#### Abstract

We study the normal families related to a Hayman conjecture of higher derivative and concerning shared values and get two normal criteria. Our results improve the related theorems which were obtained independently, respectively by Fang and Yuan (2001), Yuan et al. ((2011) and (2012)), Wang et al. (2011), and Qiu et al. (2012). Meanwhile, some examples are given to show the sharpness of our results.

#### 1. Introduction and Main Results

Let and be two nonconstant meromorphic functions in a domain , and let be a finite complex value. We say that and share CM (or IM) in provided that and have the same zeros counting (or ignoring) multiplicity in . When , the zeros of mean the poles of (see [1]). It is assumed that the reader is familiar with the standard notations and the basic results of Nevanlinna’s value-distribution theory ([1] or [2–4]).

It is very interesting to find normality criteria from the point of view of shared values. In this area, Schwick [5] first proved an interesting result that a family of meromorphic functions in a domain is normal in which every function shares three distinct finite complex numbers with their first derivative. And later, more results about normality criteria concerning shared values have emerged; for instance, see [6–8]. In recent years, this subject has attracted the attention of many researchers worldwide.

We now first introduce a normality criterion related to a Hayman normal conjecture [9].

Theorem 1. *Let be a family of holomorphic (meromorphic) functions defined in a domain, , , and . If does not vanish in for each function and , then is normal in . *

The results for the holomorphic case are due to Drasin [10] for , Pang [11] for , Chen and Fang [12] for , Ye [13] for , and Chen and Gu [14] for the generalized result with and replaced by meromorphic functions. The results for the meromorphic case are due to Li [15], Li [16], and Langley [17] for , Pang [11] for , Chen and Fang [12] for , and Zalcman [18] for , obtained independently.

When and is meromorphic, Theorem 1 is not valid in general. Fang and Yuan [19] gave an example to show this and got a special result below.

*Example 2. *The family of meromorphic functions is not normal in . This is deduced by , as and Marty’s criterion [2], although for any , .

Here denotes the spherical derivative

Theorem 3. *Let be a family of meromorphic functions in a domain , and , . If does not vanish in and the poles of are of multiplicity 3 for each , then is normal in . *

In 2008, by the ideas of shared values, Zhang [8] proved the following.

Theorem 4. *Let be a family of meromorphic (holomorphic) functions in , let be a positive integer and let , be two finite complex numbers such that . If and for every pair of functions and in , and share the value IM, then is normal in . *

*Example 5 (see [8]). *The family of meromorphic functions is not normal in . Obviously . So for each pair of , , and share the value in , but is not normal at the point , since , as .

*Remark 6. *Example 5 shows that Theorem 4 is not valid when , and the condition is best possible for meromorphic case.

In 2011, Yuan et al. [20] and Wang et al. [21] proved the following theorems, independently, respectively.

Theorem 7 (see [20, 21]). *Let be a family of meromorphic functions in and , two finite complex numbers such that . Suppose that each has no simple pole. If and share the value IM for every pair of functions and in , then is normal in . *

Theorem 8 (see [20]). *Let be a family of meromorphic functions in and and two finite complex numbers such that . Suppose that each admits zeros of multiple and the poles of multiplicity at least 3. If and share the value IM for every pair of functions and in , then is normal in . *

Lately, Yuan et al. [22] and Qiu et al. [23] studied this result, independently, respectively, in which the derivative was replaced by th derivative , and they got the following results.

Theorem 9 (see [22]). *Let be a family of meromorphic functions in , and let and be two positive integers. Let and be two finite complex numbers. If*(i)* and share IM in for every pair of functions and in ,*(ii)* has no simple pole and no zero of multiplicity less than in for every function , then is normal in .*

Theorem 10 (see [22, 23]). *Let be a family of meromorphic functions in , and be two positive integers. Let and be two finite complex numbers. If*(i)* and share IM in for every pair of functions and in ,*(ii)* has no zero of multiplicity less than in for every function , then is normal in .*

It is natural to ask whether the condition or in the previous theorems can be reduced. In this paper, we study this problem and get the following results.

Theorem 11 (main theorem). *Let be a domain in and let be a family of meromorphic functions in . Let and let , be two finite complex numbers with . Suppose that every has all its zeros of multiplicity at least and all its poles of multiplicity at least . If and share the value IM for every pair of functions of , then is a normal family in . *

*Remark 12. *When or , we have or , respectively. It follows that Theorem 11 generalizes Theorems 4, 7, 9, and 10.

