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volume 7, issue 3, article 93, 2006.

Received 21 December, 2005;

accepted 28 July, 2006.

Communicated by:H.M. Srivastava

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Journal of Inequalities in Pure and Applied Mathematics

MEROMORPHIC FUNCTIONS THAT SHARE ONE VALUE WITH THEIR DERIVATIVES

KAI LIU AND LIAN-ZHONG YANG

School of Mathematics & System Sciences Shandong University

Jinan, Shandong, 250100, P. R. China EMail:liuk@mail.sdu.edu.cn EMail:lzyang@sdu.edu.cn

URL:http://202.194.3.2/html/professor/ylzh/YangIndex.html

c

2000Victoria University ISSN (electronic): 1443-5756 373-05

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Meromorphic Functions that Share One Value with their

Derivatives Kai Liu and Lian-Zhong Yang

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J. Ineq. Pure and Appl. Math. 7(3) Art. 93, 2006

http://jipam.vu.edu.au

Abstract

In this paper, we deal with the problems of uniqueness of meromorphic functions that share one finite value with their derivatives and obtain some results that improve the results given by Rainer Brück and Qingcai Zhang.

2000 Mathematics Subject Classification:30D35.

Key words: Meromorphic functions, Uniqueness, Sharing values.

This work was supported by the NNSF of China (No.10371065) and the NSF of Shandong Province, China (No.Z2002A01).

1. Introduction and Main Results

In this paper, a meromorphic function will mean meromorphic in the finite com- plex plane. We say that two meromorphic functionsf andgshare a finite value aIM (ignoring multiplicities) whenf−aandg−ahave the same zeros. Iff−a andg−ahave the same zeros with the same multiplicities, then we say thatf andgshare the valueaCM (counting multiplicities). We say thatf andg share

∞ CM provided that1/f and 1/g share 0 CM. It is assumed that the reader is familiar with the standard symbols and fundamental results of Nevanlinna Theory, as found in [3,6].

Let f(z) be a meromorphic function. It is known that the hyper-order of f(z),denoted byσ2(f),is defined by

σ₂(f) = lim sup

r→∞

log logT(r, f) logr . In 1996, R. Brück posed the following conjecture (see [1]).

Conjecture 1.1. Let f be a non-constant entire function such that the hyper- orderσ₂(f)off is not a positive integer andσ₂(f)<+∞.Iff andf⁰ share a finite valueaCM, then

f⁰ −a f −a =c, wherecis nonzero constant.

In [1], Brück proved that the conjecture holds when a = 0. In 1998, Gun- dersen and Yang [2] proved that the conjecture is true whenf is of finite order.

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In 1999, Yang [4] confirmed that the conjecture is also true whenf⁰is replaced byf^(k)(k ≥2)andf is of finite order.

In 1996, Brück obtained the following result.

Theorem A ([1]). Letf be a nonconstant entire function. Iff andf⁰ share the value 1 CM, and ifN

r,_f¹0

=S(r, f), then

f⁰−1 f−1 ≡c

for a non-zero constantc.

In 1998, Q. Zhang proved the next two results in [7].

Theorem B. Letf be a nonconstant meromorphic function. Iff andf⁰ share the value 1 CM, and if

N(r, f) +N

r, 1 f⁰

<(λ+o(1))T(r, f⁰),

0< λ < 1 2

,

then f⁰−1

f−1 ≡c for some non-zero constantc.

Theorem C. Let f be a nonconstant meromorphic function, k be a positive integer. Iff andf^(k)share the value 1 CM, and if

2N(r, f) +N

r, 1 f⁰

+N

r, 1

f^(k)

<(λ+o(1))T(r, f^(k)), (0< λ <1),

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then

f^(k)−1 f−1 ≡c for some non-zero constantc.

The above results suggest the following question: What results can be obtained if the condition that f and f⁰ share the value 1 CM is replaced by the condition thatf andf⁰ share the value 1 IM?

In this paper, we obtained the following results.

