Some Normality Criteria of Meromorphic Functions

(1)

Volume 2010, Article ID 926302,10pages doi:10.1155/2010/926302

Research Article

Some Normality Criteria of Meromorphic Functions

Junfeng Xu and Wensheng Cao

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

Correspondence should be addressed to Junfeng Xu,[email protected] Received 2 September 2009; Revised 11 January 2010; Accepted 23 March 2010 Academic Editor: Ram N. Mohapatra

Copyrightq2010 J. Xu and W. Cao. 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.

This paper studies some normality criteria for a family of meromorphic functions, which improve some results of Lahiri, Lu and Gu, as well as Charak and Rieppo.

1. Introduction and Main Results

Let f be a nonconstant meromorphic function in the complex plane C. We shall use the standard notations in Nevanlinna’s value distribution theory of meromorphic functions such asTr, f,Nr, f, andmr, f see, e.g., 1,2. The notation Sr, fis defined to be any quantity satisfyingSr, f oTr, fasr → ∞possibly outside a set ofEof finite linear measure.

Let F be a family of meromorphic functions on a domain D ⊂ C. We say that F is normal in D if every sequence of functions {fn} ⊂ F contains either a subsequence which converges to a meromorphic functionf uniformly on each compact subset ofD or a subsequence which converges to∞uniformly on each compact subset ofD.See1,3.

The Bloch principle 3 is the hypothesis that a family of analytic meromorphic functions which have a common propertyPin a domainDwill in general be a normal family ifPreduces an analyticmeromorphicfunction in the open complex planeCto a constant.

Unfortunately the Bloch principle is not universally true. But it is also very diﬃcult to find some counterexamples about the converse of the Bloch principle, and hence it is interesting to study the problem.

In 2005, Lahiri 4 proved the following criterion for the normality, and gave a counterexample to the converse of the Bloch principle by using the result.

Theorem A. LetFbe a family of meromorphic functions in a domainD, and leta/0,bbe two finite constants. Define

(2)

E_f

z:z∈ D, fz a fz b

. 1.1

If there exists a positive numberMsuch that for everyf∈ F, one has|fz| ≥Mwheneverz∈Ef, thenFis normal.

In this direction, Lahiri and Dewan 5 as well as Xu and Zhang 6 proved the following result.

Theorem B. LetFbe a family of meromorphic functions in a domainD, and leta/0,bbe two finite constants. Suppose that

E_f

z:z∈ D, f^k−af⁻ⁿb

, 1.2

wherekandnare positive integers.

If for everyf ∈ F

iall zeros offhave multiplicity at leastk,

iithere exists a positive numberMsuch that for everyf ∈ Fone has|fz| ≥Mwhenever z∈Ef,

thenFis normal inDso long asAn≥2; orBn1 andk1.

Here, we also give a counterexample to the converse of the Bloch principle by considering Theorem B, which is similar to an example in7.

Example 1.1. Letfz cotz, thenfz 1cot²z /0 for allz∈ C. Now we can see that

fz f⁻²z 1

1cot²z₂ cot²z 4

sin²2z/0, 1.3

but Theorem B is true especially whenEf is an empty set for everyfin the family.

In the following, we continue to study the normal family whenn 1 andk ≥ 2 in Theorem B.

Theorem 1.2. LetFbe a family of meromorphic functions in a domainD, anda/0,bbe two finite constants. Suppose that

Ef

z:z∈ D, f^k−af⁻¹ b

, 1.4

wherek≥2 is a positive integer.

If for everyf ∈ F

iall zeros offhave multiplicity at leastk1,

iithere exists a positive numberMsuch that for everyf∈ F, one has|fz| ≥Mwhenever z∈E_f, thenFis normal inD.

(3)

Corollary 1.3. LetFbe a family of meromorphic functions in a domainD, all of whose zeros have multiplicity at leastk1, and leta/0,bbe two finite constants. Suppose thatf^k−af⁻¹/b, where k≥2 is a positive integer. ThenFis normal inD.

Recently, Lu and Gu8considered two related normal families.

Theorem C. Let Fbe a family of meromorphic functions in a domain D; all of whose zeros have multiplicity at leastk2. Suppose that, for eachf∈ F,ff^k/aforz∈ D, thenFis a normal family onD, whereais a nonzero finite complex number andk≥1 is an integer number.

