Uniqueness of Entire Functions Sharing Polynomials with Their Derivatives

and Applied Analysis 3 From the ideas of Theorem D to Theorem F, it is natural to ask whether the values a, b in Theorem C can be replaced by two polynomials Q1, Q2. The main purpose of this paper is to investigate this problem. We prove the following result. Theorem 1.2. Let Q1 z a1z a1,p−1zp−1 · · · a1,0 and Q2 z a2z a2,p−1zp−1 · · · a2,0 be two polynomials such that degQ1 z degQ2 z p (where p is a nonnegative integer) and a1, a2 a2 / 0 are two distinct complex numbers. Let f z be a transcendental entire function. If f z Q1 z f ′ z Q1 z and f ′ z Q2 ⇒ f z Q2 z , then f z ≡ f ′ z . Remark 1.3. The following example shows the hypothesis that f is transcendental cannot be omitted in Theorem 1.2. Example 1.4. Let f z z3, Q1 z 2z3 − 3z2 and Q2 z z3. Then f ′ z −Q1 z f z −Q1 z 2, f ′ z Q2 z ⇒ f z Q2 z . 1.4 While f z does not satisfy the result of Theorem 1.2. Remark 1.5. The case p 0 of Theorem 1.2 yields Theorem C. It seems that we cannot get the result by the methods used in 4, 5 . In order to prove our theorem, we need the following result which is interesting in its own right. Theorem 1.6. Let Q1 z a1z a1,p−1zp−1 · · · a1,0 and Q2 z a2z a2,p−1zp−1 · · · a2,0 be two polynomials such that degQ1 z degQ2 z p (where p is a nonnegative integer) and a1, a2 a2 / 0 are two distinct complex numbers. Let f z be a nonconstant entire function, and f z Q1 z ⇒ f ′ z Q1 z and f ′ z Q2 z ⇒ f z Q2 z , then f z is of finite order. 2. Some Lemmas In order to prove our theorems, we need the following lemmas. Let h be a meromorphic function in C. h is called a normal function if there exists a positive M such that h# z ≤ M for all z ∈ C, where h# z |h′ z | 1 |h z | 2.1 denotes the spherical derivative of h. Let F be a family of meromorphic functions in a domain D ⊂ C. We say that F is normal inD if every sequence {fn}n ⊆ F contains a subsequence which converges spherically and uniformly on compact subsets of D; see 9 . Normal families, in particular, of holomorphic functions often appear in operator theory on spaces of analytic functions; for example, see in 10, Lemma 3 and in 11, Lemma 4 . 4 Abstract and Applied Analysis Lemma 2.1 see 12 . Let F be a family of analytic functions in the unit disc Δ with the property that for each f z ∈ F, all zeros of f z have multiplicity at least k. Suppose that there exists a number A ≥ 1 such that |f k z | ≤ A whenever f z ∈ F and f z 0. If F is not normal in Δ, then for 0 ≤ α ≤ k, there exist 1 a number r ∈ 0, 1 , 2 a sequence of complex numbers zn, |zn| < r, 3 a sequence of functions fn ∈ F, and 4 a sequence of positive numbers ρn → 0 such that gn ξ ρ−α n fn zn ρnξ converges locally and uniformly (with respect to the spherical metric) to a nonconstant analytic function g ξ onC, and moreover, the zeros of g ξ are of multiplicity at least k, g# ξ ≤ g# 0 kA 1. Lemma 2.2 see 13 . A normal meromorphic function has order at most two. A normal entire function is of exponential type and thus has order at most one. Lemma 2.3 see 9, Marty’s criterion . A family F of meromorphic functions on a domain D is normal if and only if, for each compact subsetK ⊆ D, there exists a constantM such that f# z ≤ M for each f ∈ F and z ∈ K. Lemma 2.4 see 2 . Let f z be a meromorphic function, and let a1 z , a2 z , a3 z be three distinct meromorphic functions satisfying T r, ai S r, f , i 1, 2, 3. Then T ( r, f ) ≤ N ( r, 1 f − a1 ) N ( r, 1 f − a1 ) N ( r, 1 f − a3 ) S ( r, f ) . 