# Normal Criteria of Function Families Concerning Shared Values

Authors: Wenjun Yuan; Bing Zhu; Jianming Lin
Journal: Journal of Applied Mathematics
 (2011)

---

## Abstract

We study the normality of families of meromorphic functions concerning shared values. We consider whether a family of meromorphic functionsℱ is normal in D, if, for every pair of functions f and g in ℱ, f′−af−n and g′−ag−n share the value b, where a and b are two finite complex numbers such that a≠0, n is a positive integer. Some examples show that the conditions in our results are best possible.


---

## Body

## 1. Introduction and Main Results

Letf(z) and g(z) be two nonconstant meromorphic functions in a domain D⊆ℂ, and let a be a finite complex value. We say that f and g share a CM (or IM) in D provided that f-a and g-a have the same zeros counting (or ignoring) multiplicity in D. When a=∞, the zeros of f-a mean 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 ([2–4] or [1]).Bloch's principle [5] states that every condition which reduces a meromorphic function in the plane ℂ 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 Liouville-Picard type theorem (see [7] or [4]).It is also more interesting to find normality criteria from the point of view of shared values. In this area, Schwick [8] first proved an interesting result that a family of meromorphic functions in a domain is normal if every function shares three distinct finite complex numbers with its first derivative. And later, more results about normality criteria concerning shared values can be found, for instance, in [9–11] and so on. 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 [12].Theorem 1.1.
Letℱ be a family of holomorphic (meromorphic) functions defined in a domain  D,  n∈ℕ,a≠0,b∈ℂ. If  f′(z)+afn(z)-b≠0 for each function  f(z)∈ℱ and  n≥2(n≥3), then  ℱ is normal in  D.The results for the holomorphic case are due to Drasin [7] for n≥3, Pang [13] for n=3, Chen and Fang [14] for n=2, Ye [15] for n=2, and Chen and Gu [16] for the generalized result with a and b replaced by meromorphic functions. The results for the meromorphic case are due to Li [17], Li  [18] and Langley [19] for n≥5, Pang [13] for n=4, Chen and Fang [14] for n=3, and Zalcman [20] for n=3, obtained independently.Whenn=2 and ℱ is meromorphic, Theorem 1.1 is not valid in general. Fang and Yuan [21] gave an example to this, and moreover a result added other conditions below.Example 1.2.
The family of meromorphic functionsℱ={fj(z)=jz/(jz-1)2:j=1,2,…,} is not normal in D={z:|z|<1}. This is deduced by fj#(0)=j→∞,asj→∞ and Marty's criterion [2], although, for any fj(z)∈ℱ,fj′+fj2=j(jz-1)-4≠0.Heref#(ξ) denotes the spherical derivative
(1.1)f#(ξ)=|f′(ξ)|1+|f(ξ)|2.Theorem 1.3.
Letℱ be a family of meromorphic functions in a domain D,  and a≠0,b∈ℂ. If f′(z)+a(f(z))2-b≠0 and the poles of f(z) are of multiplicity ≥3 for each f(z)∈ℱ, then  ℱ is normal in D.In 2008, by the ideas of shared values, Zhang [11] proved the following.Theorem 1.4.
Letℱ be a family of meromorphic (holomorphic) functions in D,  n a positive integer, and a,  b two finite complex numbers such that a≠0.  If n≥4(n≥2) and, for every pair of functions f and g in ℱ,  f′-afn and g′-agn share the value b,  then ℱ is normal in D.Example 1.5 (see [11]).
The family of meromorphic functionsℱ={fj(z)=1/(j(z-(1/j))):j=1,2,…,} is not normal in D={z:|z|<1}. Obviously fj′-fj3=-z/(j(z-(1/j))3).So for each pair m,j,fj′-fj3andfm′-fm3 share the value 0 in D,but ℱ is not normal at the point z=0,since fj#(0)=2(j)3/(1+j)→∞,asj→∞.Remark 1.6.
Example1.5 shows that Theorem 1.4 is not valid when n=3, and the condition n=4 is best possible for meromorphic case.In this paper, we will consider the similar relations and prove the following results.Theorem 1.7.
Letℱ be a family of meromorphic functions in D,  n a positive integer, and a,  b two finite complex numbers such that a≠0. If  n≥2 and, for every pair of functions f and g in ℱ,  f′-af-n and g′-ag-n share the value b, then ℱ is normal in D.Example 1.8.
The family of holomorphic functionsℱ={fj(z)=j(z-(1/j)):j=1,2,…,} is not normal in D={z:|z|<1}. This is deduced by fj#(0)=jj/(j+1)→∞,asj→∞ and Marty's criterion [2], although, for any fj(z)∈ℱ,fj′+fj-1=jjz/(jz-1).Remark 1.9.
Example1.8 shows that the condition that added n≥2 in Theorem 1.7 is best possible. In Theorem 1.7 taking b=0 we get Corollary 1.10 obtained by Zhang [22].Corollary 1.10.
Letℱ be a family of meromorphic functions in D,  n≥2, and let a be a nonzero finite complex number. If, for every pair of functions f and g in ℱ,  fnf′ and gng′ share the value a,  then ℱ is normal in D.A natural problem is what conditions are added such that Theorem1.7 holds when n=1. Next we give an answer.Theorem 1.11.
Letℱ be a family of meromorphic functions in D,  and let a and b be two finite complex numbers such that a≠0.  Suppose that all of zeros are multiple for each f(z)∈ℱ.  If, for every pair of functions f and g in ℱ,  f′-af-1 and g′-ag-1 share the value b,  then ℱ is normal in D.Remark 1.12.
Example1.8 shows that the condition that all of zeros are multiple for each f(z)∈ℱ added in Theorem 1.7 is best possible. In Theorem 1.11 taking b=0 we get Corollary 1.13.Corollary 1.13.
Letℱ be a family of meromorphic functions in D, and let a be a nonzero finite complex number. Suppose that all of zeros are multiple for each f(z)∈ℱ.  If, for every pair of functions f and g in ℱ,  ff′ and gg′ share the value a,  then ℱ is normal in D.From the proof of Theorem1.7 we know that the following corollary holds.Corollary 1.14.
Letℱ be a family of meromorphic functions in D,  n be a positive integer and a,  b be two finite complex numbers such that a≠0.  If for each function f in ℱ,  f′-af-n≠b,  then ℱ is normal in D.


