Composite Holomorphic Functions and Normal Families

and Applied Analysis 3 Remark 1.5. In Theorem 1.2 setting α z constant zero and P z,w a polynomial in variable w that vanishes exactly on a finite set of holomorphic functions S, we obtain Corollary 1.6 which generalizes the famous Montel’s criterion that a holomorphic family omitting 2 or more values is normal. Corollary 1.6. Let F be a family of holomorphic functions on a domain D. Let S be a finite set of holomorphic functions with at least 2 elements. If all functions in F share the set S ignoring multiplicities, that is, if for all f z , g z ∈ F and for all z ∈ D f z ∈ S ⇔ g z ∈ S, 1.3 then F is normal in D. In 2010, we 11 obtained a normal criterion as follows. Theorem C. Let α z be an analytic function, F a family of analytic functions in a domain D, and H z a transcendental entire function. If H ◦ f z and H ◦ g z share α z IM for each pair f z , g z ∈ F and one of the following conditions holds: 1 H z − α z0 has at least two distinct zeros for any z0 ∈ D; 2 α z is nonconstant and there exists z0 ∈ D such thatH z − α z0 : z − β0 Q z has only one distinct zero β0 and suppose that the multiplicities l and k of zeros of f z − β0 and α z − α z0 at z0, respectively, satisfy k / lp, for each f z ∈ F, where Q β0 / 0; 3 there exists a z0 ∈ D such that H z − α z0 has no zero and α z is nonconstant, then F is normal in D. However, there exists a gap in the proof of Theorem C which is Theorem 1.1 in our original paper 11 . We will give the correct proof after the proof of Theorem 1.1 in Section 3. 2. Preliminary Lemmas In order to prove our result, we need the following lemmas. The first one extends a famous result by Zalcman 12 concerning normal families. Lemma 2.1 see 13 . Let F be a family of meromorphic functions on the unit disc. Then F is not normal on the unit disc if and only if there exist a a number 0 < r < 1; b points zn with |zn| < r; c functions fn ∈ F; d positive numbers ρn → 0 such that gn ζ : fn zn ρnζ converges locally uniformly to a nonconstant meromorphic function g ζ , whose order is at most 2. Remark 2.2. If F is a family of holomorphic functions on the unit disc in Lemma 2.1, then g ζ is a nonconstant entire function. 4 Abstract and Applied Analysis Lemma 2.3 is very useful in the proof of our main theorem. In order to state them, we denote byU z0, r or U0 z0, r the open or punctured disc of radius r around z0, that is, U z0, r : {z ∈ C : |z − z0| < r}, U0 z0, r : {z ∈ C : 0 < |z − z0| < r}. 2.1 Lemma 2.3 see 9 or 14 . Let {fn z } be a family of analytic functions inU z0, r . Suppose that {fn z } is not normal at z0 but is normal in U0 z0, r . Then there exists a subsequence {fnk z } of {fn z } and a sequence of points {znk} tending to z0 such that fnk znk 0, but {fnk z } tending to infinity locally uniformly onU0 z0, r . 3. Proof of the Results Proof of Theorem 1.1. Without loss of generality, we assume that D {z ∈ C, |z| < 1}. Then we consider the following two cases. Case 1. P z0, z − α z0 has at least two distinct zeros a and b for any z0 ∈ D. Suppose that F is not normal inD. Without loss of generality, we assume that F is not normal at z 0. By Lemma 2.1, there exist zn → 0, fn ∈ F, ρn → 0 such that hn ξ fn ( zn ρnξ ) −→ h ξ 3.1 uniformly on any compact subset of C, where h ξ is a nonconstant entire function. Hence Pw ◦ fn ( zn ρnξ ) − αzn ρnξ ) −→ Pw ◦ h ξ − α 0 3.2 uniformly on any compact subset of C. We claim that Pw ◦ h ξ − α 0 has at least two distinct zeros. If h ξ is a nonconstant polynomial, then both of the two equations of h ξ a and h ξ b have roots. So Pw ◦ h ξ − α 0 has at least two distinct zeros. If h ξ is a transcendental entire function, then by Picard’s theorem 3 at least one of the two equations h ξ a or h ξ b has infinitely many zeros. Thus, the claim gives that there exist ξ1 and ξ2 such that Pw ◦ h ξ1 − α 0 0, Pw ◦ h ξ2 − α 0 0 ξ1 / ξ2 . 3.3 We choose a positive number δ small enough such thatD1∩D2 ∅ and Pw ◦h ξ −α 0 has no other zeros in D1 ∪D2 except for ξ1 and ξ2, where D1 {ξ ∈ C; |ξ − ξ1| < δ}, D2 {ξ ∈ C; |ξ − ξ2| < δ}. 3.4 Abstract and Applied Analysis 5 By 3.2 and Hurwitz’s theorem 14 , for sufficiently large n there exist points ξ1n ∈ D1, ξ2n ∈ D2 such that Pw ◦ fn ( zn ρnξ1n ) − αzn ρnξ1n ) 0, Pw ◦ fn ( zn ρnξ2n ) − αzn ρnξ2n ) 0. 3.5and Applied Analysis 5 By 3.2 and Hurwitz’s theorem 14 , for sufficiently large n there exist points ξ1n ∈ D1, ξ2n ∈ D2 such that Pw ◦ fn ( zn ρnξ1n ) − αzn ρnξ1n ) 0, Pw ◦ fn ( zn ρnξ2n ) − αzn ρnξ2n ) 0. 3.5 Noting that Pw ◦ fm z and Pw ◦ fn z share α z IM, it follows that Pw ◦ fm ( zn ρnξ1n ) − αzn ρnξ1n ) 0, Pw ◦ fm ( zn ρnξ2n ) − αzn ρnξ2n ) 0. 3.6 Taking n → ∞, we obtain Pw ◦ fm 0 − α 0 0. 3.7 Since P z,w is a polynomial in variable w, we know that the zeros of