Theorem D. LetF be a family of meromorphic functions in a domainD; all of whose zeros have multiplicity at leastk1, and all of whose poles are multiple. Suppose that, for eachf∈ F,ff^k/a forz∈ D, thenFis a normal family onD, where a is a nonzero finite complex number andk ≥1 is an integer number.

In this paper, we give a simple proof and improve the above results.

Theorem 1.4. LetFbe a family of meromorphic functions in a domainD; all of whose zeros have multiplicity at leastk1. Suppose that, for eachf∈ F,ff^k/aforz∈ D, thenFis a normal family onD, whereais a nonzero finite complex number andk≥1 is an integer number.

In 2009, Charak and Rieppo7generalized Theorem A and obtained two normality criteria of Lahiri’s type.

Theorem E. LetFbe a family of meromorphic functions in a complex domainD. Leta, b ∈ Csuch thata /0. Letm₁,m₂,n₁,n₂be nonnegative integers such thatm₁n₂−m₂n₁ >0,m₁m₂≥1, and n₁n₂≥2, and put

E_f

z∈ D:

fzn1

fzm1 a fz_n₂

fz_m₂ b . 1.5

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

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

E_f

z∈ D:

fzn1

fzm1 a fz_n₂

fz_m₂ b . 1.6

Naturally, we ask whether the above results are still true whenfis replaced byf^kin Theorems E and F. We obtain the following results.

(4)

Theorem 1.5. LetFbe a family of meromorphic functions in a complex domainD; all of whose zeros have multiplicity at leastk. Leta, b∈ Csuch thata /0. Letm₁,m₂,n₁,n₂ be nonnegative integers such thatm₁n₂−m₂n₁>0,m₁m₂≥1, andn₁n₂≥2 (ifn₁n₂1,k≥5), and put

E_f

z∈ D:

fzn1

f^kz_m₁

a fz_n₂

f^kz_m₂ b . 1.7

Theorem 1.6. LetFbe a family of meromorphic functions in a complex domainD; all of whose zeros have multiplicity at leastk. Leta, b ∈ Csuch thata /0. Letm₁ ≥ 2,m₂,n₁,n₂ be nonnegative integers such thatm₁n₂m₂n₁, and put

E_f

z∈ D:

fzn1

f^kz_m₁

a fz_n₂

f^kz_m₂ b . 1.8

2. Some Lemmas

Lemma 2.1see9. LetFbe a family of functions meromorphic on the unit disc, all of whose zeros have multiplicity at leastk, then ifFis not normal, there exist, for each 0≤α < k,

aa number 0< r <1, bpointsz_n, z_n<1, cfunctionsf_n ∈ζ,

dpositive numberρ_n → ∞such thatρ^−α_n f_nznρ_nξ g_nξ → gξlocally uniformly, wheregis a nonconstant meromorphic onC, all of whose zeros have multiplicity at leastk, such thatg^#ξ≤g^#0.

Here, as usual,g^#ξ |gξ|/1|gξ|²is the spherical derivative.

Lemma 2.2. Letfbe rational in the complex plane andm, npositive integers. Iffhas only zero with multiplicity at leastk, thenfⁿf^k^mtakes on each nonzero valuea∈C.

Proof. In Lemma 6 of7, the case ofk 1 is proved. We just consider the case ofk≥ 2 by a diﬀerent way which comes from10.

Iff is a polynomial, obviously the conclusion holds. Iff is a nonpolynomial rational function, then we can set

fⁿ f^k_m

Az−α₁^m¹z−α₂^m²· · ·z−α_s^m^s z−β1

_n₁ z−β2

_n₂

· · ·

z−βt_n_t , 2.1

(5)

whereAis a nonzero constant. Sincefhas only zero with multiplicity at leastk, we find that mi≥kn i1,2, . . . , s, nj≥k1n

j1,2, . . . , t

. 2.2

For convenience, we denote

Mm₁m₂· · ·ms≥kns,

Nn1n2· · ·nt≥k1nt. 2.3

Diﬀerentiating2.1, we obtain fⁿ

f^km

z−α₁^m¹⁻¹z−α₂^m²⁻¹· · ·z−α_s^m^s⁻¹ z−β₁n11

z−β₂n21· · ·

z−β_tnt1 gz, 2.4

wheregzis a polynomial with degg≤st−1.

