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Research | Open | Published:

Some normality criteria for families of meromorphic functions

Abstract

In the note, we study the normality of families of meromorphic functions. We consider whether a family of meromorphic functions is normal in D if, for a normal family G and for each function fF, there exists gG such that f n f = a i implies g n g = a i for two distinct nonzero points a i , i=1,2 and an integer n. An example shows that the condition in our results is best possible.

MSC:30D45, 30D35.

1 Introduction and main results

Let f(z) and g(z) be two nonconstant meromorphic functions in a domain DC, and let a be a finite complex value. We say that f and g share a CM (or IM) in D provided that fa and ga have the same zeros counting (or ignoring) multiplicity in D. If g=a whenever f=a, we denote it by f=ag=a. When a=, the zeros of fa means the poles of f (see [1]). It is assumed that the reader is familiar with the standard notations and the basic results of Nevanlinna’s value-distribution theory (see [24] or [1]).

Bloch’s principle [5] states that every condition which reduces a meromorphic function in the plane to be a constant forces a family of meromorphic functions in a domain D to be normal. Although the principle is false in general (see [6]), many authors proved normality criterion for families of meromorphic functions corresponding to a Liouville-Picard type theorem (see [4]).

It is also more interesting to find normality criteria from the point of view of shared values. In this area, Schwick [7] first proved an interesting result which states that a family of meromorphic functions in a domain is normal if every function shares three distinct finite complex numbers with its first derivative therein. Later, more results about normality criteria concerning shared values have emerged; for instance, see [810]. In recent years, this subject has attracted the attention of many researchers worldwide. Studying normal families of functions is of considerable interest. For example, they are used in operator theory on spaces of analytic functions (see, for example, [11], [[12], Lemma 2.5], [[13], Lemma 2.2]).

We now first introduce a normality criterion related to a Hayman normal conjecture [14].

Theorem 1.1 Let be a family of meromorphic functions on D and nN. If each function f(z) of family satisfies f n (z) f (z)1, then is normal in D.

The proof of Theorem 1.1 is due to Gu [15] for n3, Pang [16] for n=2, Chen and Fang [17] for n=1. In 2004, by the ideas of shared values, Fang and Zalcman [18] obtained the following.

Theorem 1.2 Let be a family of meromorphic functions on D and nN. If for each pair of functions f and g in , f and g share the value 0 and f n f and g n g share a nonzero value a in D, then is normal in D.

In 2008, Zhang [10] obtained some criteria for normality of in terms of the multiplicities of the zeros and poles of the functions in and used them to improve Theorem 1.2 as follows.

Theorem 1.3 Let be a family of meromorphic functions in D satisfying that all of zeros and poles of fF have multiplicities at least 3. If for each pair of functions f and g in , f and g share a nonzero value a in D, then is normal in D.

Theorem 1.4 Let be a family of meromorphic functions on D and nN. If n2 and for each pair of functions f and g in , f n f and g n g share a nonzero value a in D, then is normal in D.

Zhang [10] gave the following example to show that Theorem 1.4 is not true when n=1, and therefore the condition n2 is best possible.

Example 1.1 The family of holomorphic functions F={ f j (z)= j (z+ 1 j ):j=1,2,} is not normal in D={z:|z|<1}. This is deduced by f j # (0)= j j j + 1 as j and Marty’s criterion [2], although for any f j (z)F, f j f j =jz+1. So, for each pair m, j, f m f m and f j f j share the value 1.

Here f # (ξ) denotes the spherical derivative

f # (ξ)= | f ( ξ ) | 1 + | f ( ξ ) | 2 .

Recently, Yuan et al. [19] improved Theorem 1.3 and used it to consider Theorem 1.4 when n=1. They proved the following theorems.

Theorem 1.5 Let be a family of meromorphic functions on D such that all of zeroes of each fF have multiplicities at least 4 and all of poles of fF are multiple. If for each pair of functions f and g in , f and g share a nonzero value a in D, then is normal in D.

Theorem 1.6 Let be a family of meromorphic functions on D such that all of zeroes of each fF are multiple. If for each pair of functions f and g in , f f and g g share a nonzero value a in D, then is normal in D.

