Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 22, 1-10; http://www.math.u-szeged.hu/ejqtde/

On Oscillation Theorems for Differential Polynomials

Abdallah EL FARISSI and Benharrat BELAÏDI

Department of Mathematics Laboratory of Pure and Applied Mathematics University of Mostaganem B. P 227 Mostaganem-(Algeria) elfarissi.abdallah@yahoo.fr belaidi@univ-mosta.dz belaidibenharrat@yahoo.fr

Abstract. In this paper, we investigate the relationship between small functions and differential polynomials g_{f} (z) = d_{2}f′′ + d_{1}f′ + d_{0}f, where d_{0} (z) , d_{1} (z) , d_{2} (z) are meromorphic functions that are not all equal to zero with finite order generated by solutions of the second order linear differential equation

f′′ + Af′ + Bf = F,

where A, B, F ≢ 0 are finite order meromorphic functions having only finitely many poles.

2000 Mathematics Subject Classification: 34M10, 30D35. Key words: Linear differential equations, Meromorphic solutions, Order of growth, Exponent of convergence of zeros, Exponent of convergence of distinct zeros, Differential polynomials.

1 Introduction and Statement of Results

Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory (see [7] , [10]). In addition, we will use λ (f) and \overline{λ} (f) to denote respectively the exponents of convergence of the zero-sequence and the sequence of distinct zeros of f, ρ (f) to denote the order of growth of f. A meromorphic function φ (z) is called a small function of a meromorphic function f (z) if T (r, φ) = o (T (r, f)) as r → +∞, where T (r, f) is the Nevanlinna characteristic function of f.

To give the precise estimate of fixed points, we define:

Definition 1.1 ([9] , [11] , [12]) Let f be a meromorphic function and let z_{1}, z_{2}, ... (|z_{j}| = r_{j}, 0 < r_{1} ≤ r_{2} ≤ ...) be the sequence of the fixed points of f, each point being repeated only once. The exponent of convergence of the sequence of distinct fixed points of f (z) is defined by

\overline{τ} (f) = inf \{τ > 0 : ∑_{j=1}^{+∞} |z_{j}|^{−τ} < +∞\} .

Clearly,

\overline{τ} (f) = \overline{lim_{r→+∞}} \frac{log \overline{N} (r, \frac{1}{f−z})}{log r},  (1.1)

where \overline{N} (r, \frac{1}{f−z}) is the counting function of distinct fixed points of f (z) in {|z| < r}.

Recently the complex oscillation theory of the complex differential equations has been investigated actively [1, 2, 3, 4, 5, 6, 8, 9, 11, 12]. In the study of the differential equation

f′′ + Af′ + Bf = F,  (1.2)

where A, B, F ≢ 0 are finite order meromorphic functions having only finitely many poles, Chen [4] has investigated the complex oscillation of (1.2) and has obtained the following results:

Theorem A [4] Suppose that A, B, F ≢ 0 are finite order meromorphic functions having only finitely many poles and F ≢ CB for any constant C. Let α > 0, β > 0 be real constants and we have ρ (B) < β, ρ (F) < β. Suppose that for any given ε > 0, there exist two finite collections of real numbers {ϕ_{m}} and {θ_{m}} that satisfy

ϕ_{1} < θ_{1} < ϕ_{2} < θ_{2} < ... < ϕ_{n} < θ_{n} < ϕ_{n+1} = ϕ_{1} + 2π  (1.3)

and

∑^{n} ; m=1 (ϕ_{m+1} − θ_{m}) < ε  (1.4)

such that

|A (z)| ≥ exp {(1 + o (1)) α |z|^{β}}  (1.5)

as z → ∞ in ϕ_{m} ≤ arg z ≤ θ_{m} (m = 1, ..., n) .

