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SOME RESULTS ON THE COMPLEX OSCILLATION THEORY OF SOME DIFFERENTIAL POLYNOMIALS

Benharrat Bela¨ıdi and Abdallah El Farissi

Abstract. In this paper, we investigate the complex oscillation of the differ-ential polynomial gf = d2f

00 + d1f

0

+ d0f, where dj (j = 0,1,2) are meromorphic

functions with finite iterated p−order not all equal to zero generated by solutions of the differential equation f00+ A (z) f = 0, where A (z) is a transcendental meromor-phic function with finite iterated p−order ρp(A) = ρ > 0.

2000 Mathematics Subject Classification: 34M10, 30D35.

1. Introduction and main results

Throughout this paper, we assume that the reader is familiar with the funda-mental results and the standard notations of the Nevanlinna’s value distribution theory (see [5, 7, 10]). In addition, we will use λ (f ) and λ (1/f ) to denote respec-tively the exponents of convergence of the zero-sequence and the pole-sequence of a meromorphic function f , ρ (f ) to denote the order of growth of f , λ (f ) and λ (1/f ) to denote respectively the exponents of convergence of the sequence of distinct ze-ros and distinct poles of f . In order to express the rate of growth of meromorphic solutions of infinite order, we recall the following definition.

Definition 1.1 [9, 11, 13] Let f be a meromorphic function. Then the hyper order ρ2(f ) of f (z) is defined by

ρ2(f ) = lim

r→+∞

log log T (r, f )

log r , (1.1)

where T (r, f ) is the Nevanlinna characteristic function of f (see [5, 10]) .

Definition 1.2 [6, 9, 11] Let f be a meromorphic function. Then the hyper exponent of convergence of the sequence of distinct zeros of f (z) is defined by

λ2(f ) = lim r→+∞

log log Nr,f1

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where Nr,f1 is the counting function of distinct zeros of f (z) in {|z| < r}. Consider the linear differential equation

f00+ A (z) f = 0, (1.3)

where A (z) is a transcendental meromorphic function. Many important results have been obtained on the fixed points of general transcendental meromorphic functions for almost four decades (see [14]). However, there are a few studies on the fixed points of solutions of differential equations. In [12] , Wang and Lu have investigated the.. fixed points and hyper order of solutions of second order linear differential equations with meromorphic coefficients and their derivatives and have obtained the following result:

Theorem A [12] Suppose that A (z) is a transcendental meromorphic function satisfying δ (∞, A) = lim

r→+∞ m(r,A)

T (r,A) > 0, ρ (A) = ρ < +∞. Then every meromorphic

solution f (z) 6≡ 0 of the equation (1.3) satisfies that f and f0, f00 all have infinitely many fixed points and

λ (f − z) = λ  f0− z= λ  f00− z= ρ (f ) = +∞, (1.4) λ2(f − z) = λ2  f0− z= λ2  f00− z= ρ2(f ) = ρ. (1.5) In [9], Theorem A has been generalized to higher order linear differential equa-tions by Liu Ming-Sheng and Zhang Xiao-Mei as follows :

Theorem B [9] Suppose that k ≥ 2 and A (z) is a transcendental meromorphic function satisfying δ (∞, A) = δ > 0, ρ (A) = ρ < +∞. Then every meromorphic solution f (z) 6≡ 0 of

f(k)+ A (z) f = 0 (1.6)

satisfies that f and f0, f00, ..., f(k) all have infinitely many fixed points and λ (f − z) = λ  f0 − z= ... = λ  f(k)− z= ρ (f ) = +∞, (1.7) λ2(f − z) = λ2  f0 − z= ... = λ2  f(k)− z= ρ2(f ) = ρ. (1.8)

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Recently, Theorem A has been generalized to differential polynomials by the first author as follows:

Theorem C [2] Let A (z) be a transcendental meromorphic function of finite order ρ (A) = ρ > 0 such that δ (∞, A) = δ > 0. Suppose, moreover, that either:

(i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (∞, f ) > 0.