## 2. Preliminary Lemmas

In order to prove our result, we need the following lemmas. The first one extends a famous result by Zalcman [23] concerning normal families.Lemma 2.1 (see [24]).
Letℱ be a family of meromorphic functions on the unit disc satisfying all zeros of functions in ℱ that have multiplicity ≥p and all poles of functions in ℱ that have multiplicity ≥q.  Let α be a real number satisfying -q<α<p.  Then ℱ is not normal at 0 if and only if there exist
(a)
anumber0<r<1;(b)
pointsznwith|zn|<r;(c)
functionsfn∈ℱ;(d)
positivenumbersρn→0such that gn(ζ):=ρ-αfn(zn+ρnζ) converges spherically uniformly on each compact subset of ℂ to a nonconstant meromorphic function g(ζ), whose all zeros have multiplicity ≥p and all poles have multiplicity ≥q and order is at most 2.Remark 2.2.
Ifℱ 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 ofg is defined by using Nevanlinna's characteristic function T(r,g):
(2.1)ρ(g)=lim⁡r→∞sup⁡log⁡T(r,g)log⁡r.Lemma 2.3 (see [25] or [26]).
Letf(z) be a meromorphic function and c∈ℂ∖{0}. If f(z) has neither simple zero nor simple pole, and f′(z)≠c, then f(z) is constant.Lemma 2.4 (see [27]).
Letf(z) be a transcendental meromorphic function of finite order in ℂ and have no simple zero, then f′(z) assumes every nonzero finite value infinitely often.


