Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations

and Applied Analysis 3 Theorem C. Let Aj z /≡ 0 j 0, 1 be entire functions with σ Aj < 1, and let a, b be complex constants such that ab / 0 and arga/ arg b or a cb 0 < c < 1 . If ψ z /≡ 0 is an entire function with finite order, then every solution f /≡ 0 of 1.2 satisfies λ f − ψ λ f ′ − ψ λ f ′′ − ψ ∞. Furthermore, let d0 z , d1 z ,and d2 z be polynomials that are not all equal to zero, and let L f d2f ′′ d1f ′ d0f . If the order of ψ is less than 1, then λ L f − ψ ∞. Belaı̈di and El Farissi 7 also studied the relation between small functions and some differential polynomials generated by solutions of the second-order nonhomogeneous linear differential equation 1.3 . They obtained the following. Theorem D. Let Aj z /≡ 0 j 0, 1 and F /≡ 0 be entire functions with max{σ Aj j 0, 1 , σ F } < 1, and let a, b be complex constants that satisfy ab / 0 and arga/ arg b or a cb 0 < c < 1 . Let d0 z , d1 z , d2 z be entire functions that are not all equal to zero with σ dj < 1 j 0, 1, 2 , and let ψ z be an entire function with finite order. If f is a solution of 1.3 , then the differential polynomial L f d2f ′′ d1f ′ d0f satisfies λ L f − ψ ∞. The main purpose of this paper is to study the growth and the oscillation of solutions of second-order linear differential equation with meromorphic coefficients. Also, we will investigate the relation between small functions and differential polynomials generated by solutions of the above equation. Our results can be stated as follows. Theorem 1.2. Let Aj z /≡ 0 (j 0, 1) and F z be meromorphic functions with max{σ F , σ Aj } < n, and let P z anz · · · a0, Q z bnz · · · b0 be polynomials with degree n (n ≥ 1), where ai, bi (i 0, 1, . . . , n), anbn / 0 are complex constants such that argan / arg bn or an cbn 0 < c < 1 , then every meromorphic solution f /≡ 0 of the equation f ′′ A1e z f ′ A0e z f F 1.4 has infinite order and satisfies λ ( f ) λ ( f ) σ ( f ) ∞. 1.5 Theorem 1.3. Under the assumption of Theorem 1.2, and let d0 z , d1 z , d2 z be meromorphic functions that are not all equal to zero with σ dj < 1 j 0, 1, 2 , and let ψ z be a meromorphic function with finite order, if f /≡ 0 is a meromorphic solution of 1.4 , then the differential polynomial L f d2f ′′ d1f ′ d0f satisfies λ L f − ψ ∞. Remark 1.4. Clearly, the method used in linear differential equations with entire coefficients cannot deal with the case of meromorphic coefficients. In addition, the proof of the results in 7, 13 relies heavily on the idea of Lemma 5 in 13 or Lemma 2.5 in 7 . However, it seems too complicated to deal with our cases. We will use an important result in uniqueness theory of meromorphic functions, that is Lemma 2.5, to prove our theorems. 