Theorem 1.2. Letf be a nonconstant meromorphic function, iff andf⁰ share the value 1 IM, and if

N(r, f) +N

r, 1 f⁰

<(λ+o(1))T(r, f⁰)

0< λ < 1 4

,

then f⁰−1

Corollary 1.3. Let f be a nonconstant entire function. If f and f⁰ share the value 1 IM, and if

N

r, 1 f⁰

<(λ+o(1))T(r, f⁰),

0< λ < 1 4

,

then f⁰−1

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Theorem 1.4. Let f be a nonconstant meromorphic function, k be a positive integer. Iff andf^(k)share the value 1 IM, and if

(3k+ 6)N(r, f) + 5N

r, 1 f

<(λ+o(1))T(r, f^k), (0< λ <1), then

Corollary 1.5. Let f be a nonconstant entire function. Iff andf^(k) share the value 1 IM, and if

N

r, 1 f

<(λ+o(1))T(r, f),

0< λ < 1 10

, then

Theorem 1.6. Let f be a nonconstant meromorphic function, k be a positive integer. Iff andf^(k)share the valuea 6= 0CM, and satisfy one of the following conditions,

(i) δ(0, f) + Θ(∞, f)> _2k+1^4k , (ii) N(r, f) +N

r, ¹_f

<(λ+o(1))T(r, f), 0< λ < _2k+1² ,

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(iii) k+¹₂

N(r, f) + ³₂N

r,_f¹

<(λ+o(1))T(r, f), (0< λ <1).

Thenf ≡f^(k).

Theorem 1.7. Letf be a nonconstant meromorphic function. Iff andf⁰share the valuea6= 0IM, and if

N(r, f) +N

r, 1 f

<(λ+o(1))T(r, f),

0< λ < 2 3

, thenf ≡f⁰.

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2. Some Lemmas

Lemma 2.1 ([7]). Letf be a nonconstant meromorphic function,kbe a positive integer. Then

N

r, 1 f^(k)

< N

r, 1 f

+kN(r, f) +S(r, f), (2.1)

N

r, f^(k) f

< kN(r, f) +kN

r, 1 f

+S(r, f), (2.2)

N

r,f^(k) f

< kN(r, f) +N

r, 1 f

+S(r, f).

(2.3)

Suppose that f andg share the value a IM, and let z₀ be aa-point of f of orderp, aa-point off^(k) of orderq. We denote byN_L

r,_f_(k)¹_−a

the counting function of thosea-points off^(k)whereq > p.

Lemma 2.2. Letfbe a nonconstant meromorphic function. Iff andf^(k)share the value 1 IM, then

(2.4) N_L

r, 1 f^(k)−1

< N

r, 1 f^(k)

+N(r, f) +S(r, f).

Lemma 2.3 ([7]). Letf be a nonconstant meromorphic function,kbe a positive integer. Iff andf^(k)share the value 1 IM, then

T(r, f)<3T(r, f^(k)) +S(r, f), specially iff is a nonconstant entire function, then

T(r, f)<2T(r, f^(k)) +S(r, f).

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3. Proof of Theorem 1.2

Let N₁₎

r,_f−a¹

denote the counting function of the simple zeros of f − a, N₍₂

r,_f_−a¹

denote the counting function of the multiplea-points of f. Each point in these counting functions is counted only once. We denote byN2

r,_f_−a¹

the counting function of the zeros off−a,where a simple zero is counted once and a multiple zero is counted twice. It follows that

(3.1) N₂

r, 1 f−a

=N₁₎

r, 1 f−a

+ 2N₍₂

r, 1 f−a

. Set

F = f⁰⁰⁰

f⁰⁰ − 2f⁰⁰ f⁰−1−

f⁰⁰

f⁰ − 2f⁰ f −1

.

We suppose thatF 6≡0. By the lemma of logarithmic derivatives, we have

(3.2) m(r, F) =S(r, f)

and

(3.3) N(r, F)≤N(r, f) +N

r, 1 f⁰

+N₍₂

r, 1 f⁰−1

+N₀

r, 1

f⁰⁰

+S(r, f),

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where N₍₂

r,_f0¹−1

denotes the counting function of multiple 1-points off⁰, and each 1-point is counted only once; N₀

r,_f¹00

denotes the counting functions off⁰⁰which are not the zeros off⁰andf⁰−1.

Sincef andf⁰ share the value 1 IM, we know thatf −1has only simple zeros. Iff⁰−1also has only simple zeros, thenf andf⁰share the value 1 CM, and Theorem1.2follows by the conclusion of TheoremB.

Now we assume thatf⁰−1has multiple zeros. By calculation, we know that the common simple zeros of f −1and f⁰ −1are the zeros of F; we denote by N_E¹⁾

r,_f−1¹

the counting function of common simple zeros off −1 and f⁰−1. It follows that

(3.4) N_E¹⁾

r, 1 f−1

≤N

r, 1 F

≤T(r, F) =N(r, F) +S(r, f).

From (3.3) and (3.4), we have (3.5) N_E¹⁾

r, 1

f −1

≤N(r, f) +N

r, 1 f⁰

+N₍₂

r, 1 f⁰−1

+N₀

r, 1

f⁰⁰

+S(r, f).