## 3. Proof of the Results

Proof of Theorem1.7.
Suppose thatℱ is not normal in D. Then there exists at least one point z0 such that ℱ is not normal at the point z0. Without loss of generality we assume that z0=0. By Lemma 2.1, there exist points zj→0, positive numbers ρj→0, and functions fj∈ℱ such that
(3.1)gj(ξ)=ρj-1/(n+1)fj(zj+ρjξ)⟹g(ξ)
locally uniformly with respect to the spherical metric, where g is a nonconstant meromorphic function in ℂ. Moreover, the order of g is ≤2.
From (3.1) we know
(3.2)gj′(ξ)=ρjn/(n+1)fj′(zj+ρjξ)⟹g′(ξ),ρjn/(n+1)(fj′(zj+ρjξ)-afj-n(zj+ρjξ)-b)=gj′(ξ)-agj-n(ξ)-ρjn/(n+1)b⟹g′(ξ)-ag-n(ξ)
in ℂ∖S locally uniformly with respect to the spherical metric, where S is the set of all poles of g(ξ).
Ifg′gn-a≡0, then -1/(n+1)gn+1≡aξ+c, where c is a constant. This contradicts with g being a meromorphic function. So g′gn-a≢0.
Ifg′gn-a≠0, by Lemma 2.3, then g is also a constant which is a contradiction with g being a nonconstant. Hence, g′gn-a is a nonconstant meromorphic function and has at least one zero.
Next we prove thatg′gn-a has just a unique zero. On the contrary, let ξ0 and ξ0* be two distinct zeros of g′gn-a, and choose δ(>0) small enough such that D(ξ0,δ)∩D(ξ0*,δ)=ϕ, where D(ξ0,δ)={ξ:|ξ-ξ0|<δ} and D(ξ0*,δ)={ξ:|ξ-ξ0*|<δ}. From (3.2), by Hurwitz′s theorem, there exist points ξj∈D(ξ0,δ), ξj*∈D(ξ0*,δ) such that for sufficiently large j(3.3)fj′(zj+ρjξj)-afj-n(zj+ρjξj)-b=0,fj′(zj+ρjξj*)-afj-n(zj+ρjξj*)-b=0.
By the hypothesis that, for each pair of functionsf and g in ℱ, f′-af-n and g′-ag-n share b in D, we know that for any positive integerm(3.4)fm′(zj+ρjξj)-afm-n(zj+ρjξj)-b=0,fm′(zj+ρjξj*)-afm-n(zj+ρjξj*)-b=0.
Fixm, take j→∞, and note zj+ρjξj→0, zj+ρjξj*→0, then fm′(0)-afm-n(0)-b=0. Since the zeros of fm′-afm-n-b have no accumulation point, so
(3.5)zj+ρjξj=0,zj+ρjξj*=0.
Hence, ξj=-zj/ρj,ξj*=-zj/ρj. This contradicts with ξj∈D(ξ0,δ),ξj*∈D(ξ0*,δ), and D(ξ0,δ)∩D(ξ0*,δ)=ϕ. So g′gn-a has just a unique zero, which can be denoted by ξ0. By Lemma 2.4, g is not any transcendental function.
Ifg is a nonconstant polynomial, then g′gn-a=A(ξ-ξ0)l, where A is a nonzero constant, l is a positive integer, because l≥n≥3. Set ϕ=(1/(n+1))gn+1, then ϕ′=A(ξ-ξ0)l+a and ϕ′′=Al(ξ-ξ0)l-1. Note that the zeros of ϕ are of multiplicity ≥4. But ϕ′′ has only one zero ξ0, so ϕ has only the same zero ξ0 too. Hence, ϕ′(ξ0)=0 which contradicts with ϕ′(ξ0)=a≠0. Therefore, g and ϕ are rational functions which are not polynomials, and ϕ′-a has just a unique zero ξ0.
Next we prove that there exists no rational function such asϕ. Noting that ϕ=(1/(n+1))gn+1, we can set