2. Preliminary Lemmas In order to prove our theorems, we need the following lemmas. 4 Abstract and Applied Analysis Lemma 2.1 see 14 . Let w z be a transcendental meromorphic function with σ f σ < ∞. Let Γ { k1, j1 , . . . , km, jm } be a finite set of distinct pairs of integers satisfying ki > ji ≥ 0 for i 1, 2, . . . , m. Also let > 0 be a given constant, then there exists a set E1 ⊂ 1,∞ that has finite logarithmic measure, such that for all z satisfying z / ∈ E ∪ 0, 1 and for all k, j ∈ Γ, one has ∣ ∣w k z ∣ ∣ ∣ ∣w j z ∣ ∣ ≤ |z| k−j σ−1 ε . 2.1 Nowwe introduced a notation, see 15 and 8, Lemma 2.3 . Let P z α βi z · · · is a nonconstant polynomial, and α, β is real constants. For θ ∈ 0, 2π , set δ P z , θ α cosnθ− β sinnθ. Lemma 2.2 see 15 . Let P z be a nonconstant polynomial of degree n. Letw z be a meromorphic function, not identically zero, of order less than n, and set g z w z e z . Then for any given ε > 0 there exists a zero measure set H1 ⊂ 0, 2π such that if θ ∈ θ ∈ 0, 2π \ H1 ∪ H2 , then for |z| > r θ , 1 if δ P, θ < 0, then exp 1 δ P, θ r ≤ |g re | ≤ exp 1 − δ P, θ r , 2 if δ P, θ > 0, then exp 1 − δ P, θ r ≤ |g re | ≤ exp 1 δ P, θ r , where H2 {θ : δ P, θ 0, 0 ≤ θ < 2π} is a finite set. Lemma 2.3 see 8, Lemma 2.5 . Let f z be an entire function, and suppose that G z : log ∣f k ∣ |z| 2.2 is unbounded on some ray arg z θ with constant ρ > 0, then there exists an infinite sequence of points zn rne n 1, 2, . . . , where rn → ∞, such that G zn → ∞ and ∣f j zn ∣ ∣f k zn ∣ ≤ 1 ( k − j! 1 o 1 r k−j n , j 0, . . . , k − 1, 2.3 as n → ∞. Lemma 2.4 see 16 . Let A0, . . . , Ak−1, F /≡ 0 be finite-order meromorphic functions. If f is an infinite-order meromorphic solution of the equation f k Ak−1f k−1 · · · A0f F, 2.4 then f satisfies λ f λ f σ f ∞. Lemma 2.5 see 17, page 79 . Suppose that f1 z , f2 z , . . . , fn z n ≥ 2 are meromorphic functions and g1 z , g2 z , . . . , gn z are entire functions satisfying the following conditions: 1 Σnj 1fj z e gj z ≡ 0, 2 gj z − gk z are not constants for 1 ≤ j < k ≤ n, Abstract and Applied Analysis 5 3 for 1 ≤ j ≤ n, 1 ≤ h < k ≤ n, T r, fj o{T r, egh−gk } r → ∞, r / ∈ E , where E has a finite measure.and Applied Analysis 5 3 for 1 ≤ j ≤ n, 1 ≤ h < k ≤ n, T r, fj o{T r, egh−gk } r → ∞, r / ∈ E , where E has a finite measure. then fj z ≡ 0 j 1, 2, . . . , n . Lemma 2.6 see 8, Lemma 2.6 . Let f z be a an entire function of order σ f σ < ∞. Suppose that there exists a set E ⊂ 0, 2π which has linear measure zero, such that log |f re | ≤ Mr for any ray arg z θ ∈ 0, 2π \ E, whereM is a