Normality Criteria of Lahiri’s Type and Their Applications

Journal of Inequalities and Applications Volume 2011, Article ID 873184,16pages doi:10.1155/2011/873184

Research Article

Normality Criteria of Lahiri’s Type and Their Applications

Xiao-Bin Zhang,

Jun-Feng Xu,

and Hong-Xun Yi

1Department of Mathematics, Shandong University, Jinan, Shandong 250100, China

2Department of Mathematics, Wuyi University, Jiangmen, Guangdong 529020, China

Correspondence should be addressed to Jun-Feng Xu,[email protected] Received 22 September 2010; Revised 9 January 2011; Accepted 9 February 2011 Academic Editor: Siegfried Carl

Copyrightq2011 Xiao-Bin Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove two normality criteria for families of some functions concerning Lahiri’s type, the results generalize those given by Charak and Rieppo, Xu and Cao. As applications, we study a problem related to R. Br ¨uck’s Conjecture and obtain a result that generalizes those given by Yang and Zhang, L ¨u, Xu and Chen.

1. Introduction and Main Results

Let denote the complex plane, and letfzbe a nonconstant meromorphic function in . It is assumed that the reader is familiar with the standard notion used in the Nevanlinna value distribution theory such as the characteristic functionTr, f, the proximity functionmr, f, the counting functionNr, f see, e.g.,1–4, andSr, fdenotes any quantity that satisfies the conditionSr, f oTr, fasr → ∞outside of a possible exceptional set of finite linear measure. A meromorphic functionazis called a small function with respect tofz, provided thatTr, a Sr, f.

Letfzandgzbe two nonconstant meromorphic functions. Letazand bzbe small functions offzandgz.fz az^ªgz bzmeansfz − azandgz − bzhave the same zeroscounting multiplicityandfz ∞^ªgz ∞means thatfand ghave the same polescounting multiplicity. Ifgz−bz 0 wheneverfz−az 0, we writefz az⇒gz bz. Iffz az⇒gz bzandgz bz⇒fz az, we writefz az⇔gz bz. Iffz az⇔gz az, then we say thatfandg sharea.

Set

P f

f^nn1···nk, M1

f, f, . . . , f^k fⁿ

f_n1

· · · f^knk

,

M2

f, f, . . . , f^k f^m

f_m1

· · · f^k_mk

,

γM1nn1· · ·nk, γM2mm1· · ·mk,

γ_M^∗

1^k−1

j1

n_j, ΓM1^k

j1

jn_j, γ_M^∗

2^k−1

j1

m_j, ΓM2^k

j1

jm_j,

1.1

where n, n1, . . . , nk, m, m1, . . . , mk are nonnegative integers. Mif, f, . . . , f^k is called the differential monomial offandγ_Mi is called the degree ofM_if, f, . . . , f^k i1, 2.

LetFbe a family of meromorphic functions defined in a domainD ⊂ .Fis said to be normal inD, in the sense of Montel, if for any sequencefn∈ F, there exists a subsequence f_nj such thatf_nj converges spherically locally uniformly inD, to a meromorphic function or

∞.

According to Bloch’s principle, every condition which reduces a meromorphic function in to a constant makes a family of meromorphic functions in a domainDnormal.

Although the principle is false in general, many authors proved normality criteria for families of meromorphic functions starting from Picard type theorems, for instance.

Theorem Asee5. Letn≥5 be an integer,a, b∈ anda /0. If, for a meromorphic functionf, fafⁿ/bfor allz∈ , thenfmust be a constant.

Theorem Bsee6,7. Letn≥3 be an integer,a, b∈ ,a /0, and letF, be a family of meromorphic functions in a domainD. Iffafⁿ/bfor allf∈ F, thenFis a normal family.

In 2005, Lahiri8got a normality criterion as follows.

Theorem C. LetFbe a family of meromorphic functions in a complex domainD. Leta, b ∈ such thata /0. Define

Ef

z∈D:fz a

fz b

. 1.2

If there exists a positive constantMsuch that|fz| ≥Mfor allf∈ Fwheneverz∈Ef, thenFis a normal family.