If the second order non-homogeneous linear differential equation (1.2) has a meromorphic solution f (z) , then

λ (f) = \overline{λ} (f) = ρ (f) = ∞.  (1.6)

Theorem B [4] Suppose that A, B, F ≢ 0 are finite order meromorphic functions having only finitely many poles. Let α > 0, β > 0 be real constants and we have ρ (B) < β ≤ ρ (F) . Suppose that for any given ε > 0, there exist two finite collections of real numbers {ϕ_{m}} and {θ_{m}} that satisfy (1.3) and (1.4) such that (1.5) holds as z → ∞ in ϕ_{m} ≤ arg z ≤ θ_{m} (m = 1, ..., n) . If equation (1.2) has a meromorphic solution f (z) , then

(a) If B ≢ 0, then (1.2) has at most one finite order meromorphic solution f_{0} and all other meromorphic solutions of (1.2) satisfy (1.6) . If B ≡ 0, then any two finite order solutions f_{0}, f_{1} of (1.2) satisfy f_{1} = f_{0} + C for some constant C. If all the solutions of (1.2) are meromorphic, then (1.2) has a solution which satisfies (1.6).

(b) If there exists a finite order meromorphic solution f_{0} in case (a), then f_{0} satisfies

ρ (f_{0}) ≤ max \{ρ (F) , ρ (A) , \overline{λ} (f_{0})\} .  (1.7)

If \overline{λ} (f_{0}) < ρ (f_{0}) , ρ (F) ≠ ρ (A) , then ρ (f_{0}) = max \{ρ (F) , ρ (A)\} .

Recently, in [6] Chen Zongxuan and Shon Kwang Ho have studied the growth of solutions of the differential equation

f′′ + A_{1} (z) e^{az}f′ + A_{0} (z) e^{bz}f = 0  (1.8)

and have obtained the following result:

Theorem C [6] let A_{j} (z) (≢ 0) (j = 0, 1) be meromorphic functions with ρ (A_{j}) < 1 (j = 0, 1) , a, b be complex numbers such that ab ≠ 0 and arg a ≠ arg b or a = cb (0 < c < 1) . Then every meromorphic solution f (z) ≢ 0 of equation (1.8) has infinite order.

In the same paper, Z. X. Chen and K. H. Shon have investigated the fixed points of solutions, their 1st and 2nd derivatives and the differential polynomials and have obtained :

Theorem D [6] Let A_{j} (z) (j = 0, 1) , a, b, c satisfy the additional hypotheses of Theorem C. Let d_{0}, d_{1}, d_{2} be complex constants that are not all equal to zero. If f (z) ≢ 0 is any meromorphic solution of equation (1.8), then:

(i) f, f′, f′′ all have infinitely many fixed points and satisfy

\overline{τ} (f) = \overline{τ} (f′) = \overline{τ} (f′′) = ∞,  (1.9)

(ii) the differential polynomial

g (z) = d_{2}f′′ + d_{1}f′ + d_{0}f  (1.10)

has infinitely many fixed points and satisfies \overline{τ} (g) = ∞.

The main purpose of this paper is to study the growth, the oscillation and the relation between small functions and differential polynomials generated by solutions of second order linear differential equation (1.2) .

Before we state our results, we denote by

α_{1} = d_{1} − d_{2}A, β_{0} = d_{2}AB − (d_{2}B)′ − d_{1}B + d0^{′},  (1.11)
α_{0} = d_{0} − d_{2}B, β_{1} = d_{2}A^{2} − (d_{2}A)′ − d_{1}A − d_{2}B + d_{0} + d1^{′},  (1.12)
h = α_{1}β_{0} − α_{0}β_{1}  (1.13)

and

ψ (z) = \frac{α_{1} (φ′ − (d_{2}F)′ − α_{1}F ) − β_{1} (φ − d_{2}F)}{h},  (1.14)

where A, B, F ≢ 0 are meromorphic functions having only finitely many poles and d_{j} (j = 0, 1, 2) , φ are meromorphic functions with finite order.