Let dj (j = 0,1,2) be polynomials that are not all equal to zero, and let ϕ (z) 6≡ 0 be

a meromorphic function of finite order. If f (z) 6≡ 0 is a meromorphic solution of (1.3) with λ



1 f



< +∞, then the differential polynomial gf = d2f

00 + d1f

0 + d0f

satisfies λ (gf− ϕ) = +∞.

The first main purpose of this paper is to improve Theorem C. The first theorem considers the case ” dj (j = 0,1,2) are meromorphic functions and h 6≡ 0” which is

different from ” dj (j = 0,1,2) are polynomials and λ

 1 f  < +∞” in Theorem C. Let us denote by α1= d1, α0 = d0− d2A, β1= −d2A + d0+ d 0 1, (1.9) β0= −d1A − (d2A) 0 + d00, (1.10) h = α1β0− α0β1, (1.11)

where A , dj (j = 0.1.2) , ϕ are meromorphic functions. We first obtain:

Theorem 1.1 Let A (z) be a transcendental meromorphic function of finite order ρ (A) = ρ > 0 such that δ (∞, A) = δ > 0. Let dj (j = 0,1,2) be meromorphic

functions with finite order not all equal to zero such that h 6≡ 0 and let ϕ (z) (6≡ 0) be a meromorphic function of finite order such that α1ϕ

0

−β1ϕ 6≡ 0. Suppose, moreover, that either:

(i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (∞, f ) > 0.

If f (z) 6≡ 0 is a meromorphic solution of (1.3) , then the differential polynomial gf = d2f 00 + d1f 0 + d0f satisfies λ (gf− ϕ) = ρ (f ) = +∞ and λ2(gf − ϕ) = ρ2(f ) = ρ (A) = ρ.

The second main purpose of this paper is to study the relation between dif-ferential polynomials generated by solutions of the difdif-ferential equation (1.3) and meromorphic functions of finite iterated order.

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Before we can state our second result, we need to give some definitions. For the definition of the iterated order of a meromorphic function, we use the same definition as in [6] , [3, p. 317] , [7, p. 129] . For all r ∈ R, we define exp1r := er

and expp+1r := exp exppr , p ∈ N. We also define for all r sufficiently large log1r := log r and logp+1r := log logpr , p ∈ N. Moreover, we denote by exp0r := r, log0r := r, log−1r := exp1r and exp−1r := log1r.

Definition 1.3 (see [6, 7]) Let f be a meromorphic function. Then the iterated p−order ρp(f ) of f is defined by

ρp(f ) = lim

r→+∞

logpT (r, f )

log r (p ≥ 1 is an integer) . (1.12) For p = 1, this notation is called order and for p = 2 hyper-order.

Definition 1.4 (see [6, 7]) The finiteness degree of the order of a meromorphic function f is defined by i (f ) =        0, for f rational,

minj ∈ N : ρj(f ) < +∞ , for f transcendental for which some j ∈ N with ρj(f ) < +∞ exists,

+∞, for f with ρj(f ) = +∞ for all j ∈ N.

(1.13)

Definition 1.5 (see [8]) Let f be a meromorphic function. Then the iterated expo-nent of convergence of the sequence of distinct zeros of f (z) is defined by

λp(f ) = lim r→+∞

logpNr,1f

log r (p ≥ 1 is an integer) . (1.14) For p = 1, this notation is called exponent of convergence of the sequence of distinct zeros and for p = 2 hyper-exponent of convergence of the sequence of distinct zeros (see [9]).

Definition 1.6 (see [8]) Let f be a meromorphic function. Then the iterated expo-nent of convergence of the sequence of distinct fixed points of f (z) is defined by

τp(f ) = λp(f − z) = lim r→+∞

logpNr,f −z1 

log r (p ≥ 1 is an integer) . (1.15) For p = 1, this notation is called exponent of convergence of the sequence of distinct fixed points and for p = 2 hyper-exponent of convergence of the sequence of distinct

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fixed points (see [9]). Thus τp(f ) = λp(f − z) is an indication of oscillation of

distinct fixed points of f (z) . We obtain the following result:

Theorem 1.2 Let A (z) be a transcendental meromorphic function of finite iterated p−order ρp(A) = ρ > 0 such that δ (∞, A) = δ > 0. Let dj (j = 0,1,2) be

mero-morphic functions with finite iterated p−order not all equal to zero such that h 6≡ 0 and let ϕ (z) (6≡ 0) be a meromorphic function of finite iterated p−order such that α1ϕ

0

− β1ϕ 6≡ 0. Suppose, moreover, that either:

(i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (∞, f ) > 0.