In 2009, Charak and Rieppo9generalized Theorem C and obtained two normality criteria of Lahiri’s type.

Theorem D. LetFbe a family of meromorphic functions in a complex domainD. Leta,b∈ such thata /0. Letm1,m2,n1,n2 be positive integers such thatm1n2 − m2n1 > 0,m1 m2 ≥ 1, n₁ n₂≥2, and put

E_f z∈D:

fzn1

fzm1 a

fz_n2

fz_m2 b

. 1.3

If there exists a positive constantMsuch that|fz| ≥Mfor all f∈ Fwheneverz∈E_f, thenFis a normal family.

Theorem E. LetFbe a family of meromorphic functions in a complex domainD. Leta,b∈ such thata / 0. Letm1,m2,n1,n2be nonnegative integers such thatm1n2m2n1, and put

Ef z∈D:

fz_n1

fz_m1

a fz_n2

fz_m2 b

. 1.4

If there exists a positive constantMsuch that|fz| ≥Mfor allf∈ Fwheneverz∈Ef, thenFis a normal family.

Very recently, Xu and Cao10 further extended Theorems D and E by replacingf withf^k; they got

Theorem F. LetFbe a family of meromorphic functions in a complex domainD, all of whose zeros have multiplicity at leastk. Leta, b∈ such thata / 0. Letm1,m2,n1,n2be nonnegative integers such thatm1n2−m2n1>0,m1m2≥1,n1n2 ≥2, (ifn1n21,k≥5), and put

Ef z∈D:

fz_n1

f^kz_m1

a fz_n2

f^kz_m2 b

. 1.5

If there exists a positive constantMsuch that|fz| ≥Mfor allf∈ Fwheneverz∈E_f, thenFis a normal family.

Theorem G. LetFbe a family of meromorphic functions in a complex domainD, all of whose zeros have multiplicity at leastk. Leta, b∈ such thata / 0. Letm₁≥2,m₂,n₁,n₂be positive integers such thatm1n2m2n1, and put

Ef z∈D:

fz_n1

f^kz_m1

a fzn2

f^kzm2 b

. 1.6

If there exists a positive constantMsuch that|fz| ≥Mfor allf∈ Fwheneverz∈Ef, thenFis a normal family.

To prove Theorems D–G, the authors used a key lemma Lemma 2.4in this paper besides Zalcman-Pang’s Lemma. It’s natural to ask whether such normality criteria of Lahiri’s

type still hold for the general differential monomialMf, f, . . . , f^k. We study this problem and obtain the following theorem.

Theorem 1.1. LetFbe a family of meromorphic functions in a complex domainD, for everyf ∈ F, all zeros off have multiplicity at least k. Let a, b ∈ such that a / 0, let m,n,k≥ 1,mj, nj j1,2, . . . , kbe nonnegative integers such that

γ_M2ΓM1−γ_M1ΓM2>0, n_km_k>0, mn≥2. 1.7

Put

Ef z∈D:M1

f, f, . . . , f^k

a M2

f, f, . . . , f^k b

. 1.8

If there exists a positive constantMsuch that|fz| ≥Mfor allf∈ Fwheneverz∈Ef, thenFis a normal family.

Theorem 1.2. LetFbe a family of meromorphic functions in a complex domainD, for everyf ∈ F, all zeros off have multiplicity at least k. Let a, b ∈ such that a / 0, let m,n,k≥ 1,mj, nj j 1,2, . . . , kbe nonnegative integers such thatmnmknkγ_M^∗

1γ_M^∗

2 > 0, (k /2 whenn 1 or m1),m/nm_j/n_jfor all positive integersm_jandn_j,1≤j≤k. Put

Ef z∈D:M1

f, f, . . . , f^k

a M₂

f, f, . . . , f^k b

. 1.9

If there exists a positive constantMsuch that|fz| ≥Mfor allf∈ Fwheneverz∈Ef, thenFis a normal family.