Theorem 1.1 Suppose that A, B, F, α, β, ε, {ϕ_{m}} and {θ_{m}} satisfy the hypotheses of Theorem A. Let d_{0} (z) , d_{1} (z) , d_{2} (z) be meromorphic functions that are not all equal to zero with ρ (d_{j}) < ∞ (j = 0, 1, 2) such that h ≢ 0, and let φ (z) be a meromorphic function with finite order. If f (z) is a meromorphic solution of (1.2) , then the differential polynomial g_{f} (z) = d_{2}f′′ + d_{1}f′ + d_{0}f satisfies

\overline{λ} (g_{f} − φ) = ρ (g_{f}) = ρ (f) = ∞.  (1.15)

Theorem 1.2 Suppose that A, B, F, α, β, ε, {ϕ_{m}} and {θ_{m}} satisfy the hypotheses of Theorem 1.1, and let φ (z) be a meromorphic function with finite order. If f (z) is a meromorphic solution of (1.2) , then we have

\overline{λ} (f − φ) = \overline{λ} (f′ − φ) = \overline{λ} (f′′ − φ) = ∞.  (1.16)

Setting φ (z) = z in Theorem 1.2, we obtain the following corollary:

Corollary 1.1 Suppose that A, B, F, α, β, ε, {ϕ_{m}} and {θ_{m}} satisfy the hypotheses of Theorem 1.1. If f (z) is a meromorphic solution of (1.2) , then f, f′, f′′ all have infinitely many fixed points and satisfy

\overline{τ} (f) = \overline{τ} (f′) = \overline{τ} (f′′) = ∞.  (1.17)

Theorem 1.3 Suppose that A, B, F, α, β, ε, {ϕ_{m}} and {θ_{m}} satisfy the hypotheses of Theorem B. Let d_{0} (z) , d_{1} (z) , d_{2} (z) be meromorphic functions that are not all equal to zero with ρ (d_{j}) < ∞ (j = 0, 1, 2) such that h ≢ 0, and let φ (z) be a meromorphic function with finite order such that ψ (z) is not a solution of (1.2). If f (z) is an infinite order meromorphic solution of (1.2) , then the differential polynomial g_{f} (z) = d_{2}f′′ + d_{1}f′ + d_{0}f satisfies

\overline{λ} (g_{f} − φ) = ρ (g_{f}) = ρ (f) = ∞.  (1.18)

Remark 1.1 In Theorem 1.1, Theorem 1.3, if we don’t have the condition h ≢ 0, then the differential polynomial can be of finite order. For example if d_{2} (z) ≢ 0 is of finite order meromorphic function and d_{0} (z) = Bd_{2} (z) , d_{1} (z) = Ad_{2} (z) , then g_{f} (z) = d_{2} (z) F is of finite order.

In the next, we investigate the relation between the solutions of a pair non-homogeneous linear differential equations and we obtain the following result :

Theorem 1.4 Suppose that A, B, d_{0} (z) , d_{1} (z) , d_{2} (z) , α, β, ε, {ϕ_{m}} and {θ_{m}} satisfy the hypotheses of Theorem 1.1. Let F_{1} ≢ 0 and F_{2} ≢ 0 be meromorphic functions having only finitely many poles such that max{ρ (F_{1}) , ρ (F_{2})} < β, F_{1} − CF_{2} ≢ C_{1}B for any constants C, C_{1}, and let φ (z) be a meromorphic function with finite order. If f_{1} is a meromorphic solution of equation

f′′ + Af′ + Bf = F_{1},  (1.19)

and f_{2} is a meromorphic solution of equation

f′′ + Af′ + Bf = F_{2},  (1.20)

then the differential polynomial g_{f_{1}−Cf_{2}} (z) = d_{2} (f1^{′′} − Cf2^{′′})+d_{1} (f1^{′} − Cf2^{′})+

d_{0} (f_{1} − Cf_{2}) satisfies

\overline{λ} (g_{f_{1}−Cf_{2}} − φ) = ρ (g_{f_{1}−Cf_{2}}) = ∞  (1.21)

for any constant C.

2 Auxiliary Lemmas

Lemma 2.1 [5] Let A_{0}, A_{1}, ..., A_{k−1}, F ≢ 0 be finite order meromorphic functions. If f is a meromorphic solution with ρ (f) = +∞ of the equation

f^{(k)} + A_{k−1}f^{(k−1)} + ... + A_{1}f′ + A_{0}f = F,  (2.1)

then \overline{λ} (f) = λ (f) = ρ (f) = +∞.