Suppose thatfⁿf^k^m−ahas no zero, then we can write

fⁿ f^km

−aA B

z−β1

_n₁ z−β2

_n₂

· · ·

z−βt_n_t, 2.5

whereBis a nonzero constant.

Diﬀerentiating2.5, we obtain fⁿ

f^k_m

Bg₁z z−β₁_n₁₁

z−β₂_n₂₁

· · ·

z−β_t_n_t₁, 2.6

whereg₁zis a polynomial of the form−BNz^t−1B_t−2z^t−2· · ·B₀, in whichB₀,. . .,B_t−2are constants.

Comparing2.1and2.5, we can obtainMN. From2.4and2.6, we have s

i1

mi−1 M−s≤deg g1z

t−1,

M≤st−1

≤ M

kn N

k1n−1 M

kn M

k1n−1

1

kn 1

k1n

M−1.

2.7

It is a contradiction withn≥1 andk≥2. This proves the lemma.

(6)

Lemma 2.3 see 11. Letf be a transcendental meromorphic function all of whose zeros have multiplicity at leastt, thenff^kassumes every finite nonzero value infinitely often, wheretk1 ifk≤4, andt5 ifk≥5.

Remark 2.4. The lemma was first proved by Wang ast5 ifk 5 andt 6 ifk ≥6 in12.

Recently, the result is improved by11.

Lemma 2.5. Letfbe a meromorphic function all of whose zeros have multiplicity with at leastk1 in the complex plane, thenff^k−amust have zeros for any constanta /0,∞.

Proof. Iffis rational, then byLemma 2.2the conclusion holds.

Iff is transcendental, supposing thatff^k−ahas no zeros, then byLemma 2.3, we can get a contradiction. This completes the proof of the lemma.

Lemma 2.6. Letfbe meromorphic in the complex plane, and leta /0 be a constant, for any positive integerk; ifff^k≡a, thenfis a constant.

Proof. Iffis not a constant, and froma /0, we know thatf /0, then with the identityff^k≡ a, we can get that, ifr → ∞,

T

r, 1 f

m

r,1

f

≤log 1

|a|m

r,f^k f

o

T r, f

, 2.8

andr /∈EwithEbeing a set ofrvalues of finite linear measure. It is a contradiction.

Lemma 2.7see13. Letf be a transcendental meromorphic function, and letn ≥ 2,nk ≥ 1 be two integers. Then for any nonzero valuec, the functionfⁿf^kⁿ^k −chas infinitely many zeros.

Lemma 2.8see14. Letfbe a transcendental meromorphic function, and letn≥2 be an integer.

Then for any nonzero valuec, the functionff^kⁿ−chas infinitely many zeros.

Lemma 2.9. LetFbe a family of meromorphic functions in a complex domainD. Leta, b ∈ Csuch thata /0. Letm₁≥2,m₂,n₁,n₂be nonnegative integers such thatm₁n₂m₂n₁, and put

E_f

⎧⎨

⎩z∈ D:

fzn1

f^kzm1

a fzn2

f^kzm2n2/n1 b

⎫⎬

⎭. 2.9

has a finite zero.

Proof. The algebraic complex equation

x a

xⁿ²^/n¹ −b0 2.10

has always a nonzero solution; sayx₀∈ C. By14, Corollary 3or15, Lemmas2.2,2.7, and 2.8, the meromorphic functionfⁿ¹f^k^m¹ cannot avoid it and thus there existsz0 ∈ Csuch thatfz0ⁿ¹f^kz0^m¹x₀.

(7)

By assumption, we may writem2 n2/n1m1andn2 n2/n1n1. Consequently

Ψz₀ fz0n1

f^kz0m1

a fz0n1

f^kz0m1n2/n1 −b 2.11

and we complete the proof of the lemma.

Remark 2.10. Ifm₁ 1, we needk≥5 whenn₁1 byLemma 2.3. We can get a similar result.