If f (z) 6≡ 0 is a meromorphic solution of (1.3) , then the differential polynomial gf = d2f 00 + d1f 0 + d0f satisfies λp(gf− ϕ) = ρp(f ) = +∞ and λp+1(gf− ϕ) = ρp+1(f ) = ρp(A) = ρ.

Setting p = 1 and ϕ (z) = z in Theorem 1.2, we obtain the following corollary: Corollary 1.1 Let A (z) be a transcendental meromorphic function of finite order ρ (A) = ρ > 0 such that δ (∞, A) = δ > 0, let dj (j = 0,1,2) be meromorphic

functions with finite order not all equal to zero such that h 6≡ 0 and α1− β1z 6≡ 0.

Suppose, moreover, that either:

(i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (∞, f ) > 0.

If f (z) 6≡ 0 is a meromorphic solution of (1.3) , then the differential polynomial gf = d2f

00 + d1f

0

+ d0f has infinitely many fixed points and satisfies τ (gf) = ρ (f ) =

+∞, τ2(gf) = ρ2(f ) = ρ (A) = ρ.

2. Several Lemmas

We need the following lemmas in the proofs of our theorems.

Lemma 2.1 (see Remark 1.3 of [6]) . If f is a meromorphic function with i (f ) = p ≥ 1, then ρp(f ) = ρp

 f0

 .

Lemma 2.2 [8] If f is a meromorphic function with 0 < ρp(f ) < ρ (p ≥ 1) , then ρp+1(f ) = 0.

Lemma 2.3 [4] Let A0, A1, ..., Ak−1, F 6≡ 0 be finite order meromorphic functions.

If f is a meromorphic solution with ρ (f ) = +∞ of the equation f(k)+ Ak−1f(k−1)+ ... + A1f

0

+ A0f = F, (2.1)

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Lemma 2.4 [2] Let A0, A1, ..., Ak−1, F (6≡ 0) be finite order meromorphic functions.

If f is a meromorphic solution of the equation (2.1) with ρ (f ) = +∞ and ρ2(f ) = ρ, then f satisfies λ2(f ) = λ2(f ) = ρ2(f ) = ρ.

Lemma 2.5 [2] Let k ≥ 2 and A (z) be a transcendental meromorphic function of finite order ρ (A) = ρ > 0 such that δ (∞, A) = δ > 0. Suppose, moreover, that either:

(i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (∞, f ) > 0.

Then every meromorphic solution f (z) 6≡ 0 of (1.6) satisfies ρ (f ) = +∞ and ρ2(f ) = ρ (A) = ρ.

Lemma 2.6 [1] Let k ≥ 2 and A (z) be a transcendental meromorphic function of finite iterated p−order ρp(A) = ρ > 0 such that δ (∞, A) = δ > 0. Suppose, moreover, that either:

(i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (∞, f ) > 0.

Then every meromorphic solution f (z) 6≡ 0 of (1.6) satisfies i (f ) = p + 1 and ρp(f ) = +∞, ρp+1(f ) = ρp(A) = ρ.

Lemma 2.7 [1] Let A0, A1, ..., Ak−1, F 6≡ 0 be finite iterated p−order meromorphic

functions. If f is a meromorphic solution with ρp(f ) = +∞ and ρp+1(f ) = ρ < +∞ of the equation (2.1) , then λp(f ) = ρp(f ) = +∞ and λp+1(f ) = ρp+1(f ) = ρ.

Lemma 2.8 Let A (z) be a transcendental meromorphic function with finite iter-ated p−order ρp(A) = ρ > 0 such that δ (∞, A) = δ > 0, let dj (j = 0,1,2) be

meromorphic functions with finite iterated p−order not all equal to zero such that h 6≡ 0. Suppose, moreover, that either:

(i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (∞, f ) > 0.