Notice that

(3.6) N

r, 1

f⁰−1

=N_E¹⁾

r, 1 f −1

+N₍₂

r, 1

f⁰−1

.

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By the second fundamental theorem, we have (3.7) T(r, f⁰)< N(r, f) +N

r, 1

f⁰

+N

r, 1 f⁰−1

−N₀

r, 1 f⁰⁰

+S(r, f).

From Lemma2.2, (3.8) N₍₂

r, 1

f⁰−1

=N_L

r, 1 f⁰−1

< N(r, f) +N

r, 1 f⁰

+S(r, f).

Combining (3.5), (3.6), (3.7) and (3.8), we obtain T(r, f⁰)≤N(r, f) +N

r, 1

f⁰

+N_E¹⁾

r, 1 f−1

+N_L

r, 1 f⁰−1

−N₀

r, 1 f⁰⁰

+S(r, f)

≤N(r, f) +N

r, 1 f⁰

+N(r, f) +N

r, 1 f⁰

+ 2N_L

r, 1 f⁰−1

+S(r, f)

≤4N(r, f) + 4N

r, 1 f⁰

+S(r, f),

which contradicts the condition of Theorem1.2. Therefore, we haveF ≡0.By integrating twice, we have

1

f−1 = A

f⁰−1+B,

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whereA6= 0andB are constants.

We distinguish the following three cases.

Case 1. IfB 6= 0,−1, then

f = (B+ 1)f⁰+ (A−B−1) Bf⁰+ (A−B) ,

f⁰ = (B −A)f+ (A−B−1 Bf −(B+ 1) , and so

N r, 1

f⁰+ ^A−B_B

!

=N(r, f)

By the second fundamental theorem T(r, f⁰)< N(r, f⁰) +N

r, 1

f⁰

+N r, 1 f⁰+ ^A−B_B

!

+S(r, f)

<2N(r, f) +N

r, 1 f⁰

+S(r, f), which contradicts the assumption of Theorem1.2.

Case 2. IfB =−1, then

f = A

−f⁰+ (A−1)

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and so

N

r, 1

f⁰−(A+ 1)

=N(r, f).

We also get a contradiction by the second fundamental theorem.

Case 3. IfB = 0, it follows that

f⁰−1 f −1 =A, and the proof of Theorem1.2is thus complete.

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4. Proof of Theorem 1.4

Let

F = f^(k+2)

f^(k+1) −2 f^(k+1)

f^(k)−1 −f⁰⁰

f⁰ + 2 f⁰ f−1

We suppose thatF 6≡0.Since the common zeros (with the same multiplicities) off −1andf^(k)−1are not the poles ofF,and the common simple zeros of f −1andf^(k)−1are the zeros ofF,we have

(4.1) N_E¹⁾

r, 1 f−1

≤N

r, 1 F

≤T(r, F) =N(r, F) +S(r, f).

and

(4.2) N(r, F)≤N(r, f) +N_L

r, 1 f −1

+N_L

r, 1 f^(k)−1

+N₍₂(

r, 1 f

+N₍₂

r, 1

f^(k)

+N₀

r, 1 f⁰

+N₀

r, 1

f^(k+1)

+S(r, f),

whereN₀

r,_f(k+1)¹

denotes the counting function of the zeros off^(k+1)which are not the zeros off^(k) andf^(k)−1, N0

r,_f¹0

denotes the counting function

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of the zeros off⁰which are not the zeros off. Since N

r, 1 f^(k)−1

=N

r, 1 f −1

(4.3)

=N_E¹⁾

r, 1 f −1

+N⁽²_E

r, 1

f −1

+N_L

r, 1 f−1

+N_L

r, 1 f^(k)−1

,

we obtain from (4.1), (4.2) and (4.3) that N

r, 1 f^(k)−1

≤N(r, f) + 2N_L

r, 1 f −1

+N⁽²_E

r, 1

f−1

+N₍₂

r, 1 f

+N

0r, 1 f⁰

+ 2N_L

r, 1 f^(k)−1

+N₍₂

r, 1 f^(k)

+N₀

r, 1

f^(k+1)

+S(r, f)

≤N(r, f) + 2N

r, 1 f⁰

+ 2N_L

r, 1 f^(k)−1

+N₍₂

r, 1 f^(k)

+N₀

r, 1

f^(k+1)

+S(r, f),

whereN⁽²_E

r,_f−1¹

is the counting function of common multiple zeros off−1 andf^(k)−1,each point is counted once. By the second fundamental theorem

