#### Abstract

Let be a family of meromorphic functions defined in , let , be holomorphic functions in , and let be a positive integer. Suppose that, for every function , , and, for every pair functions , , share , then is normal in .

#### 1. Introduction and Main Results

Let be complex plane. Let be a domain in . Let be a family meromorphic functions defined in the domain . is said to be normal in , in the sense of Montel, if for any sequence , there exists a subsequence such that converges spherically locally uniformly in , to a meromorphic function or .

Let and be two meromorphic functions, let be a finite complex number. If and have the same zeros, then we say they share or share IM (ignoring multiplicity) (see [1–3]).

*Definition 1.1. *Let be analytic in , let be nonnegative integers, set
where is called a differential monomial of , the degree of , and the weight of .

From Definition 1.1, we give Definition 1.2.

*Definition 1.2. *Let be differential monomials of . Set
where is called the differential polynomial of , the degree of , and the weight of ,
In 1979, Gu [4] proved the following result.

Theorem A. *Let be a family of meromorphic functions defined in , let be a positive integer, and let be a nonzero constant. If, for each function , , in , then is normal in .*

Yang [5] and Schwick [6] proved that Theorem A still holds if is replaced by a holomorphic function in Theorem A.

Xu [7] improved Theorem A by the ideas of shared values and obtained the following result.

Theorem B. *Let be a family of meromorphic functions defined in , let be a holomorphic functions and with only simple zeros in , and let be a positive integer. Suppose that, for every function , has all multiple poles and . If, for every pair of functions and , and share in , then is normal in .*

Recently, Xu [7] did not know whether the condition has only simple zero in and has all multiple poles are necessary or not in Theorem B.

In 2007, Fang and Chang considered the case in Theorem A. In this note, Fang and Chang [8] proved the following result.

Theorem C. *Let be a family of meromorphic functions defined in , and let be a positive integer, and let be a nonzero complex number. If, for each , , and the zeros of have multiplicity at least , then is normal in .*

*Remark 1.3. *The number is sharp, as is shown by the examples in [8].

In 2009, Xia and Xu [9] replaced the constant 1 by a function in Theorem C. They obtained the following result.

Theorem D. *Let be a family of meromorphic functions defined in , let , be holomorphic functions in , and let be a positive integer. Suppose that, for every function , , and all zeros of have multiplicity at least . If, for , has only zeros with multiplicities at most 2 and, for , has only simple zeros, then is normal in .*

It is natural to ask whether Theorem D can be improved by the ideas of shared values. In this paper, we investigate the problem and obtain the following results.

Theorem 1.4. *Let be a family of meromorphic functions defined in , let , be holomorphic functions in , and let be a positive integer. Suppose that, for every function , , and, for every pair functions , and share , then is normal in .*

By Theorem 1.4, we immediately deduce.

Corollary 1.5. *Let be a family of meromorphic functions defined in , let , be holomorphic functions in , and be a positive integer. Suppose that, for every function , , and for every pair functions , and share , then is normal in .*

*Remark 1.6. *By the ideas of sharing values, Theorem 1.4 and Corollary 1.5 yield the number can be omitted.

*Remark 1.7. *Obviously, Corollary 1.5 omitted the conditions with only simple zeros, and, for every function , has all multiple poles in Theorem D. But the condition for every function , is additional. Hence, Corollary 1.5 improves Theorem B in some sense.

The condition in Theorem 1.4 is necessary. For example, we consider the following families.

*Example 1.8. *, obviously, any satisfies , . For distinct positive integers , , , and share 0 IM. However, the families are not normal at .

*Remark 1.9. *Some ideas of this paper are based on [7, 9, 10].

#### 2. Preliminary Lemmas

In order to prove our theorems, we need the following lemmas.

The well-known Zalcman’s lemma is a very important tool in the study of normal families. It has also undergone various extensions and improvements. The following is one up-to-date local version, which is due to Pang and Zaclman [11].

Lemma 2.1 (see [11, 12]). *Let be a family of meromorphic functions in the unit disc with the property that, for each , all zeros are of multiplicity at least . Suppose that there exists a number such that whenever and . If is not normal in , then, for , there exist *(1)*a number ;*(2)*a sequence of complex numbers , ;*(3)*a sequence of functions ;*(4)*a sequence of positive numbers ;**such that converge locally uniformly (with respect to the spherical metric) to a nonconstant meromorphic function on , and, moreover, the zeros of are of multiplicity at least , . In particular, has order at most 2.*

Here, as usual, is the spherical derivative.

Lemma 2.2 (see [1]). *Let be a transcendental meromorphic function in , let be a integer, and let be a nonzero finite value, then or has infinite zeros.*

Lemma 2.3 (see [7]). * Let be a nonconstant rational function. Let be an integer, and let be a non-zero finite value. If , then has at least two distinct zeros in the plane.*

Lemma 2.4. *Let be a nonconstant rational function. Let be an integer, and let be a positive integer. If , , then has at least two distinct zeros in the plane.*

*Proof. *Since and , then is a nonpolynomial rational function and has the form
where is a constant, and are positive integers. Set . Then,
where are constants. For , by mathematical induction, we have
where are constants. Since , we deduce that , and thus
*Case 1 (if has exactly one zero ). *From (2.5), we set
Obviously, is a nonzero constant and .

From (2.6), we obtain
where . By (2.4), we deduce
where is nonzero constant.

Comparing (2.7) and (2.8), we obtain that is impossible.*Case 2 (if ). *By (2.5), clearly Case 2 is impossible.

Lemma 2.4 is proved.

Lemma 2.5 (see [7]). *Let be a family of meromorphic functions defined in , let be a positive integer, and let be a holomorphic function in . If, for any satisfying and if share IM for every pair of functions , then is normal in .*

In this paper, by the same method of [7], we consider the differential polynomial in Lemma 2.5 and prove a more general result.

