1f i) + α i 1,1 f i) A i α k 1,1 f i) = α 0,1 ) α k 1,1 A 0 f + α i,1 + α i 1,1 A i α k 1,1 f i). α i,2 = 4.6) 4.7) g k b = α i,2 f i), 4.8) α i,1 + α i 1,1 A i α k 1,1, if i = 1,..., k 1, α 0,1 A 0α k 1,1, if i = )

17 NONHOMOGENEOUS LINEAR DIFFERENTIAL POLYNOMIALS 61 By using the same method as above we can easily deduce that g j) k b j) = α i,j f i), j = 0, 1,..., k 1, 4.10) where the coefficients a i,j are determined by 1.4). By 4.3) 4.10) we obtain the system of equations g k b = α 0,0 f + α 1,0 f + + α k 1,0 f k 1), g k b = α 0,1 f + α 1,1 f + + α k 1,1 f k 1), g k b = α 0,2 f + α 1,2 f + + α k 1,2 f k 1), g k 1) k b k 1) = α 0,k 1 f + α 1,k 1 f + + α k 1,k 1 f k 1). By Cramer s rule, since h k 0, we have g k b α 1,0... α k 1,0 g k b α 1,1... α k 1, g k 1) k b k 1) α 1,k 1... α k 1,k 1 f =. h k Then f = C 0 g k b) + C 1 g k b ) + + C k 1 g k 1) k b k 1)), 4.11) where C j are finite [p, q]-order meromorphic functions in depending on α i,j, where α i,j are defined in 1.4). If ρ [p,q] g k ) < +, then by 4.11), Lemmas , we obtain ρ [p,q] f) < +, this is a contradiction. Hence ρ [p,q] g k ) = ρ [p,q] f) = +. Now, we prove that ρ [p+1,q] g k ) = ρ [p+1,q] f) = ρ. By 4.2), Lemma 3.1 Lemma 3.2, we get ρ [p+1,q] g k ) ρ [p+1,q] f), by 4.11) we have ρ [p+1,q] f) ρ [p+1,q] g k ). This yields ρ [p+1,q] g k ) = ρ [p+1,q] f) = ρ. Furthermore, if f is a finite [p, q]-order meromorphic solution in of equation 1.1) such that 2.1) holds, then ρ [p,q] f) > max ρ [p,q] α i,j ): i = 0,..., k 1, j = 0,..., k 1 }. 4.12) So by 4.2) we have ρ [p,q] g k ) ρ [p,q] f). To prove the equality ρ [p,q] g k ) = ρ [p,q] f), we suppose that ρ [p,q] g k ) < ρ [p,q] f). Then, by 4.11) 4.12), ρ [p,q] f) max ρ [p,q] C j ) j = 0,..., k 1), ρ [p,q] g k ) } < ρ [p,q] f) which is a contradiction. Hence ρ [p,q] g k ) = ρ [p,q] f).

18 62 BENHARRAT BELAÏDI Remark 4.1. From 4.11) it follows that the condition h k 0 is equivalent to the condition that g k b, g k b,..., g k 1) k b k 1) are linearly independent over the field of meromorphic functions of finite [p, q]-order. As it was noted in the paper by Laine Rieppo [18], one may assume that d k 0. Note that the linear dependence of g k b, g k b,..., g k 1) k b k 1) implies that f satisfies a linear differential equation of order smaller than k with appropriate coefficients, vise versa e.g., Theorem 2.3 in the paper of Laine Rieppo [18]). Proof of Theorem 2.2. Suppose that f is an infinite [p, q]-order meromorphic solution of equation 1.1) with ρ [p+1,q] f) = ρ. Set w z) = g k ϕ. Since ρ [p,q] ϕ) <, by Lemma 3.2 Theorem 2.1 we have ρ [p,q] w) = ρ [p,q] g k ) = ρ [p+1,q] w) = ρ [p+1,q] g k ) = ρ. To prove λ [p,q] g k ϕ) = λ [p,q] g k ϕ) = λ [p+1,q] g k ϕ) = λ [p+1,q] g k ϕ) = ρ, we must show that λ [p,q] w) = λ [p,q] w) = λ [p+1,q] w) = λ [p+1,q] w) = ρ. By the equalities g k = w + ϕ 4.11), where f = C 0 w + C 1 w + + C k 1 w k 1) + ψ k z), 4.13) ψ k z) = C 0 ϕ b) + C 1 ϕ b ) + + C k 1 ϕ k 1) b k 1)). Substituting 4.13) into 1.1), we obtain 2k 2 C k 1 w 2k 1) + j=0 φ j w j) = = H, ) ψ k) k + A k 1 z) ψ k 1) k + + A 0 z) ψ k where C k 1 φ j j = 0,..., 2k 2) are meromorphic functions in with finite [p, q]-order. Since ψ k z) is not a solution of 1.1), it follows that H 0. Thus 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 k ϕ) = λ [p,q] g k ϕ) = λ [p+1,q] g k ϕ) = λ [p+1,q] g k ϕ) = ρ. Suppose that f is a finite [p, q]-order meromorphic solution in of equation 1.1) such that 2.2) holds. Set w z) = g k ϕ. Since ρ [p,q] ϕ) < ρ [p,q] f), by Lemma 3.2 Theorem 2.1, we have ρ [p,q] w) = ρ [p,q] g k ) = ρ [p,q] f). In order to prove that λ [p,q] g k ϕ) = λ [p,q] g k ϕ) = ρ [p,q] f), we must show that λ [p,q] w) = λ [p,q] w) = ρ [p,q] f). Using the same reasoning as above, we get 2k 2 C k 1 w 2k 1) + j=0 φ j w j) = ) ψ k) k + A k 1 z) ψ k 1) k + + A 0 z) ψ k