As an application ofTheorem 1.1, we obtain the following theorem.

Theorem 1.3. LetFbe a family of holomorphic functions in a domainD, for everyf∈ F, all zeros of f have multiplicity at leastk. Leta, b/ 0be two finite values andn, k, n1, . . . , nkbe nonnegative integers withn ≥ 1,k ≥ 1,n_k ≥ 1. For everyf ∈ F, all zeros off have multiplicity at leastk, if Pf a⇔M1f, f, . . . , f^k b, thenFis normal inD.

Example 1.4. LetD {z : |z| < 1}andF {fm}. Ifa 0, letfm : e^mz. For each function f ∈ F,PfandM₁f, f, . . . , f^kshare 0 inD. However, it can be easily verified thatFis not normal inD.Example 1.4shows that the conditionb / 0 inTheorem 1.3is sharp.

Example 1.5. LetD {z : |z| < 1}andF {fm}. Ifa / 0, letfm : me^λz−e^−λz, where λis the root ofz² b/a. For each functionf ∈ F,f b/af,fⁿ¹ a ⇔ fⁿf bin D. However, it can be easily verified thatFis not normal inD.Example 1.5shows that the multiplicity restriction on zeros offinTheorem 1.3is sharpat least fork2.

2. Preliminary Lemmas

Lemma 2.1see11. LetFbe a family of meromorphic functions on the unit discΔ, all of whose zeros have the multiplicity at leastk, then ifFis not normal, there exist, for each 0≤α < k

aa numberr, 0< r <1, bpointszn,|zn|< r,

cfunctionsfn ∈ F, and dpositive numbersρn → 0

such thatρ_n^−αfnznρnξ gnξ → gξlocally uniformly with respect to the spherical metric, wheregξis a nonconstant meromorphic function on , all of whose zeros have multiplicity at least k, such thatg^#ξ≤g^#0. Here, as usual,g^#z |gz|/1|gz|²is the spherical derivative.

Lemma 2.2see1, page 158. LetF {f}be a family of meromorphic functions in a domain D⊂ . ThenFis normal inDif and only if the spherical derivatives of functionsf∈ Fare uniformly bounded on each compact subset ofD.

Lemma 2.3see12. Letf be an entire function andMa positive integer. Iff^#z ≤ Mfor all z∈ , thenfhas the order at most one.

Lemma 2.4see13. Take nonnegative integersn, n₁, . . . , n_kwithn≥1,n₁ n₂ · · ·n_k≥1 and definednn1n2· · ·nk. Letfbe a transcendental meromorphic function with the deficiency δ0, f>3/3d1. Then for any nonzero valuec, the functionfⁿfⁿ¹· · ·f^knk−chas infinitely many zeros. Moreover, ifn≥2, the deficient condition can be omitted.

The following two lemmas can be seen as supplements ofLemma 2.4.

Lemma 2.5. Take nonnegative integersn, n₁, . . . , n_kwith n ≥ 1,n_k ≥ 1 and defined nn₁ n₂· · ·n_k. Letfbe a transcendental meromorphic function whose zeros have multiplicity at leastk.

Then for any nonzero valuec, the functionfⁿfⁿ¹· · ·f^knk−chas infinitely many zeros, provided thatn₁ n₂ · · · n_k−1≥1 andk / 2 whenn1. Specially, iffis transcendental entire, the functionfⁿfⁿ¹· · ·f^knk −chas infinitely many zeros.

Proof. If n1n2 · · ·n_k−1 0, then fⁿfⁿ¹· · ·f^knk fⁿf^knk, this case has been consideredsee5,12–20.

Ifn₁ n₂ · · · n_k−1 ≥ 1 and ifn ≥ 2, we immediately get the conclusion from Lemma 2.4. Next we consider the casen1.

LetΨ fⁿfⁿ¹· · ·f^knk. Using the proof ofLemma 2.4see13, page 161–163 , we obtain

3d−2T r, f

≤3dN

r,1 f

N

r, 1

f

4N

r, 1 Ψ−c

N

r,Ψ−c Ψ

−3N

r,Ψ−c Ψ

S

r, f .