3. Proof of Theorems

Proof ofTheorem 1.2. Let α k/2 < k. Suppose that F is not normal at z₀ ∈ D. Then by Lemma 2.1, there exist a sequence of functionsfj ∈ Fj 1,2, . . ., a sequence of complex numbersz_j → z₀,andρ_j>0 → 0 such that

g_jζ ρ^−α_j f_j

z_jρ_jζ

3.1 converges spherically and locally uniformly to a nonconstant meromorphic functiongζin C. Also the zeros ofgzare of multiplicity at least≥k1. So g^k/≡0. ApplyingLemma 2.5 to the functiongz, we know that

gζ0g^kζ0−a0, g^kζ0− a

gζ0 0 3.2

for some ζ₀ ∈ C. Clearly ζ₀ is neither a zero nor a pole ofg. So in some neighborhood of ζ₀,g_jζconverges uniformly togζ. Now in some neighborhood ofζ₀we see thatg^kζ− agζ⁻¹is the uniform limit of

g_j^kζ0−agjζ0⁻¹−ρ^α_jbρ^k/2_j f_j^k

zjρjζ0

−af_j⁻¹

zjρjζ0

−b

. 3.3

By3.2and Hurwitz’s theorem, there exists a sequenceζ_j → ζ₀such that for all large values ofj

f_j^k

z_jρ_jζ_j

−af_j⁻¹

z_jρ_jζ_j

b. 3.4

Therefore for all large values ofj, it follows from the given condition that|gjζj| ≥ |fjzj ρ_jζ_j|/ρ^α_j ≥M/ρ^α_j.

Since ζ0 is not a pole of g, there exists a positive number K such that in some neighborhood ofζ₀we get|gζ| ≤K.

(8)

Sincegjζ converges uniformly togζin some neighborhood of ζ0, we get for all large values ofjand for allζin that neighborhood ofζ₀

gjζ−gζ<1. 3.5

Sinceζj → ζ, we get for all large values ofj

K≥g

ζ_j≥g_j

ζ_j−g ζ_j

−g_j

ζ_j> M

ρ^α_j −1, 3.6

which is a contradiction. This proves the theorem.

Proof ofTheorem 1.4. IfFis not normal atz₀ ∈ D. We assume without loss of generality that z0 0, then byLemma 2.1, forαk/2, there exist a sequence of pointszn → z0, a sequence of positive numbersρ_n → 0,and a sequence of functions{fn}ofFsuch that

gjζ ρ^−α_j fj

zjρjζ

−→gz 3.7

spherically uniformly on compact subsets ofC, wheregz is a nonconstant meromorphic function onC; all of whose zeros have multiplicityk1 at least. By3.7,

gjζg^k_j ζ−afjζf_j^kζ−a /0. 3.8

It follows thatgζg^kζ/aorgζg^kζ≡aby Hurwitz’s theorem. FromLemma 2.6, we obtain thatgg^k/a. ByLemma 2.5, we get a contradiction. This completes the proof of the theorem.

Proof ofTheorem 1.5. Suppose thatFis not normal atz0∈ D. Then byLemma 2.1, for 0≤α <

k, there exist a sequence of functionsf_j ∈ Fj 1,2, . . ., a sequence of complex number z_j → z₀, andρ_j>0 → 0 such that

gjζ ρ^−α_j fj

zjρjζ

3.9

converges spherically and locally uniformly to a nonconstant meromorphic functiongζin C. Also the zeros ofgzare of multiplicity at least≥k. Sog^k/≡0. By Lemmas2.2,2.3,2.7, and2.8, we get

gζ0n1

g^kζ0_m₁

a gζ0_n₂

g^kζ0_m₂ 0, 3.10

(9)

for someζ0 ∈ C. Clearlyζ0 is neither a zero nor a pole ofg. So in some neighborhood ofζ0, g_jζconverges uniformly togζ. Now in some neighborhood ofζ₀we have

g_jζ_n₁

g_j^kζm1

a g_jζn2

g^k_j ζ_m₂ −ρ^αk_j ²^−km²b ρ^−αk_j ¹^km¹

f_jn1 f_j^k_m₁

a ρ_j^−αk²^km²

fj_n₂

f_j^km2 −ρ_j^αk²^−km²b

ρ^αk_j ²^−km²

⎛

⎜⎝ρ^−αk_j ¹^k²^km¹^m² fjn1

f_j^k_m₁

a fj_n₂

f_j^km2 −b

⎞

⎟⎠,

3.11

wheref_jzjρ_jζis replaced byf_jandk_jn_jm_j,j1,2.