positive constant depending on θ, while ρ is a positive constant independent of θ. Then σ f ≤ ρ. Lemma 2.7. Under the assumption of Theorem 1.2, and let f be a meromorphic solution of 1.4 . Set w f ′′ A1e z f ′ A0e z f . If f /≡ 0 is of finite order, then σ w ≥ n. Proof. Suppose the contrary that σ w < n, we will deduce a contradiction. First, if f z ≡ C/ 0, then w CA0e z . Clearly, σ w n, this is a contradiction. Now suppose that f /≡C. If σ f < n, then f ′′ A1e z f ′ A0e z f −w 0. 2.5 By Lemma 2.5, we have A0 ≡ 0, and this is a contradiction. Hence, σ f ≥ n. Since f is a meromorphic solution of 1.4 , we know that the poles of f can occur only at the poles of Aj j 0, 1 and F. Let f g z /d z , where d z is the canonical product formed with the nonzero poles of f z , with σ d ≤ max{σ F , σ Aj , j 0, 1} < n, and g is an entire function with n ≤ σ g σ f σ ≤ ∞. Substituting f g/d into 2.5 , by some calculation we can get dw g ′′ g ′ [ A1e P z − 2 ( d′ d )] g [ A0e Q z −A1e z d ′ d 2 ( d′ d )2 − d ′′ d ] . 2.6 Now, we rewrite 2.6 into dw g − g ′′ g − [ A1e P z − 2 ( d′ d )] g ′ g [ A0e Q z −A1e z d ′ d 2 ( d′ d )2 − d ′′ d ] . 2.7 Set max{σ w , σ Aj , j 0, 1} β < n. By Lemma 2.1, for any given ε 0 < ε < 1 − β , there exists a set E2 ∈ 0, 2π which has linear measure zero, such that if θ ∈ 0, 2π \E2, then there is a constant R1 r1 θ > 1 such that for all z satisfying arg z θ and |z| ≥ R1, we have ∣g i z ∣ ∣g z ∣ ≤ |z| 2 σ−1 ε , ∣d i z ∣ |d z | ≤ |z| 2 β−1 ε , i 1, 2. 2.8 6 Abstract and Applied Analysis Case 1. Suppose that an cbn 0 < c < 1 , then by Lemmas 2.1 and 2.2, there exists a ray arg z θ ∈ 0, 2π \E2 ∪H1 ∪H2,H1 andH2 being defined in Lemma 2.2, such that δ P, θ cδ Q, θ > 0, and for the above ε and sufficiently r, ∣ ∣ ∣∣ ∣ A0e Q z −A1e z d ′ d 2 ( d′ d )2 − d ′′ d ∣ ∣ ∣∣ ∣ ≥ ∣ ∣A0eQ z ∣ ∣∣ − ∣ ∣A1eP z ∣ ∣∣ ∣ ∣ ∣ ∣ d′ d ∣ ∣ ∣ ∣ − ∣ ∣ ∣∣ ∣ 2 ( d′ d )2∣ ∣∣ ∣ − ∣ ∣ ∣ ∣ d′′ d ∣ ∣ ∣ ∣ ≥ 1 2 exp{ 1 − ε δ Q, θ r}. 2.9 Also, by Lemmas 2.1 and 2.2, we have ∣∣ ∣A1e P z − 2 ( d′ d )∣∣ ∣ ≤ ∣ ∣A1eP z ∣ ∣ ∣∣ ∣ 2d′ d ∣∣ ∣ ≤ M exp{ 1 ε cδ Q, θ r}, 2.10 where M is a constant. Now we claim that log ∣g z ∣ |z| ε 2.11 is bounded on the ray arg z θ. Otherwise, by Lemma 2.3, there exists a sequence of points zm rme, such that rm → ∞ log ∣g zm ∣ r β ε m −→ ∞. 