2.1

Suppose thatz0 is a zero of f of multiplicityp≥ k, then z0 is a zero of Ψof multiplicity dp−Σ^k_j1jnj, and thus is a pole ofΨ−c/Ψof multiplicitydp−Σ^k_j1jnj−1. Thereby, from 2.1we get

3d−2T r, f

≤

⎛

⎝3 k j1

jn_j5

⎞

⎠N

r, 1 f

4N

r, 1

Ψ−c

S r, f

≤ 3_k

j1jn_j5

k N

r, 1

f

4N

r, 1 Ψ−c

S

r, f .

2.2

Note thatn1, we deduce from2.2that

k−53_k−1

j1

k−j nj

k T

r, f

≤4N

r, 1 Ψ−c

S

r, f

. 2.3

Ifk1, thenΨ fⁿfⁿ¹; this case has been proved as mentioned abovesee13–16.

Ifk≥5, then we havek−53_k−1

j1k−jnj>0; the conclusion is evident.

If 3≤k≤4, note thatn₁ n₂ · · · n_k−1≥1 and we deduce thatk − 5 3_k−1

j1k− jnj >0, thus the conclusion holds.

Iffis a transcendental entire function, we only need to consider the casek ≥2. Note thatsee Hu et al.21, page 67

dT r, f

≤dN

r,1 f

N

r, 1

Ψ−c

−N

r,Ψ−c Ψ

S

r, f

. 2.4

With similar discussion as above, we obtain

⎛

⎝n _k−1

j1

k−j nj−1 k

⎞

⎠T r, f

≤N

r, 1 Ψ−c

S

r, f

. 2.5

In view ofn≥1 andk≥2, we getn _k−1

j1k−jnj−1/k >0, thus we immediately obtain the conclusion. This completes the proof ofLemma 2.5.

Lemma 2.6. Take nonnegative integers n, n₁, . . . , n_k, k with n ≥ 1, n_k ≥ 1, k ≥ 1 and define dn n1 n2 · · · nk. Letfbe a nonconstant rational function whose zeros have multiplicity at leastk. Then for any nonzero valuec, the functionfⁿfⁿ¹· · ·f^knk−chas at least one finite zero.

Proof. Since the casek 1 has been proved by Charak and Rieppo 9, we only need to considerk≥2.

Suppose thatfⁿfⁿ¹· · ·f^knk −chas no zero.

Case 1. Iff is a nonconstant polynomial, since the zeros offhave multiplicity at leastk, we know thatfⁿfⁿ¹· · ·f^knkis also a nonconstant polynomial, sofⁿfⁿ¹· · ·f^knk−chas at least one zero, which contradicts our assumption.

Case 2. Iffis a nonconstant rational function but not a polynomial. Set fz Az−a₁^m1z−a₂^m2· · ·z−a_s^ms

z−b1^l1z−b2^l2· · ·z−bt^lt , 2.6 whereAis a nonzero constant andm_i≥ki1,2, . . . , s,l_j≥1j 1,2, . . . , t.

Then by mathematical induction, we get

f^kz Az−a1^m1−kz−a2^m2−k· · ·z−as^ms−kgkz

z−b₁^l1kz−b₂^l2k· · ·z−b_t^ltk , 2.7 whereg_kz M−NM−N−1· · ·M−N−k1z^kst−1c_mz^kst−1−1· · ·c₀,c_m, . . . , c₀ are constants and

m1m2· · ·msM≥ks, 2.8

l1l2· · ·ltN≥t. 2.9

It is easily obtained that

deg g_k

≤kst−1. 2.10

Combining2.6and2.7yields

fⁿ f_n1

· · · f^k_nk

A^dz−a₁^dm1−kj1jnj· · ·z−a_s^dms−kj1jnjgz

z−b1^dl1kj1jnj· · ·z−bt^dltkj1jnj , 2.11 wheregz _k

j1g_j^njzwith degg≤_k

j1jn_jst−1.