Theorem 13 (main theorem). *Let be a domain in and let be a family of meromorphic functions in . Let and , be two finite complex numbers with . Suppose that every has all its zeros of multiplicity at least and all its poles of multiplicity at least . If and share the value IM for every pair of functions of , then is a normal family in .*

*Remark 14. *When , it follows that Theorem 13 generalizes Theorem 8.

*Example 15 (see [23]). *Let be two positive integers and a nonzero complex constant. The family of meromorphic functions is , . Obviously, for each pair of , , and share the value in , but is not normal.

*Example 16. *Let be a positive integer and two nonzero complex constants such that . The family of meromorphic functions is , . Obviously, for each pair of and share the value in , but is normal since as .

*Remark 17. *Example 15 shows that the condition that admits zeros of multiplicity at least is best in Theorem 11. For the case , , Example 5 shows that the condition that admits poles of multiplicity at least is sharp in Theorem 11. For the case , Example 16 shows that the condition that admits poles of multiplicity at least is sharp in Theorem 13. For the case , , Example 2 shows that the condition that admits zeros of multiplicity at least in Theorem 13 is sharp.

#### 2. Preliminary Lemmas

In order to prove our results, we need the following lemmas. The first is the extended version Zalcman’s [24] concerning normal families.

Lemma 1 (see [25]). *Let be a family of meromorphic functions on the unit disc satisfying all zeros of functions in having multiplicity ≥ p and all poles of functions in having multiplicity ≥q. Let be a real number satisfying . Then is not normal at if and only if there exist *(a)

*a number ;*(b)

*points ;*(c)

*functions ;*(d)

*positive numbers*

*such that converges spherically uniformly on each compact subset of to a nonconstant meromorphic function , whose all zeros have multiplicity ≥*

*p*and all poles have multiplicity ≥*q*and order is at most 2.Lemma 2. *Let be a meromorphic function such that and , with , . If all zeros of are of multiplicity at least and all poles of are of multiplicity at least , then
**
where , as , possibly outside a set with finite linear measure.*

*Proof. *Set

Since , we have . Thus,
Hence,
So that

On the other hand, (4) gives
where denotes the counting function of zeros of both and .

We obtain
By (4), we have
From (6)~(9), we obtain

Since all zeros and poles of are multiplicities at least and , we get
So that

Lemma 3. *Let be a transcendental meromorphic function such that . Let and . If all zeros of are of multiplicity at least and all poles of are of multiplicity at least , then has infinitely many zeros.*

*Proof. *Suppose that has only finitely many zeros; then . Clearly, an arbitrary zero of is a zero of since all zeros of are of multiplicity at least ; then we can deduce that has only finite zeros, so .

Set
Similarly, with the proof of Lemma 2, we can get

Since all poles of are multiplicities at least , we obtain
so that
This is contradicting with the fact that is transcendental.

Hence, Lemma 3 is proved completely.

Lemma 4. *Let be a nonconstant rational function such that . Let , and with and . If and all poles of are of multiplicity at least , then has at least two zeros.*

*Proof. *Suppose, to the contrary, that has at most one zero.

Since , we get that is a rational but not a polynomial.*Case 1.* If has only zero with multiplicity , set
where is a nonzero constant and .

For the sake of simplicity, we denote

From (17), we have
where is a polynomial such that .

From (17) and (19), we get

By the assumption that has exactly one zero with multiply , we have
where is a nonzero constant. Thus,

Differentiating (22), we obtain

For the sake of simplicity, we denote
Hence,

Since , we have . But , a contradiction.*Case 2.* If has no zeros, then for (21). We have
where is a nonzero constant. Thus,

Obviously, is not a constant, a contradiction.

Lemma 4 is proved.

Lemma 5. *Let be a rational function and , and with and . If all zeros of are of multiplicity at least and all poles of are of multiplicity at least , then has at least two distinct zeros.*

*Proof. *Suppose, to the contrary, that has at most one zero.*Case I.* When is a nonconstant polynomial, noting that all zeros of have multiplicity at least , we know that must have zeros. We claim that has exactly one zero. Otherwise, we can get that has at least two zeros, which contradicts our assumption.