(3.6)ϕ(ξ)=A(ξ-ξ1)m1(ξ-ξ2)m2⋯(ξ-ξs)ms(ξ-η1)n1(ξ-η2)n2⋯(ξ-ηt)nt,
where A is a nonzero constant, s≥1,t≥1,mi≥n+1≥3(i=1,2,…,s),nj≥n+1≥3(j=1,2,…,t). For stating briefly, denote
(3.7)m=m1+m2+⋯+ms≥3s,N=n1+n2+⋯+nt≥3t.
From (3.6),
(3.8)ϕ′(ξ)=A(ξ-ξ1)m1-1(ξ-ξ2)m2-1⋯(ξ-ξs)ms-1h(ξ)(ξ-η1)n1+1(ξ-η2)n2+1⋯(ξ-ηt)nt+1=p1(ξ)q1(ξ),
where
(3.9)h(ξ)=(m-N-t)ξs+t-1+as+t-2ξs+t-2+⋯+a0,p1(ξ)=A(ξ-ξ1)m1-1(ξ-ξ2)m2-1⋯(ξ-ξs)ms-1h(ξ),q1(ξ)=(ξ-η1)n1+1(ξ-η2)n2+1⋯(ξ-ηt)nt+1
are polynomials. Since ϕ′(ξ)+a has only a unique zero ξ0, set
(3.10)ϕ′(ξ)+a=B(ξ-ξ0)l(ξ-η1)n1+1(ξ-η2)n2+1⋯(ξ-ηt)nt+1,
where B is a nonzero constant, so
(3.11)ϕ′′(ξ)=(ξ-ξ0)l-1p2(ξ)(ξ-η1)n1+2(ξ-η2)n2+2⋯(ξ-ηt)nt+2,
where p2(ξ)=B(l-N-2t)ξt+bt-1ξt-1+⋯+b0 is a polynomial. From (3.8) we also have
(3.12)ϕ′′(ξ)=(ξ-ξ1)m1-2(ξ-ξ2)m2-2⋯(ξ-ξs)ms-2p3(ξ)(ξ-η1)n1+2(ξ-η2)n2+2⋯(ξ-ηt)nt+2,
where p3(ξ) is also a polynomial.
Letdeg(p) denote the degree of a polynomial p(ξ).
From (3.8) and (3.9),
(3.13)deg⁡(h)≤s+t-1,deg⁡(p1)≤m+t-1,deg⁡(q1)=N+t.
Similarly from (3.11), (3.12) and noting (3.13),
(3.14)deg⁡(p2)≤t,(3.15)deg⁡(p3)≤deg⁡(p1)+t-1-(m-2s)≤2t+2s-2.
Note thatmi≥3(i=1,2,…,s), it follows from (3.8) and (3.10) that ϕ′(ξi)=0(i=1,2,…,s) and ϕ′(ξ0)=a≠0. Thus, ξ0≠ξi(i=1,2,…,s), and then (ξ-ξ0)l-1 is a factor of p3(ξ). Hence, we get that l-1≤deg(p3). Combining (3.11) and (3.12) we also have m-2s=deg(p2)+l-1-deg(p3)≤deg(p2). By (3.14) we obtain
(3.16)m-2s≤deg⁡(p2)≤t.
Sincem≥3s, we know by (3.16) that
(3.17)s≤t.
Ifl≥N+t, by (3.15), then
(3.18)4t-1≤N+t-1≤l-1≤deg⁡(p3)≤2t+2s-2.
Noting (3.17), we obtain 1≤0; a contradiction.
Ifl<N+t, from (3.8) and (3.10), then deg(p1)=deg(q1). Noting that deg(h)≤s+t-1, deg(p1)≤m+t-1, and deg(q1)=N+t, hence m≥N+1≥3t+1. By (3.16), 2t+1≤2s. From (3.17), we obtain 1≤0; a contradiction.
The proof of Theorem1.7 is complete.Proof of Theorem1.11.
The proof of this theorem is the same as the proof of Theorem1.7, some different places are stated as follows.
The zeros ofg are multiple;
(3.19)l≥2n+1=3.
The zeros ofϕ are of multiplicity ≥4:
(3.20)mi≥2(n+1)=4(i=1,2,…,s),nj≥n+1=2(j=1,2,…,t);((3.7)′)m=m1+m2+⋯+ms≥4s,N=n1+n2+⋯+nt≥2t.
Notingm≥4s, by (3.16) we have
((3.17)′)2s≤t.
Ifl≥N+t, by (3.15), then
(3.21)3t-1≤N+t-1≤l-1≤deg⁡(p3)≤2t+2s-2.
Noting (3.17), we obtain 1≤0; a contradiction.
Ifl<N+t, from (3.8) and (3.10), then deg(p1)=deg(q1). Noting that deg(h)≤s+t-1, deg(p1)≤m+t-1, and deg(q1)=N+t, hence m≥N+1≥2t+1. By (3.16), 2t+1≤2s+t. From ((3.17)′), we obtain 1≤0; a contradiction.
The proof of Theorem1.11 is complete.



---

*Source: 790942-2011-11-14.xml*