Moreover,gzis not a constant, or else, we getgjis a constant forj 1, . . . , k. The leading coefficient ofg_jisM−N−j−1st.

Ifg1is a constant, then we get

MN. 2.12

Ifgkis a constant, then we get

k−1st 0, 2.13 which impliesk1, a contradiction with the assumptionk≥2.

Then from2.11, we obtain

fⁿ

fn1· · · f^k

_nk

A^dz−a₁^{dm1−kj1jnj−1}· · ·z−a_s^{dms−kj1jnj−1}hz

z−b1^dl1kj1jnj1· · ·z−bt^dltkj1jnj1 , 2.14 wherehzis a polynomial withst−1≤degh≤_k

j1jnj1 st−1.

Sincefⁿfⁿ¹· · ·f^knk−c / 0, we obtain from2.11that fⁿ

f_n1

· · · f^knk

c B

z−b1^dl1kj1jnj· · ·z−bt^dltkj1jnj, 2.15 whereBis a nonzero constant. Then

fⁿ

f_n1

· · ·

f^k_nk

B·Hz

z−b₁^dl1kj1jnj1· · ·z−b_t^dltkj1jnj1, 2.16 whereHzis a polynomial with degH t−1.

From2.14and2.16, we deduce that

dM−

⎛

⎝^k

j1

jnj1

⎞

⎠sdegh degH t−1, 2.17

in view of degh≥st1, together with2.8, we have

dks≤^k

j1

jn_js, 2.18

namely

nks^k

j1

k−j

njs≤0. 2.19

which is a contradiction sincen≥1.

Hencefⁿfⁿ¹· · ·f^knk −chas at least one finite zero.

This provesLemma 2.6.

Remark 2.7. Lemma 2.6is a generalization of Lemma 2.2 in10. The proof ofLemma 2.6is quite different from its proof. Actually, the proof of Lemma 2.2 in10is incorrect. The main problem appears at2.2in10. Sincef has only zero with multiplicity at leastk, then each zero offⁿis of multiplicity at leastnk, but this does not mean that each zero offⁿf^kmis of multiplicity at leastnkbecause the zeros off^kmay not be the zeros off, and thus their multiplicity may less thannk. Therefore, the inequality of2.2in10is not valid, which implies that the proof of Lemma 2.2 in10is not correct.

Lemma 2.8. Leta,b∈ such thata / 0. Letm,n,k≥1,mj,nj j 1,2, . . . , kbe nonnegative integers such thatmnmknkγ_M^∗

1γ_M^∗

2>0, (k / 2 whenn1 orm1),m/nmj/njfor all positive integersm_jandn_j,1≤j≤k. Letfbe a meromorphic function in ; all zeros offhave multiplicity at leastk. Define

Φz M1

f, f, . . . , f^k

a M₂

f, f, . . . , f^k−b. 2.20

ThenΦzhas a finite zero.

Proof. The algebraic complex equation

x a

x^m/n b 2.21

has always a nonzero solution, say x0 ∈ . By Lemmas 2.5 and 2.6, the differential monomial M1f, f, . . . , f^k cannot avoid it and thus there exists z0 ∈ such that M₁fz0, fz0, . . . , f^kz0 x₀.

Under the assumptions, for all positive integersm,n,mj,nj, we have mnm

n, mjnjm

n. 2.22

Thus

Φz₀ M₁

fz0, fz0, . . . , f^kz0

a M₁^m/n

fz0, fz0, . . . , f^kz0−b0. 2.23

This provesLemma 2.8.

Lemma 2.9see2, page 51. Iffis an entire function of orderσf, then

σ f

lim sup

r→ ∞

logν r, f

logr , 2.24

whereνr, fdenotes the central-index offz.

Lemma 2.10see22, page 187–199or2, page 51. Ifgis a transcendental entire function, let 0 < δ < 1/4 andzbe such that|z| rand that|gz| Mr, gνr, g^−1/4δ holds. Then there exists a setF ⊂of finite logarithmic measure, that is,

Fdt/t <∞such that g^mz

gz

ν r, g

z m

1o1 2.25 holds for allm≥0 and allr /∈F.