2.2 Lemma 2.5 see 5 . Let F be a family of functions holomorphic on a domain D, and let a and b be two finite complex numbers such that b / a, 0. If for each f ∈ F, f z a ⇒ f ′ z a and f ′ z b ⇒ f z b, then F is normal in D. 3. Proof of Theorem 1.6 If Q1 ≡ 0, by degQ1 degQ2, we obtain p 0, a1 0, Q2 ≡ a2 a2 / 0 . From the conditions of Theorem 1.6, we obtain f z 0 ⇒ f ′ z 0 and f ′ z a2 ⇒ f z a2. By Lemmas 2.5 and 2.3 we obtain that f is a normal function in D. By Lemma 2.2 we obtain that f is a finite order function. If Q1 /≡ 0, by degQ1 degQ2 and a2 / 0, we obtain a1 / 0. Now we consider the function F f/Q1 − 1, and we distinguish two cases. Case 1. If there exists a constant M such that F# z ≤ M, by Lemmas 2.3 and 2.2, then F is of finite order. Hence f F 1 Q1 is of finite order as well. Case 2. If there does not exist a constantM such that F# z ≤ M, then there exists a sequence wn n such that wn → ∞ and F# wn → ∞ for n → ∞. Abstract and Applied Analysis 5and Applied Analysis 5 Since Q1 is a polynomial, there exists an r1 such that ∣ ∣ ∣ ∣ ∣ Q′ 1 z Q1 z ∣ ∣ ∣ ∣ ∣ ≤ 2p |z| ∀z ∈ C satisfying |z| ≥ r1. 3.1 Obviously, if z → ∞, then 2p/|z| → 0. Let r > r1, and D {z : |z| ≥ r}, then F is analytic in D. Without loss of generality, wemay assume |wn| ≥ r 1 for all n. We defineD1 {z : |z| < 1} and Fn z F wn z f wn z Q1 wn z − 1. 3.2 Let z ∈ D1 be fixed; from the above equality, if F wn z 0, then f wn z Q1 wn z . Noting that f Q1 ⇒ f ′ Q1, then we obtain the following: if n → ∞, then ∣F ′ n z ∣ ∣∣∣∣ ( f wn z Q1 wn z )∣∣∣∣ ∣∣∣∣ f ′ wn z Q1 wn z − f wn z Q1 wn z Q′ 1 wn z Q1 wn z ∣∣∣∣ ≤ ∣∣∣∣ f ′ wn z Q1 wn z ∣∣∣∣ ∣∣∣∣ f wn z Q1 wn z ∣∣∣∣ ∣∣∣∣ Q′ 1 wn z Q1 wn z ∣∣∣∣ < 2. 3.3 Obviously, Fn z are analytic in D1 and F# n 0 F # wn → ∞ as n → ∞. It follows from Lemma 2.3 that Fn n is not normal at z 0. Therefore, we can apply Lemma 2.1, with α k 1 andA 2 . Choosing an appropriate subsequence of Fn n if necessary, we may assume that there exist sequences zn n and ρn n, such that zn → 0 and ρn → 0 and such that the sequence gn n defined by gn ξ ρ−1 n Fn ( zn ρnξ ) ρ−1 n { f ( wn zn ρnξ ) Q1 ( wn zn ρnξ ) − 1 } −→ g ξ 3.4 converges locally and uniformly in C where g ξ is a nonconstant analytic function and g# ξ ≤ g# 0 A 1 3. By lemma 2.2, the order of g ξ is at most 1. First, we will prove that g 0 ⇒ g ′ 1 on C. Suppose that there exists a point ξ0 such that g ξ0 0. Then by Hurwitz’s theorem, there exist ξn, ξn → ξ0 as n → ∞ such that for n sufficiently large gn ξn ρ−1 n { f ( wn zn ρnξn ) Q1 ( wn zn ρnξn ) − 1 } 0. 3.5 This implies f wn zn ρnξn Q1 wn zn ρnξn . From the above, we obtain g ′ n ξ f ′ ( wn zn ρnξ ) Q1 ( wn zn ρnξ ) − f ( wn zn ρnξ ) Q1 ( wn zn ρnξ ) Q′ 1 ( wn zn ρnξ ) Q1 ( wn zn ρnξ ) . 3.6 6 Abstract and Applied Analysis Let Gn ξ f ′ wn zn ρnξ /Q1 wn zn ρnξ , by 3.1 , 3.3 and 3.4 , it is easy to obtain limn→∞Gn ξ limn→∞g ′ n ξ g ′ ξ . Noting that f Q1 ⇒ f ′ Q1, we have Gn ξn f ′ ( wn zn ρnξn ) Q1 ( wn zn ρnξn ) 1 n → ∞ 3.7