The Normal Family of Meromorphic Functions Concerning Shared Analytic Function

YANG Qi,YUAN Wen-jun2,TIAN Hong-gen

(1.School of Mathematics Science,Xinjiang Normal University,Urumqi 830054,China;2.South China Institute of Software Engineering,Guangzhou University,Guangzhou 510990,China)

Abstract:In this paper,we study the normal criterion of meromorphic functions concerning shared analytic function.We get some theorems concerning shared analytic function,which improves some earlier related results.

Keywords:Meromorphic function;Shared function;Normal family

§1.Introduction and main results

LetFbe a meromorphic function in C,andDbe a domain in C.Fis said to be normal inDif any sequence{fn}⊂Fcontains a subsequencefnjsuch thatfnjconverges spherically locally uniformly inD,to a meromorphic function or∞[1,6-8].

Letf(z)andg(z)denote two meromorphic functions inD,iff(z)−ψ(z)andg(z)−ψ(z)have the same zeros(or ignoring multiplicity),then we sayf(z)andg(z)shareψ(z)CM(or IM).

In 2009,D.W.Meng and P.C.Hu[2]proved:

Theorem A.Take positive integers n and k≥2,and take a non-zero complex number a.Let F be a family of meromorphic functions in the plane domain D such that each f∈F has only zeros of multiplicity at least k.For each pair(f,g)∈F,if f(f(k))n and g(g(k))n share a IM,then F is normal in D.

Lately,Y.B.Jiang and Z.S.Gao[3]extended Theorem A as follows:

Theorem B.Let m≥0,n≥2m+2,k≥2be three positive integers and m be divisible by n+1.Suppose that a(z)(/≡0)is a holomorphic function with zeros of multiplicity m in a domain D.Let F be a family of meromorphic functions in a domain D,for each f∈F,f has only zeros of multiplicitymax{k+m,2m+2}at least.For each pair(f,g)∈F,f(f(k))n and g(g(k))n share a(z)IM,then F is normal in D.

Recently,Q.Yang[9]improved Theorem B as follows:

Theorem C.Let F be a family of meromorphic functions defined in a domain D,and m≥0,n≥2m+2,k≥2,d≥1,p≥1be five integers and m be divisible by n+d.Letψ(z)/≡0be ananalytic function with zeros of multiplicitymin a domain D.For every f∈F,f has only zeroswith multiplicity at least p≥max{k+2m+2}.If f d(f(k))n and gd(g(k))n shareψ(z)in Dfor every pair of function f,g∈F,then F is normal in D.

A natural question is:Whether the conditionn≥2m+2 in Theorem C can be reduced.In this paper,we answer this question and prove the following theorem.

Theorem 1.1.Let F be a family of meromorphic functions defined in a domain D,and m≥0,d≥2,n≥2,k≥1be four integers and m be divisible by n+d.Letψ(z)/≡0be an analytic function with zeros of multiplicity m in a domain D.For every f∈F,f has only zeros withmultiplicity at least k++1.If f d(f(k))n andgd(g(k))n shareψ(z)in D for every pair of function(f,g)∈F,then F is normal in D.

In 2005,Lahiri[4]proved the following normality theorem.

Theorem D.Let F be a family of meromorphic functions in a complex domain D.Let a,b∈Cand a/=0.Define

If there exists a positive constant M such that|f(z)|≥M for al l f∈F whenever z∈Ef,then F is normal in D.

Recently,Y.B.Jiang and Z.S.Gao[3]proved the following Lahiri type normality theorem.

Theorem E.m,a(z)are supposed as in Theorem B.Let n≥2m+2,k be two positive integers.Let F be a family of meromorphic functions in a domain D,for each f∈F,f has only zeros of multiplicity at leastmax{k+m,2m+2}and poles of multiplicity at least 2.Define

If there exists a positive constant M such that|f(z)|≥M for al l f∈F whenever z∈Ef,then F is normal in D.

In this paper,we prove the following Lahiri type normality theorem.

Theorem 1.2.Let F be a family of meromorphic functions defined in a domain D,and m≥0,d≥2,n≥2,k≥1be four integers and m be divisible by n+d.Letψ(z)/≡0be an analyticfunction with zeros of multiplicity m in a domain D.For every f∈F,f has only zeros with multiplicity at least k++1.Define

If there exists a positive constant M such that|f(z)|≥M for al l f∈F whenever z∈Ef,then F is normal in D.