3. Proof of Theorem 1.1

Without loss of generality, we may assumeD Δ {z : |z| < 1}. Suppose thatFis not normal atz₀ ∈D. ByLemma 2.1, for 0≤ α < k, there existr < 1,z_j ∈Δsuch thatz_j → z₀, fj ∈ Fandρj → 0 such thatgjξ ρ^−α_j fjzjρjξ → gξlocally uniformly with respect to the spherical metric, wheregξis a nonconstant meromorphic function on , all of whose zeros have multiplicity at leastk. For simplicity, we denotefjzjρjξbyfj. By Lemmas2.4 and2.6, there existsξ0∈ such that

gξ0ⁿ

gξ0_n1

· · ·

g^kξ0nk

a gξ0^m

gξ0_m1

· · ·

g^kξ0_mk 0. 3.1 Obviously,gξ0 / 0,∞, so in some neighborhood ofξ0,gjξconverges uniformly togξ.

We have g_jξⁿ

g_jξ_n1

· · ·

g_j^kξ_nk

a gjξ^m

g_jξ_m1

· · ·

g^k_j ξ_mk −ρ_j^{αγM2−ΓM2}b ρ_j^{−αγM1ΓM1}f_jⁿ

f_jn1

· · · f_j^k_nk

a ρ^−αγ_j ^M2ΓM2f_j^m

f_j

_m1

· · · f_j^k

_mk −ρ^αγ_j ^M2−ΓM2b

ρ_j^{αγM2−ΓM2}

⎡

⎢⎣ρ_j^{−αγM1γM2ΓM1ΓM2}f_jⁿ f_jn1

· · · f_j^k_nk

a f_j^m

f_jm1

· · ·

f_j^k_mk −b

⎤

⎥⎦. 3.2 Letα ΓM1 ΓM2/γM1γM2< k, and under the assumptionγM2ΓM1 −γM1ΓM2 >0, we obtain

gⁿ g_n1

· · · g^knk

a g^m

g_m1

· · ·

g^k_mk 3.3

is the uniform limit of

ρ^γ_j^{M2ΓM1−γM1ΓM2/γM1γM2}

⎡

⎢⎣f_jⁿ f_jn1

· · · f_j^knk

a f_j^m

f_jm1

· · ·

f_j^k_mk −b

⎤

⎥⎦ 3.4

in some neighborhood ofξ0.

By3.1and Hurwitz’s theorem, there exists a sequenceξ_j → ξ₀such that for all large values ofjandζ_jz_jρ_jξ_j,

fj

ζj

_n f_j

ζj

ⁿ¹

· · · f_j^k

ζj

^nk

a fj

ζj

_m f_j

ζj

^m1

· · · f_j^k

ζj

^mk b. 3.5

Hence for all large values ofj,ζjzjρjξj ∈Ef, it follows from the condition that g_j

ξ_j f_j ζ_j ρ^α_j ≥ M

ρ^α_j . 3.6

Sinceξ₀is not a pole ofg, there exists a positive numberKsuch that in some neighborhood ofξ₀we get|gξ| ≤K.

Sincegjξconverges uniformly togξin some neighborhood ofξ0, we get for all large values ofjand for allξin that neighborhood ofξ0

g_jξ−gξ<1. 3.7

By3.7, we get

K≥g

ξ_j≥g_j

ξ_j−g_j ξ_j

−g

ξ_j≥ M

ρ^α_j −1, 3.8

which is a contradiction sinceρj → 0 asj → ∞.

This completes the proof ofTheorem 1.1.