If f (z) 6≡ 0 is a meromorphic solution of (1.3) , then the differential polynomial gf = d2f 00 +d1f 0 +d0f satisfies i (gf) = p+1 and ρp(gf) = ρp(f ) = +∞, ρp+1(gf) = ρp+1(f ) = ρp(A) = ρ.

Proof. Suppose that f (6≡ 0) is a meromorphic solution of equation (1.3) . Then by Lemma 2.6, we have ρp(f ) = +∞ and ρp+1(f ) = ρp(A) = ρ. Set gf = d2f

00 + d1f

0 + d0f, we need to prove ρp(gf) = +∞ and ρp+1(gf) = ρ. By substituting f

00 = −Af into gf, we get gf = d1f 0 + (d0− d2A) f. (2.18)

Differentiating both a sides of equation (2.18), we obtain g0f = (−d2A + d0+ d 0 1)f 0 + (−d1A − (d2A) 0 + d 0 0)f. (2.19)

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Then by (1.9) , (1.10), (2.18) and (2.19) , we have α1f 0 + α0f = gf, (2.20) β1f0+ β0f = g0f. (2.21) Set h = α1β0− β1α0. (2.22)

By the condition h 6≡ 0 and (2.20) − (2.22) , we get

f = α1g 0

f− β1gf

h . (2.23)

If ρp(gf) < +∞, then by (2.23) and Lemma 2.1 we obtain ρp(f ) < +∞, and this is

a contradiction. Hence ρp(gf) = ρp(f ) = +∞.

Now we prove that ρp+1(gf) = ρp+1(f ) = ρp(A) = ρ. By (2.18), Lemma 2.1 and

Lemma 2.2 we have ρp+1(gf) 6 ρp+1(f ) and by (2.23) we get ρp+1(f ) 6 ρp+1(gf) .

Hence ρp+1(gf) = ρp+1(f ) = ρp(A) = ρ.

Lemma 2.9 Let A (z) be a transcendental meromorphic function with finite order ρ (A) = ρ > 0 such that δ (∞, A) = δ > 0, let dj (j = 0,1,2) be meromorphic

functions with finite order not all equal to zero such that h 6≡ 0. Suppose, moreover, that either:

(i) all poles of f are of uniformly bounded multiplicity or that (ii) δ (∞, f ) > 0.

If f (z) 6≡ 0 is a meromorphic solution of (1.3) , then the differential polynomial gf = d2f

00 + d1f

0

+ d0f satisfies ρ (gf) = ρ (f ) = +∞ and ρ2(gf) = ρ2(f ) = ρ (A) =

ρ.

Proof. Suppose that f (6≡ 0) is a meromorphic solution of equation (1.3) . Then by Lemma 2.5, we have ρ (f ) = +∞ and ρ2(f ) = ρ (A) = ρ. By using a similar arguments as in the proof of Lemma 2.8, we can prove Lemma 2.9.

3. Proof of Theorem 1.1

Suppose that f (z) 6≡ 0 is a meromorphic solution of (1.3) . Then by Lemma 2.5 we have ρ (f ) = +∞ and ρ2(f ) = ρ (A) = ρ. Set w (z) = d2f

00 + d1f

0

+ d0f − ϕ.

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ρ2(w) = ρ2(gf) = ρ2(f ) = ρ (A) = ρ. In order to prove λ (gf − ϕ) = +∞ and

λ2(gf − ϕ) = ρ (A) = ρ we need to prove only λ (w) = +∞ and λ2(w) = ρ (A) = ρ.

Substituting gf = w + ϕ into (2.23), we get

f = α1w 0 − β1w h + ψ, (3.1) where ψ = α1ϕ 0 − β1ϕ h . (3.2)

Substituting (3.1) into equation (1.3) , we obtain α1 h w 000 + φ2w00+ φ1w0+ φ0w = −  ψ00+ A (z) ψ  = W, (3.3)

where φj (j = 0, 1, 2) are meromorphic functions with ρ φj < +∞ (j = 0, 1, 2). By ρ (ψ) < +∞ and the condition ψ 6≡ 0, it follows by Lemma 2.5 that W 6≡ 0. Then by Lemma 2.3 and Lemma 2.4, we obtain λ (w) = λ (w) = ρ (w) = +∞ and λ2(w) = λ2(w) = ρ2(w) = ρ, i.e., λ (gf− ϕ) = ρ (f ) = +∞ and λ2(gf − ϕ) =

ρ2(f ) = ρ (A) = ρ.