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and Lemma2.2, we have

T(r, f^(k))≤N(r, f) +N

r, 1 f^(k)

+N

r, 1 f^(k)−1

−N₀

r, 1 f^(k+1)

+S(r, f)

≤N(r, f) +N

r, 1 f^(k)

+N(r, f) + 2N

r, 1 f⁰

+ 2NL

r, 1 f^(k)−1

+N(2

r, 1

f^(k)

+S(r, f)

≤2N(r, f) +N

r, 1 f^(k)

+ 2N

(r, 1

f⁰

+ 2N

r, 1 f^(k)

+ 2N(r, f) +S(r, f)

≤(3k+ 6)N(r, f) + 5N

r, 1 f

+S(r, f),

which contradicts the assumption of Theorem1.2. HenceF ≡0.

By integrating twice, we get 1

f −1 = C

f^(k)−1 +D,

whereC 6= 0andDare constants. By arguments similar to the proof of Theo- rem1.2, Theorem1.4follows.

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Remark 1. Let f be a non-constant entire function. Then we obtain from Lemma2.3that

1

2T(r, f)≤T(r, f^(k)) +S(r, f).

By Theorem1.4, Corollary1.5holds.

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5. Proof of Theorem 1.6 and Theorem 1.7

Suppose thatf 6≡f^(k). Let

F = f f^(k). Then

(5.1) T(r, F) =m

r, 1 F

+N

r, 1

F

=N

r,f^(k) f

+S(r, f).

Sincef andf^(k)share the valuea 6= 0CM, we have N

r, 1

f−a

≤N

r, 1 f−f^(k)

(5.2)

≤N

r, 1 F −1

≤T(r, F) +O(1).

By the lemma of logarithmic derivatives and the second fundamental theorem, we obtain

(5.3) m

r, 1

f

+m

r, 1 f −a

< m

r, 1 f^(k)

+S(r, f),

and

(5.4) T r, f^(k)

< N

r, 1 f^(k)

+N(r, f^(k)) +N

r, 1 f^(k)−a

+S(r, f),

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from (5.4), we have

(5.5) m

r, 1

f^(k)

< N(r, f) +N

r, 1 f^(k)−a

+S(r, f).

Combining with (5.1), (5.2), (5.3), (5.4), (5.5) and (2.2) of Lemma 2.1, we obtain

2T(r, f)≤m

r, 1 f^(k)

+N

r, 1

f

+N

r, 1 f −a

+S(r, f)

≤N(r, f) +N

r, 1 f^(k)−a

+N

r, 1

f

+N

r, 1 f −a

+S(r, f)

≤N(r, f) +N

r, 1 f

+ 2N

r, 1

f−a

+S(r, f)

≤N(r, f) +N

r, 1 f

+ 2N

r,f^(k)

f

+S(r, f)

≤N(r, f) + 2(kN(r, f) +kN

r, 1 f

+N

r, 1

f

+S(r, f)

≤(2k+ 1)N(r, f) + (2k+ 1)N

r, 1 f

+S(r, f),

which contradicts the assumptions (i) and (ii) of Theorem1.6. Hencef ≡f^(k). Similarly, by the above inequality and (2.3) of Lemma 2.1, and suppose that (iii) is satisfied, then we get a contradiction if f 6≡ f^(k), and we complete the proof of Theorem1.6.

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Remark 2. For a nonconstant meromorphic function f, if f and f⁰ share the valuea 6= 0IM andf 6≡f^(k), since aa-point off is not a zero off⁰, we know thatf−ahas only simple zeros, and we have

N

r, 1 f−a

≤N

r, 1 F −1

≤T(r, F) +O(1),

whereF =f /f⁰.By the arguments similar to the proof of Theorem1.6, Theo- rem1.7follows.

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References

[1] R. BRÜCK, On entire functions which share one value CM with their first derivatives, Result in Math., 30 (1996), 21–24.

[2] G.G. GUNDERSEN AND L.-Z. YANG, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl., 223 (1998), 88–95.

[3] C.C. YANGANDH.-X. YI, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, 2003.

[4] L.-Z. YANG, Solution of a differential equation and its applications, Kodai Math. J., 22 (1999), 458–464.

[5] H.-X. YI, Meromorphic function that share one or two value II, Kodai Math.

J., 22 (1999), 264–272.

[6] H.-X. YIANDC.C. YANG, Uniqueness Theory of Meromorphic Functions, Science Press, Beijing, 1995. [In Chinese].

[7] Q.-C. ZHANG, The uniqueness of meromorohic function with their deriva- tives, Kodai Math. J., 21 (1998), 179–184.