4. Proof of Theorem 1.2

Without loss of generality, we may assumeD Δ {z : |z| < 1}. Suppose thatFis not normal inD. ByLemma 2.1, for 0≤α < k, there existr <1,z_j ∈Δ,f_j ∈ Fandρ_j → 0such thatg_jξ ρ^−α_j f_jzj ρ_jξ → gξlocally uniformly with respect to the spherical metric, wheregξis a nonconstant meromorphic function on , all of whose zeros have multiplicity at leastk. ByLemma 2.8, we get

gξ0ⁿ

gξ0_n1

· · ·

g^kξ0nk

a gξ0^m

gξ0_m1

· · ·

g^kξ0_mk −b0, 4.1 for someξ0 ∈ .

We can arrive at a contradiction by using the same argument as in the latter part of proof ofTheorem 1.1.

This completes the proof ofTheorem 1.2.

5. Applications

Proof ofTheorem 1.3. We shall divide our argument into two cases.

Case 1a / 0. LetMbe a positive constant withM ≤ ^γM1

|a|; under the assumptions, we have

Ef

z∈D:M1

f, f, . . . , f^k b

5.1

and|fz| ≥Mfor allf∈ Fwheneverz∈Ef; by Lemmas2.5and2.6, using the similar proof ofTheorem 1.1, we obtain the conclusion.

Case 2a0. Forf∈ F, iffz0 0 forz₀∈D, sincePf 0⇒M₁f, f, . . . , f^k b, we haveb0, which is a contradiction, hencef / 0.

IfM1fz0, fz0, . . . , f^kz0 bforz0∈D, sinceM1f, f, . . . , f^k b⇒Pf 0, we immediately get fz0 0 and hence M₁f, f, . . . , f^k b 0, which is still a contradiction, henceM1f, f, . . . , f^k / b.

Suppose thatFis not normal inD, byLemma 2.1, there existr < 1,zj ∈ Δ,fj ∈ F, andρj → 0 such thatgjξ ρ^−Γ_j ^M1/γM1fjzjρjξ → gξlocally uniformly with respect to the spherical metric, wheregξis a nonconstant entire function, all of whose zeros have multiplicity at leastk. By Hurwitz’s theorem, we have

ig≡0 org / 0, and

iigⁿgⁿ¹· · ·g^knk ≡borgⁿgⁿ¹· · ·g^knk / b.

Sincegis not a constant, we haveg / 0. ByLemma 2.3,ghas the order at most 1, sogξ e^cξd, wherec/0,dare two constants. Thus

gⁿξ g_n1

ξ· · · g^knk

ξ c^ΓM1e^γM1cξd. 5.2

Ifgⁿgⁿ¹· · ·g^knk ≡b, we immediately get a contradiction. Hence gⁿ

g_n1

· · · g^k_nk

/b, 5.3

but by Lemmas2.5and2.6we get a contradiction again.

This provesTheorem 5.1.

Further more, usingTheorem 1.3, we obtain a uniqueness theorem related to R. Br ¨uck’s Conjecture. Firstly, we recall this conjecture.

R. Br ¨uck’s Conjecture

Letfbe a nonconstant entire function such that the hyper-orderσ2fis not a positive integer and σ2f<∞. If fandfshare a finite valueaCM, then

f−a

f−a c, 5.4

wherecis a nonzero constant and the hyper-orderσ2fis defined as follow:

σ2

f

lim sup

r→ ∞

loglogT r, f

logr . 5.5

Since then, many results related to this conjecture have been obtained. We refer the reader to Br ¨uck23, Gundersen and Yang24, Yang25, Chen and Shon26, Li and Gao 27, and Wang28.

It’s interesting to ask what happens if f is replaced by fⁿ in Br ¨uck’s Conjecture.

Recently, Yang and Zhang29considered this problem and got the following theorem.

Theorem H. Letfbe a nonconstant entire function.n≥7 be an integer, and letF fⁿ. IfFandF share 1 CM, thenFF, andfassumes the form

fz ce^z/n, 5.6

wherecis a nonzero constant.

L ¨u et al.30improves Theorem H and obtained the following theorem.

Theorem I. LetQ1/≡0be a polynomial, and letn ≥ 2 be an intege; letfzbe a transcendental entire function, and letFz fzⁿ. IfFzandFzshareQ1CM, then

fz Ae^z/n, 5.7

whereAis a nonzero constant.