Remark 1.7 Example 1.1 shows that the condition that all of zeros of fF are multiple in Theorem 1.6 is best possible.

In this paper, we will consider the related problems concerning two families. Our main results are as follows.

Theorem 1.8 Let and G be two families of meromorphic functions in DC, k be a positive integer and a i (i=1,2) be two distinct nonzero constants. Suppose that for each function fF, all its zeros are of multiplicity at least k+1 and all its poles are multiple.

If G is normal and for each function fF, there exists gG such that f ( k ) = a i g ( k ) = a i for i=1,2, then is normal in D.

Theorem 1.9 Let and G be two families of meromorphic functions on D and a i (i=1,2) be two distinct nonzero constants. If all of zeroes of fF have multiplicities at least 3, G is normal and for each function fF, there exists gG such that f = a i g = a i for i=1,2, then is normal in D.

Example 1.2 The family of meromorphic functions F={ f j (z)= 1 j z :j=1,2,} is not normal in D={z:|z|<1}. This is deduced by f j # (0)=j as j and Marty’s criterion [2]. The family of holomorphic functions G={ g j (z):j=1,2,} is normal in D, where

g j (z)= z 2 k + 1 ( 2 k + 1 ) ( 2 k ) ( k + 2 ) + ( 1 ( 1 ) k k ! j ) z k k ! +3.

However, for any f j (z)F and g j (z)G, f j (z)0 and g j (z)0 and f j ( k ) (z)=1 if and only if g j ( k ) (z)=1 in D.

Noting that normality of families of and F ={ 1 f |fF} is the same by famous Marty’s criterion, we obtain the following criteria.

Theorem 1.10 Let and G be two families of meromorphic functions on D and a i (i=1,2) be two distinct nonzero constants. Suppose that n (n0, −1 and −2) is an integer number. If G is normal and for each function fF, there exists gG such that f n f = a i g n g = a i for i=1,2, then is normal in D.

Theorem 1.11 Let and G be two families of meromorphic functions on D and a i (i=1,2) be two distinct nonzero constants. If all of poles of fF have multiplicities at least 3, G is normal and for each function fF, there exists gG such that f 2 f = a i g 2 g = a i for i=1,2, then is normal in D.

Remark 1.12 Example 1.2 shows that the condition in Theorem 1.8 and Theorem 1.9 that for each fF, there exists gG such that f ( k ) = a i g ( k ) = a i for a i 0, i=1,2, is best possible.

2 Preliminary lemmas

In order to prove our result, we need the following lemmas. The first one is an extension of a result by Zalcman in [20] concerning normal families.

Lemma 2.1 [21]

Let be a family of meromorphic functions on the unit disc such that all zeros of functions in have multiplicity greater than p and all poles of functions in have multiplicity greater than q. Let α be a real number satisfying q<α<p. Then is not normal at 0 if and only if there exist

  1. (a)

    a number 0<r<1;

  2. (b)

    points z n with | z n |<r;

  3. (c)

    functions f n F;

  4. (d)

    positive numbers ρ n 0

such that g n (ζ):= ρ α f n ( z n + ρ n ζ) converges spherically uniformly on each compact subset of to a non-constant meromorphic function g(ζ), its all zeros are of multiplicity greater than p and its all poles are of multiplicity greater than q and its order is at most 2.

Remark 2.2 If is a family of holomorphic functions on the unit disc in Lemma 2.1, then g(ζ) is a nonconstant entire function whose order is at most 1.

The order of g is defined by using the Nevanlinna’s characteristic function T(r,g):

ρ(g)= lim r sup log T ( r , g ) log r .

Lemma 2.3 ([22] or [23])

Let f(z) be a meromorphic function of finite order in the plane, k be a positive integer and aC{0}. Suppose that all its zeros are of multiplicity at least k+1 and all its poles are multiple. If f ( k ) (z)a, then f(z) is constant.

Lemma 2.4 [24]

Let f be a meromorphic function of finite order in the plane and aC{0}. If all its zeros are of multiplicity at least 3 and f (z)a, then f(z) is a constant.