Set
where , is a nonzero constant. Then

Since , we obtain that has at least one zero which is not from (29). Therefore, has at least two distinct zeros, a contradiction.*Case II.* When is rational but not a polynomial, we consider two cases.*Case 1.* Suppose that has only zero with multiplicity at least . If , by Lemma 4, we get a contradiction. So has zeros, and then we can deduce that is the only zero of . Otherwise, has at least two distinct zeros, a contradiction.

We set
where is a nonzero constant and , .

For the sake of simplicity, we denote

From (31), we have
where is a polynomial with .

From (30) and (32), we get

By assumption that has exactly one zero with multiplicity , we have
where is a nonzero constant. Thus,
*Case 1.1.* If , from (35), we can deduce that is a zero of , a contradiction.*Case 1.2.* If , from (35), it follows that

Differentiating (36), we have

For the sake of simplicity, we denote
Thus,

Since , we get , but , a contradiction.*Case 2.* If has no zeros, then for (34). Similarly, as the proof of Case 1, we also have a contradiction.

The proof is completed.

#### 3. Proofs of Theorems

*Proof. *In Theorem 11, suppose that is not normal in . Then there exists at least one point such that is not normal at the point . Without loss of generality, we assume that . By Lemma 1, there exist points , positive numbers , and functions such that
locally uniformly with respect to the spherical metric, where is a nonconstant meromorphic function in and whose poles and zeros are of multiplicity at least and , respectively. Moreover, the order of is at most 2.

From (40), we know that
also locally uniformly with respect to the spherical metric.*Case 1.* If , by Theorems 4 and 7, the Theorem 11 assumes. *Case 2.* If , , by Theorems 9 and 10, the Theorem 11 assumes. *Case 3.* If , or , (If , by Theorems 9 and 10, we can get Theorem 11).

If , since all poles of have multiplicity at least , we have
Therefore, is a constant, a contradiction. So .

By Lemma 2, we have
Then
If , then (46) gives that is also a constant. Hence, is a nonconstant meromorphic function and has at least one zero.

Next we prove that has just a unique zero. On the contrary, let and be two distinct zeros of , and choose small enough such that , where and . From (42), 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 , and . So has just a unique zero, which can be denoted by .

Noting that has poles and zeros of multiplicities at least and , respectively, (46) deduces that is a rational function with degree at most .

If is a polynomial, noting that and the multiplicities of zeros are at least , we have and . Hence, there exist and such that , and then has 6 distinct zeros, a contradiction.

Suppose that is not a polynomial, and . Then the multiplicities of poles of are at least , which implies that , a contradiction.

Suppose that is not a polynomial, and ; we distinguish two cases.*Case i.* If has zeros, since all zeros of have multiplicity at least , it follows that , a contradiction. *Case ii.* If , then (46) should be as follows

From (50), we can see that is a rational function with degree at most . Since all poles of have multiplicity at least , which gives that , a contradiction.

This completes the proof of Theorem 11.

*Proof. *In Theorem 13, suppose that is not normal in . Then there exists at least one point such that is not normal at the point . Without loss of generality, we assume that . By Lemma 1, there exist points , positive numbers , and functions such that
locally uniformly with respect to the spherical metric, where is a nonconstant meromorphic function in and whose poles and zeros are of multiplicity at least and , respectively. Moreover, the order of is at most 2.

From (51), we know
also locally uniformly with respect to the spherical metric.

If , since all poles of have multiplicity at least , we can deduce that is an entire function easily. Thus,
Therefore, is a constant, a contradiction. So . By Lemmas 3, 4, and 5, has at least two distinct zeros. Proceeding as in the later proof of Theorem 11, we will get a contradiction. The proof is completed.

Similarly, as the proof of Theorem 11, when is a holomorphic function, we can get the following theorem which has been gotten in [23].

Theorem 18 (see [23]). *Let be a domain in and let be a family of holomorphic functions in . Let , and let , be two finite complex numbers with . Suppose that every has all its zeros of multiplicity at least . If and share the value IM for every pair of functions of , then is a normal family in . *

#### Acknowledgments

This paper is supported by the Nature Science Foundation of China (11271090), Nature Science Foundation of Guangdong Province (S2012010010121), and Graduate Research and Innovation Projects of Xinjiang Province (XJGRI2013131). This work was supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University. The second author would like to express his hearty gratitude to Chern Institute of Mathematics which provided very comfortable research environments to him where he worked as a Visiting Scholar.