Lemma 2.2 Suppose that A, B, F, α, β, ε, {ϕ_{m}} and {θ_{m}} satisfy the hypotheses of Theorem A or Theorem B. Let d_{0} (z) , d_{1} (z) , d_{2} (z) be meromorphic functions that are not all equal to zero with ρ (d_{j}) < ∞ (j = 0, 1, 2) such that h ≢ 0, where h is defined in (1.13). If f (z) is an infinite order meromorphic solution of (1.2) , then the differential polynomial g_{f} (z) = d_{2}f′′ + d_{1}f′ + d_{0}f satisfies

ρ (g_{f}) = ρ (f) = ∞.  (2.2)

Proof. Suppose that f is a meromorphic solution of equation (1.2) with ρ (f) = ∞. First we suppose that d_{2} ≢ 0. Substituting f′′ = F − Af′ − Bf into g_{f}, we get

g_{f} − d_{2}F = (d_{1} − d_{2}A) f′ + (d_{0} − d_{2}B) f.  (2.3)

Differentiating both sides of equation (2.3) and replacing f′′ with f′′ = F − Af′ − Bf, we obtain

gf^{′} − (d_{2}F)′ − (d_{1} − d_{2}A) F = [d_{2}A^{2} − (d_{2}A)′ − d_{1}A − d_{2}B + d_{0} + d1^{′}] f′

+ [d_{2}AB − (d_{2}B)′ − d_{1}B + d0^{′}] f.  (2.4)

Set

α_{1} = d_{1} − d_{2}A, α_{0} = d_{0} − d_{2}B,  (2.5)
β_{1} = d_{2}A^{2} − (d_{2}A)′ − d_{1}A − d_{2}B + d_{0} + d1^{′},  (2.6)
β_{0} = d_{2}AB − (d_{2}B)′ − d_{1}B + d0^{′}.  (2.7)

Then we have

α_{1}f′ + α_{0}f = g_{f} − d_{2}F,  (2.8)
β_{1}f′ + β_{0}f = gf^{′} − (d_{2}F)′ − α_{1}F.  (2.9)

Set

h = α_{1}β_{0} − α_{0}β_{1} = (d_{1} − d_{2}A) (d_{2}AB − (d_{2}B)′ − d_{1}B + d0^{′})

− (d_{0} − d_{2}B) (d_{2}A^{2} − (d_{2}A)′ − d_{1}A − d_{2}B + d_{0} + d1^{′}) .  (2.10)

By h ≢ 0 and (2.8) − (2.10) , we obtain

f = \frac{α_{1} (gf^{′} − (d_{2}F)′ − α_{1}F ) − β_{1} (g_{f} − d_{2}F)}{h}  (2.11)

If ρ (g_{f}) < ∞, then by (2.11) we get ρ (f) < ∞ and this is a contradiction. Hence ρ (g_{f}) = ∞.

Now suppose d_{2} ≡ 0, d_{1} ≢ 0 or d_{2} ≡ 0, d_{1} ≡ 0 and d_{0} ≢ 0. Using a similar reasoning to that above we get ρ (g_{f}) = ∞.

3 Proof of Theorem 1.1

Suppose that f is a meromorphic solution of equation (1.2). Then by Theorem A, we have ρ (f) = ∞. It follows by Lemma 2.2, ρ (g_{f}) = ρ (f) = ∞. Set w (z) = d_{2}f′′ + d_{1}f′ + d_{0}f − φ. Since ρ (φ) < ∞, then ρ (w) = ρ (g_{f}) = ρ (f) = ∞. In order to the prove \overline{λ} (g_{f} − φ) = ∞, we need to prove only \overline{λ} (w) = ∞. Using g_{f} = w + φ, we get from (2.11)

f = \frac{α_{1}w′ − β_{1}w}{h} + ψ,  (3.1)

where ψ is defined in (1.14) with ρ (ψ) < ∞. Substituting (3.1) into equation (1.2) , we obtain