4. Proof of Theorem 1.2

Suppose that f (z) 6≡ 0 is a meromorphic solution of (1.3) . Then by Lemma 2.6 we have ρp(f ) = +∞ and ρp+1(f ) = ρp(A) = ρ. Set w (z) = d2f

00 + d1f

0

+ d0f − ϕ.

Since ρp(ϕ) < +∞, then by Lemma 2.8 we have ρp(w) = ρp(gf) = ρp(f ) = +∞

and ρp+1(w) = ρp+1(gf) = ρp+1(f ) = ρp(A) = ρ. In order to prove λp(gf− ϕ) =

+∞ and λp+1(gf− ϕ) = ρp(A) = ρ, we need to prove only λp(w) = +∞ and

λp+1(w) = ρp(A) = ρ. Substituting gf = w + ϕ into (2.23) and using a similar

reasoning as in the proof of Theorem 1.1, we get that α1

h w 000

+ φ2w00+ φ1w0+ φ0w = −ψ00+ A (z) ψ= W, (4.1) where φj (j = 0, 1, 2) are meromorphic functions with ρp φj < ∞ (j = 0, 1, 2). By ρp(ψ) < +∞ and the condition ψ 6≡ 0, it follows by Lemma 2.6 that W 6≡ 0. By Lemma 2.7, we obtain λp(w) = ρp(w) = +∞ and λp+1(w) = ρp+1(w) = ρ, i.e.,

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References

[1] B. Bela¨ıdi, Oscillation of fixed points of solutions of some linear differential equations, Acta. Math. Univ. Comenianae, Vol 77, N◦ 2, 2008, 263-269.

[2] B. Bela¨ıdi, Growth and oscillation theory of solutions of some linear differ-ential equations, Mat. Vesnik 60 (2008), no. 4, 233–246.

[3] L. G. Bernal, On growth k−order of solutions of a complex homogeneous linear differential equations, Proc. Amer. Math. Soc. 101 (1987), 317-322.

[4] Z. X. Chen, Zeros of meromorphic solutions of higher order linear differential equations, Analysis, 14 (1994), 425-438.

[5] W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford, 1964. [6] L. Kinnunen, Linear differential equations with solutions of finite iterated order, Southeast Asian Bull. Math., 22: 4 (1998), 385-405.

[7] I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, New York, 1993.

[8] I. Laine and J. Rieppo, Differential polynomials generated by linear differential equations, Complex Var. Theory Appl. 49 (2004), N◦ 12, 897–911.

[9] M. S. Liu and X. M. Zhang, Fixed points of meromorphic solutions of higher order Linear differential equations, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 31(2006), 191-211.

[10] R. Nevanlinna, Eindeutige analytische Funktionen, Zweite Auflage. Reprint. Die Grundlehren der mathematischen Wissenschaften, Band 46. Springer-Verlag, Berlin-New York, 1974.

[11] J. Wang and H. X. Yi, Fixed points and hyper order of differential polyno-mials generated by solutions of differential equation, Complex Var. Theory Appl. 48 (2003), N◦ 1, 83–94.

[12] J. Wang and W. R. L¨u, The fixed points and hyper-order of solutions of second order linear differential equations with meromorphic functions, Acta. Math. Appl. Sinica, 2004, 27 (1), 72-80 (in Chinese) .

[13] H. X. Yi, C. C. Yang, Uniqueness theory of meromorphic functions, Math-ematics and its Applications, 557. Kluwer Academic Publishers Group, Dordrecht, 2003.

[14] Q. T. Zhang and C. C. Yang, The Fixed Points and Resolution Theory of Meromorphic Functions, Beijing University Press, Beijing, 1988 (in Chinese).

Benharrat Bela¨ıdi and Abdallah El Farissi Department of Department of Mathematics Laboratory of Pure and Applied Mathematics University of University of Mostaganem Address B. P. 227 Mostaganem-(Algeria)

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