19 NONHOMOGENEOUS LINEAR DIFFERENTIAL POLYNOMIALS 63 = F, where C k 1 φ j j = 0,..., 2k 2) are meromorphic functions in with finite [p, q]-order ρ [p,q] C k 1 ) < ρ [p,q] f), ρ [p,q] φ j ) < ρ [p,q] f) j = 0,..., 2k 2), ψ k z) = C 0 ϕ b) + C 1 ϕ b ) + + C k 1 ϕ k 1) b k 1)), ρ [p,q] F ) < ρ [p,q] f). Since ψ k z) is not a solution of 1.1), it follows that F 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 k ϕ) = λ [p,q] g k ϕ) = ρ [p,q] f). Proof of Corollary 2.5. Suppose that f is a nontrivial meromorphic solution of 2.3). Then, by Lemma 3.10, we have ρ [p,q] f) = ρ [p,q] A) ρ [p+1,q] f) = ρ M,[p+1,q] f) ρ [p,q] A) + 1. Furthermore, if p > q, then ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ [p,q] A). By the same reasoning as before we obtain that g2 b = α 0,0 f + α 1,0 f, g 2 b = α 0,1 f + α 1,1 f, where α 0,0 = d 0 d 2 A, α 1,1 = d 2 A + d 0 + d 1 α 0,1 = d 2 A) d 1 A + d 0, α 1,0 = d 1. First, we suppose that d 2 0. We have h 2 = α 0,0 α 1,0 α 0,1 α 1,1 = d2 2A 2 d 2d 1 + d 1d 2 + 2d 0 d 2 d 2 1) A + d 1 d 2 A d 0d 1 + d 0 d 1 + d ) Since d 2 0, A 0, by Lemma 3.11 we have ρ [p,q] h 2 ) = ρ [p,q] A) > 0. Hence h 2 0. Now suppose that d 2 0, d 1 0; then h 2 = d 2 1A d 0d 1 + d 0 d 1 + d 2 0,, by Lemma 3.2, we have ρ [p,q] h 2 ) = ρ [p,q] A) > 0. Hence h 2 0. Finally, if d 2 0, d 1 0 d 0 0, then h 2 = d By 4.14), since h 2 0, we obtain f = α 1,0 g 2 b ) + α 1,1 g 2 b) h )

20 64 BENHARRAT BELAÏDI It is clear that ρ [p,q] g 2 ) ρ [p,q] f) ρ [p+1,q] g 2 ) ρ [p+1,q] f)), by 4.15), 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 2.6. Set w z) = d 2 f + d 1 f + d 0 f + b ϕ. Then, since 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). To prove that 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 only need to prove that 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). Since g 2 = w + ϕ, from 4.15) it follows that where f = α 1,0w + α 1,1 w h 2 + ψ 2, 4.16) ψ 2 z) = α 1,0 ϕ b ) + α 1,1 ϕ b). 4.17) h 2 Substituting 4.16) into equation 2.3), we obtain α ) 1,0 w + φ 2 w + φ 1 w + φ 0 w = ψ 2 + A z) ψ 2 = F, h 2 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; then by 4.17), since ϕ b 0, we obtain that 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) } = η <, by Lemma 3.7, it follows that mr, A) mr, 1 ) + mr, d 0 ) + mr, d d 1) + mr, d 1 ) 2 }) 1 + O exp p 1 η + ε) log q + O1) 1 r for all r outside a set E 2 [0, 1) with dr E 2 <. Thus δ 2 T r, A) mr, A) T r, d 2) + T r, d 0 ) + T r, d 1) + T r, d 1 ) }) 1 + O exp p 1 η + ε) log q + O1) r / E 2 ). 1 r By d 2 0 Lemma 3.8 we obtain the contradiction ρ [p,q] A) max j=0,1,2 ρ [p,q] d j ).