\frac{α_{1}}{h} w′′′ + ϕ_{2}w′′ + ϕ_{1}w′ + ϕ_{0}w

= F − (ψ′′ + A (z) ψ′ + B (z) ψ) = W,  (3.2)

where ϕ_{j} (j = 0, 1, 2) are meromorphic functions with ρ (ϕ_{j}) < ∞ (j = 0, 1, 2). Since ψ (z) is of a meromorphic function of finite order, then by Theorem A, it follows that W ≢ 0. Then by Lemma 2.1, we obtain \overline{λ} (w) = λ (w) = ρ (w) = ∞, i.e., \overline{λ} (g_{f} − φ) = ∞.

Now suppose d_{2} ≡ 0, d_{1} ≢ 0 or d_{2} ≡ 0, d_{1} ≡ 0 and d_{0} ≢ 0. Using a similar reasoning to that above we get \overline{λ} (w) = λ (w) = ρ (w) = ∞, i.e., \overline{λ} (g_{f} − φ) = ∞.

4 Proof of Theorem 1.2

Suppose that f is a meromorphic solution of equation (1.2). Then by Theorem A, we have ρ (f) = ρ (f′) = ρ (f′′) = ∞. Since ρ (φ) < ∞, then ρ (f − φ) = ρ (f′ − φ) = ρ (f′′ − φ) = ∞. By using similar reasoning to that in the proof of Theorem 1.1, the proof of Theorem 1.2 can be completed.

5 Proof of Theorem 1.3

By hypothesis of Theorem 1.3, ψ (z) is not a solution of equation (1.2) . Then

F − (ψ′′ + A (z) ψ′ + B (z) ψ) ≢ 0.

By using a similar reasoning to that in the proof of Theorem 1.1, we can prove Theorem 1.3.

6 Proof of Theorem 1.4

Suppose that f_{1} is a meromorphic solution of equation (1.19) and f_{2} is a meromorphic solution of equation (1.20). Set w = f_{1} − Cf_{2}. Then w is a solution of equation w′′ + Aw′ + Bw = F_{1} − CF_{0}. By F_{1} − CF_{2} ≢ C_{1}B, ρ (F_{1} − CF_{0}) < β and Theorem A, we have ρ (w) = ∞. Thus, by Lemma 2.2, we have ρ (g_{f_{1}−Cf_{2}}) = ρ (f_{1} − Cf_{2}) = ∞. Let φ be a finite order meromorphic function. Then by Theorem 1.1, we get

\overline{λ} (g_{f_{1}−Cf_{2}} − φ) = ρ (g_{f_{1}−Cf_{2}}) = ∞.

References

[1] B. Belaïdi and S. Abbas, Growth and oscillation theory of nonhomogeneous complex differential equations with entire coefficients, International Math. F., N^{∘}1, 38 (2006), 1849-1860.

[2] B. Belaïdi, Oscillation of fixed points of solutions of some linear differential equations, Acta. Math. Univ. Comenianae, Vol 77, N^{∘} 2, 2008, 263-269.

[3] T. B. Cao and H. X. Yi, On the complex oscillation of higher order linear differential equations with meromorphic functions, J. Syst. Sci. Complex. 20 (2007), N^{∘}1, 135-148.

[4] Z. X. Chen, The zero and order of some second order linear differential equations with transcendental meromorphic coefficients, Appl. Math.- JCU, 9B (1994), 365–374.

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

[6] Z. X. Chen and K. H. Shon, On the growth and fixed points of solutions of second order differential equations with meromorphic coefficients, Acta Math. Sinica Engl. Ser., Vol. 21, N^{∘}4 (2005), 753-764.

[7] W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford, 1964.

[8] I. Laine and J. Rieppo, Differential polynomials generated by linear differential equations, Complex Variables, 49(2004), 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 polynomials generated by solutions of differential equations, Complex Variables, 48(2003), 83-94.

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

(Received December 31, 2008)