3 Proofs of the results

Proof of Theorem 1.8 Suppose that is not normal in D. Then there exists at least one point z 0 such that is not normal at the point z 0 . Without loss of generality, we assume that z 0 =0. By Lemma 2.1, there exist points z j 0, positive numbers ρ j 0 and functions f j F such that

F j (ξ)= ρ j k f j ( z j + ρ j ξ)F(ξ),
(3.1)

locally uniformly with respect to the spherical metric, where F is a non-constant meromorphic function in satisfying all of whose zeros are of multiplicity at least k+1 and all of whose poles are multiple. Moreover, the order of F is less than two.

From (3.1) we know

F j ( k ) (ξ)= f j ( k ) ( z j + ρ j ξ) F ( k ) (ξ)
(3.2)

also locally uniformly with respect to the spherical metric.

Combining with Lemma 2.3, we know that F ( k ) (ξ) takes two distinct nonzero finite values { a 1 , a 2 }. Set ξ 0 and ξ 0 to be two zeros of F ( k ) a 1 and F ( k ) a 2 , respectively. Obviously, ξ 0 ξ 0 , and then choose δ (>0) small enough such that D( ξ 0 ,δ)D( ξ 0 ,δ)=ϕ, where D( ξ 0 ,δ)={ξ:|ξ ξ 0 |<δ} and D( ξ 0 ,δ)={ξ:|ξ ξ 0 |<δ}. From (3.2), by Hurwitz’s theorem, there exist points ξ j D( ξ 0 ,δ), ξ j D( ξ 0 ,δ) such that for sufficiently large j,

f j ( k ) ( z j + ρ j ξ j ) a 1 =0, f j ( k ) ( z j + ρ j ξ j ) a 2 =0.
(3.3)

It follows from (3.3) and the hypothesis of Theorem 1.8 that

g j ( k ) ( z j + ρ j ξ j )= a 1 , g j ( k ) ( z j + ρ j ξ j ) = a 2 .
(3.4)

Since G is normal, without loss of generality, we assume that

g j (z)g(z),
(3.5)

locally uniformly with respect to the spherical metric.

By (3.4), (3.5) and taking j, we have

0<| a 1 a 2 |= | g j ( k ) ( z j + ρ j ξ j ) g j ( k ) ( z j + ρ j ξ j ) | | g ( k ) ( 0 ) g ( k ) ( 0 ) | .

It is a contradiction.

The proof of Theorem 1.8 is complete. □

Proof of Theorem 1.9 By Lemma 2.4, similar to the proof of Theorem 1.8, we can give the proof of Theorem 1.9. We omit the details.

The proof of Theorem 1.9 is complete. □

Proof of Theorem 1.10 and Theorem 1.11 Set F ={ f n + 1 n + 1 |fF}, G ={ g n + 1 n + 1 |gG}.

Noting that for each function f F , all of whose zeros and poles are multiple if n{2,1,0}, and all of whose zeros have multiplicities at least 3 if n=2. For each f F , there exists g in G such that ( f ) = a i ( g ) = a i for i=1,2 in D, we know that F is normal in D by Theorem 1.8 and Theorem 1.9. Therefore, is normal in D.

The proof of Theorem 1.10 and Theorem 1.11 is complete. □

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Acknowledgements

The authors would like to express their hearty thanks to Professors Fang Mingliang, Liao Liangwen and Yang Degui for their helpful discussions and suggestions. This work was completed with the support of the NSF of China (No. 11271090), the NSF of Guangdong Province (S2012010010121) and the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University. The first author would like to express his hearty thanks to Chern Institute of Mathematics for providing him with very comfortable research environment. The authors wish to thank the referees and editors for their very helpful comments and useful suggestions.

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Correspondence to Jianming Lin.

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The authors declare that they have no competing interests.

Authors’ contributions

WY and JL carried out the design of the study and performed the analysis. WX participated in its design and coordination. All authors read and approved the final manuscript.

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Keywords

  • holomorphic function
  • normal family
  • meromorphic function
  • shared value