21 NONHOMOGENEOUS LINEAR DIFFERENTIAL POLYNOMIALS 65 Hence ψ 2 0. It is clear now that ψ 2 0 cannot be a solution of 2.3), because ρ [p,q] ψ 2 ) <. Thus, by Lemma 3.3, λ [p,q] g 2 ϕ) = λ [p,q] g 2 ϕ) = ρ [p,q] f) = ρ [p,q] A) λ [p+1,q] g 2 ϕ) = λ [p+1,q] g 2 ϕ) = ρ [p+1,q] f) Furthermore, if p > q, then = ρ M,[p+1,q] f) ρ [p,q] A) + 1. λ [p+1,q] g 2 ϕ) = λ [p+1,q] g 2 ϕ) = ρ [p+1,q] f) = ρ M,[p+1,q] f) = ρ [p,q] A). Acknowledgments The author is grateful to the referee whose comments suggestions lead to an improvement of this paper. References [1] S. Bank, General theorem concerning the growth of solutions of first-order algebraic differential equations, Compos. Math ), [2] B. Belaïdi, Growth of solutions to linear differential equations with analytic coefficients of [p, q]-order in the unit disc, Electron. J. Differential Equations 2011, No. 156, [3] B. Belaïdi, Growth oscillation theory of [p, q]-order analytic solutions of linear differential equations in the unit disc, J. Math. Anal. 31) 2012), [4] B. Belaïdi, On the [p, q]-order of analytic solutions of linear differential equations in the unit disc, Novi Sad J. Math. 421) 2012), [5] B. Belaïdi, Differential polynomials generated by meromorphic solutions of [p, q]-order to complex linear differential equations, Rom. J. Math. Comput. Sci. 51) 2015), [6] L. G. Bernal, On growth k-order of solutions of a complex homogeneous linear differential equation, Proc. Amer. Math. Soc. 1012) 1987), [7] T.-B. Cao, H.-Y. Xu, C.-X. Zhu, On the complex oscillation of differential polynomials generated by meromorphic solutions of differential equations in the unit disc, Proc. Indian Acad. Sci. Math. Sci. 1204) 2010), [8] 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. 3191) 2006), [9] T.-B. Cao, C.-X. Zhu, K. Liu, On the complex oscillation of meromorphic solutions of second order linear differential equations in the unit disc, J. Math. Anal. Appl. 3741) 2011), [10] Y.-M. Chiang H. K. Hayman, Estimates on the growth of meromorphic solutions of linear differential equations, Comment. Math. Helv. 793) 2004), [11] W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, [12] J. Heittokangas, On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss ), 1 54.

22 66 BENHARRAT BELAÏDI [13] O. P. Juneja, G. P. Kapoor, S. K. Bajpai, On the p,q)-order lower p,q)- order of an entire function, J. Reine Angew. Math ), [14] O. P. Juneja, G. P. Kapoor, S. K. Bajpai, On the p,q)-type lower p,q)-type of an entire function, J. Reine Angew. Math ), [15] L. Kinnunen, Linear differential equations with solutions of finite iterated order, Southeast Asian Bull. Math. 224) 1998), [16] I. Laine, Nevanlinna Theory Complex Differential Equations, de Gruyter Studies in Mathematics 15. Walter de Gruyter & Co., Berlin New York, [17] I. Laine, Complex differential equations, in: Hbook of Differential Equations: Ordinary Differential Equations. Vol. IV, Elsevier/North-Holl, Amsterdam, 2008, pp [18] I. Laine J. Rieppo, Differential polynomials generated by linear differential equations, Complex Var. Theory Appl. 4912) 2004), [19] Z. Latreuch B. Belaïdi, Linear differential equations with analytic coefficients of [p,q]-order in the unit disc, Sarajevo J. Math. 921) 2013), [20] Z. Latreuch B. Belaïdi, Properties of higher order differential polynomials generated by solutions of complex differential equations in the unit disc, J. Math. Appl. 37, 2014), [21] L.-M. Li T.-B. Cao, Solutions for linear differential equations with meromorphic coefficients of [p, q]-order in the plane, Electron. J. Differential Equations 2012, No. 195, [22] J. Liu, J. Tu, L.-Z. Shi, Linear differential equations with entire coefficients of [p, q]-order in the complex plane, J. Math. Anal. Appl ), [23] J. Tu H.-X. Huang, Complex oscillation of linear differential equations with analytic coefficients of [p, q]-order in the unit disc, Comput. Methods Funct. Theory 152) 2015), [24] J. Tu Z.-X. Xuan, Complex linear differential equations with certain analytic coefficients of [p, q]-order in the unit disc, Adv. Difference Equ. 2014, 2014:167, 12 pp. [25] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea, New York, [26] H.-Y. Xu, J. Tu, Z.-X. Xuan, The oscillation on solutions of some classes of linear differential equations with meromorphic coefficients of finite [p, q]-order, Sci. World J. 2013, Art. ID , 8 pp. Department of Mathematics, Laboratory of Pure Applied Mathematics, University of Mostaganem UMAB), B. P. 227 Mostaganem, Algeria address: belaidibenharrat@yahoo.fr; benharrat.belaidi@univ-mosta.dz