2.12 From 2.12 and the definition of order, we see that ∣∣∣∣ d zm w zm g zm ∣∣∣ −→ 0, 2.13 for m is large enough. By 2.7 , 2.8 , 2.9 , 2.10 , and 2.13 , we get 1 2 exp{ 1 − ε δ Q, θ r m} ≤ ∣∣∣∣ A0e Q z −A1e z d ′ d 2 ( d′ d )2 − d ′′ d ∣∣∣∣ ≤ ∣∣∣ dw g ∣∣∣ ∣∣∣∣ ( A1e P z − 2 ( d′ d )) g ′ g ∣∣∣ ∣∣∣∣ g ′′ g ∣∣∣ ≤ M1 exp{ 1 ε cδ Q, θ r m}, 2.14 Abstract and Applied Analysis 7 where M1 is a constant. Clearly, we can choose ε such that 0 < ε < 1 − c / 1 c . Then by 2.14 , we can obtain a contradiction. Therefore,and Applied Analysis 7 where M1 is a constant. Clearly, we can choose ε such that 0 < ε < 1 − c / 1 c . Then by 2.14 , we can obtain a contradiction. Therefore, log ∣ ∣g z ∣ ∣ |z| ε 2.15 is bounded, and we have |g z | ≤ M exp{rβ ε} on the ray arg z θ. Case 2. Suppose that argan / arg bn. By Lemma 2.2, there exists a ray arg z θ ∈ 0, 2π \E2∪ H1 ∪H2, where E2, H1, and H2 are defined, respectively, as in Case 1, such that δ Q, θ > 0, δ P, θ < 0. 2.16 Then, for any given ε 0 < ε < n− β , by Lemma 2.2 and 2.7 , we have, for sufficiently large |z| r, ∣∣∣∣ A0e Q z −A1e z d ′ d 2 ( d′ d )2 − d ′′ d ∣∣∣∣ ≥ ∣∣∣A0eQ z ∣∣ − ∣∣∣A1eP z ∣∣ ∣∣∣ d′ d ∣∣∣ − ∣∣∣∣ 2 ( d′ d )∣∣∣∣ − ∣∣∣ d′′ d ∣∣∣ ≥ 1 2 exp{ 1 − ε δ Q, θ r}, 2.17 ∣∣∣∣A1e P z − 2 ( d′ d )∣∣∣∣ ≤ ∣∣∣A1eP z ∣∣ 2 ∣∣∣∣ ( d′ d )∣∣∣∣ ≤ expβ ε exp{ 1 − ε δ P, θ r} r2 β−1 ε . 2.18 As in Case 1, we prove that log ∣g z ∣ |z| ε 2.19 is bounded on the ray arg z θ. Otherwise, similarly as in Case 1, there exists a sequence of points zm rme, such that rm → ∞, log ∣g zm ∣ r β ε m −→ ∞. 2.20 Further, we have ∣∣∣ d zm w zm g zm ∣∣∣∣ −→ 0, 2.21 for m is large enough. 8 Abstract and Applied Analysis By 2.7 , 2.8 , 2.17 , 2.18 , and 2.21 , we get 1 2 exp{ 1 − ε δ Q, θ r m} ≤ ∣ ∣ ∣ ∣ ∣ A0e Q z −A1e z d ′ d 2 ( d′ d )2 − d ′′ d ∣ ∣ ∣ ∣ ∣ ≤ ∣ ∣ ∣ ∣ dw g ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ( A1e P z − 2 ( d′ d )) g ′ g ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ g ′′ g ∣ ∣ ∣ ∣ ≤ exp{ 1 − ε δ P, θ r m} r β−1 ε m . 2.22 Since δ Q, θ > 0 and δ P, θ < 0, we obtain a contradiction. So log ∣ ∣g z ∣ ∣ |z| ε 2.23 is bounded, and we have ∣g z ∣ ≤ M exp { r ε } 2.24 on the ray arg z θ. Combining Cases 1 and 2, for any given ray arg z θ ∈ 0, 2π \E, E of linear measure zero, we have 2.24 on the ray arg z θ, provided that r is sufficiently large. Thus by Lemma 2.6, we get σ g ≤ β ε < n, which is a contradiction. Then σ w ≥ n. Lemma 2.8. Under the assumption of Theorem 1.3, let f z be an infinite-order meromorphic solution of 1.4 , then σ L f ∞. Proof. Suppose that f z is a meromorphic solution of 1.4 , then by Theorem 1.2, we have σ f ∞. Now suppose that d2 /≡ 0. Substituting f ′′ F −A1e z f ′ −A0e z into L f , we have L ( f ) − d2F ( d1 − d2A1e z ) f ′ ( d0 − d2A0e z ) f. 2.25 Differentiating both sides of 2.25 , and replacing f ′′ with f ′′ F −A1ef ′ −A0ef , we obtain L ( f )′ − d2F ′ − ( d1 − d2A1e ) F [ d2A 2 1e 2P − ( d2A1 ′ P ′d2A1 d1A1 ) e − d2A0e d0 d′ 1 ] f ′ [ d2A0A1e P Q − ( d2A0 ′ Q′d2A0 d1A0 ) e d′ 0 ] f. 2.26 Abstract and Applied Analysis 9and Applied Analysis 9