Lemma 2.6. *Let be a family of meromorphic functions defined in , let be a positive integer, and let be a holomorphic function in . If, for any satisfying and if share IM for every pair of functions , where is a differential polynomial of as the definition 1 satisfying , and , then is normal in , where , are as in Definitions 1.1 and 1.2.*

*Proof. *We may assume that . Suppose that is not normal in . Without loss of generality, we assume that is not normal at . Then, by Lemma 2.1, there exists a number ; a sequence of complex numbers , ; a sequence of functions ; a sequence of positive numbers such that converges uniformly with respect to the spherical metric to a nonconstant meromorphic functions in . Moreover, is of order at most 2. Hurwitz’s theorem implies that .

We have
Considering is analytic on , we have
for sufficiently large .

Hence, we deduce from that
converges uniformly to 0 on every compact subset of which contains no poles of .

Thus, we have
on every compact subset of which contains no poles of .

Next, we will prove that has just a unique zero. By way of contradiction, let and be two distinct solutions of , and choose small enough such that where and . By Hurwitz’s theorem, there exist points , such that, for sufficiently large ,

By the hypothesis that for each pair of functions and in , and share in , we know that, for any positive integer ,
Fix , take , and note , then
Since the zeros of have no accumulation point, so , .

Hence,
This contradicts with , , and . So has just a unique zero. By Hurwitz’s theorem, we know has just a unique zero.

By Lemmas 2.2 and 2.3, we know has at least two distinct zeros. From the definition of , we deduce that has more than two distinct zeros, a contradiction.

So is normal in . Lemma 2.6 is proved.

By Lemma 2.6, we immediately deduce the following lemma.

Lemma 2.7. *Let be a family of meromorphic functions defined in , let , be holomorphic functions in , and let be a positive integer. Suppose that, for every function , , and, for every pair functions , , share , then is normal in .*

Lemma 2.8 (see [1]). * Let be a meromorphic function. Let be a positive integer. If , then , then is a constant.*

Lemma 2.9 (see [13, 14]). * Let be a transcendental meromorphic function in , and let be a polynomial. Let be a positive integer. If all zeros (except at most finite zeros) of have the multiplicity at least 3, then has infinite zeros.*

#### 3. Proof of Theorem 1.4

*Proof. *Since normality is a local property, without loss of generality, we may assume , and
where is a positive integer, , on . By Lemma 2.6, we only need to prove that is normal at .

If , , then there exists such that on . By condition of Theorem, for every , we know on . By theorem D, is normal on , so is normal on .

Now, we consider . Suppose on the neighborhood (where is a small positive number) (otherwise, on the neighborhood , by condition of theorem, for every , we also obtain . So . By Theorem D, is normal at . So Theorem 1.4 is proved), there exists such that on (). So, for every , we obtain
By Theorem D, is normal on .

Next, we will prove is normal at . Suppose, on the contrary, that is not normal at , then there exists a sequence functions (we also denote ) that has no any normal subsequence on .

Consider the family . Since for , we have that for each .

We first prove that is normal in . Suppose, on the contrary, that is not normal at . By Lemma 2.1, there exist a sequence of functions , a sequence of complex numbers , and a sequence of positive numbers , such that
converges spherically uniformly on compact subsets of where is a nonconstant meromorphic function on , and .

We distinguish two cases.*Case 1 (). *By a simple calculation, for , we have
where if , for and .

Thus, from (3.4), we have
On the other hand, we have
for . Noting that is locally bounded on minus the set of poles of since . Therefore, on every subset of which contains no poles of , we have
for , and thus
since are analytic in .

By , we know . In fact, if , by Hurwitz’s theorem, then exists , for sufficiently large,
By the condition of theorem, for every positive number , we obtain . We know , and, for sufficiently large , . However, (otherwise, , so , a contradiction), so for sufficiently large , . This contradicts with (3.2).

So and , by Lemma 2.8, we obtain is a constant, a contradiction.*Case 2. * is a finite complex number. Then,
Obviously, , and is a pole of with order at least .

Set
Then,
Noting that , thus
uniformly on compact subsets of . Since has a pole of order at least at , we have , so that .

From (3.11), we get
spherically uniformly on compact subsets of minus the set of poles of . As the above, on every compact subset of minus the set of poles of , we have
locally uniformly on .

By the assumption of Theorem and (3.16), Hurwitz’s theorem implies .

Next, we proof that if , then .

First, , otherwise , which contradicts with . If there exists a such that , by Hurwitz’s theorem and (3.16), there exists such that . By the assumption of Theorem 1.4, for every positive such that . However, for sufficiently large, , all of these contradict with (3.2). So if , then .

Noting , By Lemma 2.9, we know must be a rational function. If is not a constant, By Lemma 2.4, we know has at least two distinct zeros, a contradiction. So must be a nonzero constant, also contradicts with . Now, we have proved the is normal on .

It remains to show that is normal at . Since is normal in , then the family is equicontinuous on with respect to the spherical distance. On the other hand, for each , so there exists such that for all and each . Suppose that is not normal at . Since is normal in , the family is normal in , but it is not normal at . Then, there exists a sequence which converges locally uniformly in , but not in . Noting that in , is holomorphic in for each . The maximum modulus principle implies that in . Thus, converges locally uniformly in , and hence so does , where . But for each , a contradiction. This finally completes the proof of Theorem 1.4.

#### Acknowledgments

This paper is supported by Leading Academic Discipline Project 10XKJ01, by Key Development Project 12C102 of Shanghai Dianji University, by the National Natural Science Foundation of China (11171184), and by National Natural Science Youth Fund Project (51008190).