We obtain a more general result as follows.

Theorem 5.1. Letn, k, n1, . . . , nkbe nonnegative integers withn≥1,k≥1,nk≥1, anda,bbe two finite nonzero values. Letfbe a nonconstant entire function whose zeros have multiplicity at leastk.

Iff^nn1···nk a^ªfⁿfⁿ¹· · ·f^knk b, then

fⁿ f_n1

· · ·

f^knk−b

f^nn1···nk−a c, 5.8

wherecis a nonzero constant. Specially, ifab, thenf c₁e^ωz, wherec₁is a nonzero constant,ω is the root oft^ΓM1 1.

Proof ofTheorem 5.1. First we assert thatσf≤1. Let F g_ωz fzω, ω∈ !

, z∈D Δ. 5.9

Under the assumptions of Theorem 1.3, we get thatF is a normal family of holomorphic functions inD. ByLemma 2.2, there exists a constantMsuch that

f^#ω fω

1fω² g_ω0

1gω0² g_ω^#0≤M, 5.10 for allω∈ . Hence byLemma 2.3,fhas the order at most 1.

Sincef^nn1···nk a ^ª fⁿfⁿ¹· · ·f^knk b, f must be a transcendental entire function and

fⁿ f_n1

· · ·

f^knk−b

f^nn1···nk−a e^αz. 5.11

From5.11, we haveTr, e^αz OTr, f, henceσe^α≤σf≤1 andαzis a polynomial with degα ≤ 1. Note that f is transcendental, we haveMr, f → ∞, as r → ∞. Let Mrn, f fzn, wherez_nr_ne^iθn, we deduce

rnlim→ ∞

1

fzn lim

rn→ ∞

1 M

r_n, f 0. 5.12

By Lemma 2.10, there exists a subset F1 ⊂ 1,∞ of finite logarithmic measure, namely

F1dt/t < ∞such that for some pointz_n r_ne^iθn θn ∈ 0,2πsatisfying|zn| r_n ∈/ F₁ andMrn, f |fzn|, we obtain

f^kzn fzn

ν rn, f

z_n k

1o1, 5.13

asr → ∞.

Rewrite5.11as

f/fn1· · ·

f^k/fnk−b/f^nn1···nk

1−a/f^nn1···nk e^αz, 5.14

it follows from5.12–5.14and Lemma 2.8 that

|αzn|loge^αznΓM1

logν

rn, f

−log

rne^iθn o1 ΓM1

logν r_n, f

−logr_n−iθrn o1

≤O logr_n

,

5.15

asr_n → ∞. Sinceαzis a polynomial, from5.15, we deduce thatαzis a constant. Let e^α c, thencis a nonzero constant. Thus

fⁿ f_n1

· · · f^k_nk

−b

f^nn1···nk−a c. 5.16

Specially, ifab, suppose thatfhas a zeroz0, by lettingzz0in5.16, we getc1; hence f^n1···nk

f_n1

· · · f^knk

. 5.17

Suppose that z0 is a zero of f with multiplicity p≥ k, thenz0 is a zero off^n1···nk with multiplicityn1· · ·nkp, and a zero offⁿ¹· · ·f^knkwith multiplicityn1· · ·nkp−ΓM1, which is a contradiction. Sofhas no zero, note thatfis a transcendental entire function and σf≤ 1, we havef c₁e^tz, wherec₁andtare two finite nonzero values. In view of5.16 andab, we deduce that

c^Γ₁^M1

t^ΓM1 −c

e^γM1tzb1−c; 5.18

hencec1 andt^ΓM1 c1.fc1e^ωz,ωis the root oft^ΓM1 1.

This completes the proof ofTheorem 5.1.

Acknowledgments

The authors thank the referees for reading the manuscript very carefully and making a number of valuable suggestions to improve the readability of the paper. The authors were supported by NSF of Chinano. 10771121, NSF of Shandong Provinceno. Z2008A01and NSF of Guangdong Provinceno. 9452902001003278.

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