Theorem 1.3.Let F be a family of meromorphic functions defined in a domain D,and m≥0,d≥2,n≥2,k≥1be four integers and m be divisible by n+d.Letψ(z)/≡0be an analyticfunction with zeros of multiplicity m in a domain D.For every f∈F,f has only zeros with multiplicity at least k++1.Define

If there exists a complex number b/=0such that f(z)=b for al l f∈F whenever z∈Ef,then F is normal in D.

The conditionψ(z)/≡0 in Theorem 1.1,1.2 and 1.3 is necessary.This fact can be illustrated by the following example:

Example 1.1.Let D={z∈C||z|<1}.Letψ(z)≡0and

Example 1.2.Let D={z∈C||z|<1}.Letψ(z)=1and

§2.Some lemmas

In order to prove our theorems,we require the following results.

Lemma 2.1.[10]Let F be a family of meromorphic functions on the unit discΔsatisfying al l zeros of functions in F have multiplicity≥p and al l poles of functions in F have multiplicity≥q.Letαbe a real number satisfying−p<α

(i)points zn∈Δ,zn→z0,

(ii)positive numbersρn,ρn→0,

(iii)functions fn∈F,

such thatspherical ly uniformly on each compact subset ofC,where g(ζ)is a a nonconstant meromorphic function satisfying the zeros of g are of multiplicities≥p and the poles of g are of multiplicities≥q.Moreover,the order of g is at most 2.If g is holomorphic,then g is of exponential type and the order of g is at most 1.

The proof method is similar to the Lemma 2.2 in reference[9],we can get the following result.

whereBis a non-zero constant andlis a positive integer.

Here,we distingguish two subcases.

Subcase 1.1.m≥l.

Sincem≥l,the expression(2.5)together with(2.7)implies that

Therefore,we haveN

Differentiating both sides of(2.7),we obtain

whereg3(z)is a polynomial with deg(g3)≤(m+1)t−(m−l+1).

From(2.6)and(2.8),we get

Subcase 1.2.m

Differentiating both sides of(2.7),we have

whereg4(z)is a polynomial with deg(g4)≤(m+1)t.

Differentiating both sides of(2.7)step by step formtimes.We can getz0is a zero of(f d(f(k))n)(m)=H(m),asH(m)=am/=0,thenz0/=αi(i=1,2,···,s).Hence(z−z0)l−m−1is a factor of the polynomialg2in(2.6).By(2.6)and(2.10),we conclude that

In the following,we will discuss in two subcases.

Subcase 1.2.1.l/=(n+d)N+nk t+m.

Then(2.5)and(2.7)implies

Subcase 1.2.2.l=(n+d)N+nk t+m.

We further distinguish the following two subcases.

(i)IfN

(ii)IfN≥M,it follows from(2.6)and(2.10),we obtain

which is a contradiction.

Case 2.Iff d(f(k))n−H(z)has no zero.Thenl=0 in(2.7).Proceeding as in the proof of case 1,we get a contradiction.Lemma 2.3 is proved.

From the references[5]and[11],we can get the following result.

Lemma 2.4.Suppose that f(z)is a transcendental meromorphic function,n,k,d be threepositive integers.When k≥1,d≥1,n≥2,then f d(f(k))n−φ(z)has infinitely many zeros,whereφ(z)/≡0,T(r,φ)=S(r,f).

§3.Proof of theorems

3.1.The proof of Theorem 1.1

Proof.For any pointz0∈D,eitherψ(z0)=0 orψ(z0)/=0.

Case 1.Whenψ(z0)=0.We may assumez0=0 andψ(z)=z m+am+1z m+1+···=z m h(z),wheream+1,am+2···are constants,andh(0)=1,andmcan be divisible byn+d.

locally uniformly on compact subsets of C,whereg(ξ)is a non-constant meromorphic function in C,g(ξ)has order at most 2.Here we distinguish the following two cases.

Similar to the proof of Case 1.1,we get a contradiction.Then,F1is normal at 0.

From Case 1.1 and Case 1.2,we knowF1is normal at 0,there existsΔ={z:|z|<ρ}and a subsequence ofFj,we may still denote it asFj,such thatFjconverges spherically locally uniformly to a meromorphic functionF(z)or∞inΔ.

Next,we distinguish two cases,

spherically locally uniformly in C disjoint from the poles ofg.Hence by Lemmas 2.2,2.3 and 2.4,similar to the proof of Case 1.2,we get a contradiction.ThusFis normal atz0.

3.2.The proof of Theorem 1.2 and Theorem 1.3

Proof.For any pointz0∈D,eitherψ(z0)=0 orψ(z0)/=0.

Case 1.Whenψ(z0)=0.We may assumez0=0.