On oscillation theorems for differential polynomials, Electron

In this paper, we investigate the relationship between small functions and differential polynomials gf (z) = d2f 00 + d1f 0 + d0f, where d0 (z),d1 (z),d2 (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 00 +Af 0 +Bf = F, whereA, B, F 6� 0 are finite order meromorphic functions having only finitely many poles.


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 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.

EJQTDE, 2009 No. 22, p. 1
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 Clearly, where N r, 1 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 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 If the second order non-homogeneous linear differential equation (1.2) has a meromorphic solution f (z) , then (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) 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 Recently, in [6] Chen Zongxuan and Shon Kwang Ho have studied the growth of solutions of the differential equation 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 : , 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 (ii) the differential polynomial has infinitely many fixed points and satisfies 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 ) and 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 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 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 (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 EJQTDE, 2009 No. 22, p. 5 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 and f 2 is a meromorphic solution of equation for any constant C.
Proof.Suppose that f is a meromorphic solution of equation (1.2) with ρ