Takingα m1m₂k/k1k₂and using the assumptionm₁n₂−n₁m₂ >0, we see that

gⁿ¹ g^km1

a gⁿ²

g^km2 3.12

is the uniform limit of

ρ^m_j ¹ⁿ²⁻ⁿ¹^m²^/k¹^k²^k

⎛

⎜⎝f_jⁿ¹ f_j^km1

a f_jⁿ²

f_j^k_m₂ −b

⎞

⎟⎠ 3.13

in some neighborhood ofζ0. By3.10and Hurwitz’s theorem, there exists a sequenceζj → ζ0

such that for all large values ofj f_j

z_jρ_jζ_jn1 f_j^k

z_jρ_jζ_j^m¹

a f_j

z_jρ_jζ_jn2 f_j^k

z_jρ_jζ_j^m² −b. 3.14

Hence, for all largej, it follows from the given condition that gj

ζj≥ f_j

z_jρ_jζ_j ρ^α_j M

ρ_j^α. 3.15

In the following, we can get a contradiction in a similar way with the proof of the last part ofTheorem 1.2. This completes the proof of the theorem.

Proof ofTheorem 1.6. Suppose thatFis not normal atz0∈ D. Then byLemma 2.1, for 0≤α <

k, there exist a sequence of functionsf_j ∈ Fj 1,2, . . ., a sequence of complex numbers zj → z₀, andρj>0 → 0 such that

gjζ ρ^−α_j fj

zjρjζ

3.16

(10)

converges spherically and locally uniformly to a nonconstant meromorphic functiongζin C. Also the zeros ofgzare of multiplicity at least≥k. Sog^k/≡0. ByLemma 2.9, we get

gζ0_n₁

g^kζ0_m₁

a gζ0n2

g^kζ0m2 −b0, 3.17

for someζ₀∈ C.

In the following, we can get a contradiction in a similar way with the proof of the last part ofTheorem 1.5. This completes the proof of the theorem.

Acknowledgments

The authors would like to thank Professor Lahiri for supplying the electronic file of the paper 4. The authors were supported by NSF of China No. 10771121, No. 10801107, NSF of Guangdong Province No. 9452902001003278, No. 8452902001000043, and Department of Education of GuangdongNo. LYM08097.

References

1 L. Yang, Value Distribution Theory, Springer, Berlin, Germany, 1993.

2 C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, vol. 557 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.

3 J. F. Schiﬀ, Normal Families, Universitext, Springer, New York, NY, USA, 1993.

4 I. Lahiri, “A simple normality criterion leading to a counterexample to the converse of the Bloch principle,” New Zealand Journal of Mathematics, vol. 34, no. 1, pp. 61–65, 2005.

5 I. Lahiri and S. Dewan, “Some normality criteria,” Journal of Inequalities in Pure and Applied Mathematics, vol. 5, no. 2, article 35, 2004.

6 J. F. Xu and Z. L. Zhang, “Note on the normal family,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 4, article 133, 2006.

7 K. S. Charak and J. Rieppo, “Two normality criteria and the converse of the Bloch principle,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 43–48, 2009.

8 Q. Lu and Y. Gu, “Zeros of diﬀerential polynomialfzf^kz−aand its normality,” Chinese Quarterly Journal of Mathematics, vol. 24, no. 1, pp. 75–80, 2009.

9 X. C. Pang and L. Zalcman, “Normal families and shared values,” The Bulletin of the London Mathematical Society, vol. 32, no. 3, pp. 325–331, 2000.

10 P.-C. Hu and D.-W. Meng, “Normality criteria of meromorphic functions with multiple zeros,” Journal of Mathematical Analysis and Applications, vol. 357, no. 2, pp. 323–329, 2009.

11 W. L. Zou and Q. D. Zhang, “On the zero ofϕff^k,” Journal of Sichuan Normal University, vol. 31, no.

6, pp. 662–666, 2008.

12 J.-P. Wang, “On the value distribution offf^k,” Kyungpook Mathematical Journal, vol. 46, no. 2, pp.

169–180, 2006.

13 C. C. Yang and P. C. Hu, “On the value distribution offf^k,” Kodai Mathematical Journal, vol. 19, no.

2, pp. 157–167, 1996.

14 A. Alotaibi, “On the zeros ofaff^kⁿ−1 forn≥2,” Computational Methods and Function Theory, vol.

4, no. 1, pp. 227–235, 2004.

15 Z. F. Zhang and G. D. Song, “On the zeros offf^k−az,” Chinese Annals of Mathematics, Series A, vol.

19, no. 2, pp. 275–282, 1998.