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Journal of Applied Mathematics
VolumeΒ 2011, Article IDΒ 790942, 9 pages
http://dx.doi.org/10.1155/2011/790942
Research Article

Normal Criteria of Function Families Concerning Shared Values

1School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2College of Computer Engineering Technology, Guangdong Institute of Science and Technology, Zhuhai 519090, China
3School of Economic and Management, Guangzhou University of Chinese Medicine, Guangzhou 510006, China

Received 10 July 2011; Accepted 7 September 2011

Academic Editor: Hui-ShenΒ Shen

Copyright Β© 2011 Wenjun Yuan 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.

Abstract

We study the normality of families of meromorphic functions concerning shared values. We consider whether a family of meromorphic functions β„± is normal in 𝐷, if, for every pair of functions 𝑓 and 𝑔 in β„±, π‘“ξ…žβˆ’π‘Žπ‘“βˆ’π‘› and π‘”ξ…žβˆ’π‘Žπ‘”βˆ’π‘› share the value 𝑏, where π‘Ž and 𝑏 are two finite complex numbers such that π‘Žβ‰ 0, 𝑛 is a positive integer. Some examples show that the conditions in our results are best possible.

1. Introduction and Main Results

Let 𝑓(𝑧) and 𝑔(𝑧) be two nonconstant meromorphic functions in a domain π·βŠ†β„‚, and let π‘Ž be a finite complex value. We say that 𝑓 and 𝑔 share π‘Ž CM (or IM) in 𝐷 provided that π‘“βˆ’π‘Ž and π‘”βˆ’π‘Ž have the same zeros counting (or ignoring) multiplicity in 𝐷. When π‘Ž=∞, the zeros of π‘“βˆ’π‘Ž 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 β„‚ 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. Let β„± be a family of holomorphic (meromorphic) functions defined in a domain 𝐷, π‘›βˆˆβ„•,π‘Žβ‰ 0,π‘βˆˆβ„‚. If π‘“ξ…ž(𝑧)+π‘Žπ‘“π‘›(𝑧)βˆ’π‘β‰ 0 for each function 𝑓(𝑧)βˆˆβ„± and 𝑛β‰₯2(𝑛β‰₯3), then β„± is normal in 𝐷.

The results for the holomorphic case are due to Drasin [7] for 𝑛β‰₯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 π‘Ž and 𝑏 replaced by meromorphic functions. The results for the meromorphic case are due to Li [17], Li [18] and Langley [19] for 𝑛β‰₯5, Pang [13] for 𝑛=4, Chen and Fang [14] for 𝑛=3, and Zalcman [20] for 𝑛=3, obtained independently.

When 𝑛=2 and β„± 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 functions β„±={π‘“π‘—βˆš(𝑧)=𝑗𝑧/(π‘—π‘§βˆ’1)2βˆΆπ‘—=1,2,…,} is not normal in 𝐷={π‘§βˆΆ|𝑧|<1}. This is deduced by 𝑓#𝑗(0)=π‘—β†’βˆž, as π‘—β†’βˆž and Marty's criterion [2], although, for any 𝑓𝑗(𝑧)βˆˆβ„±,π‘“ξ…žπ‘—+𝑓2π‘—βˆš=𝑗(π‘—π‘§βˆ’1)βˆ’4β‰ 0.

Here 𝑓#(πœ‰) denotes the spherical derivative 𝑓#||𝑓(πœ‰)=ξ…ž||(πœ‰)||||1+𝑓(πœ‰)2.(1.1)

Theorem 1.3. Let β„± be a family of meromorphic functions in a domain 𝐷, and π‘Žβ‰ 0,π‘βˆˆβ„‚. If π‘“ξ…ž(𝑧)+π‘Ž(𝑓(𝑧))2βˆ’π‘β‰ 0 and the poles of 𝑓(𝑧) are of multiplicity β‰₯3 for each 𝑓(𝑧)βˆˆβ„±, then β„± is normal in 𝐷.

In 2008, by the ideas of shared values, Zhang [11] proved the following.

Theorem 1.4. Let β„± be a family of meromorphic (holomorphic) functions in 𝐷, 𝑛 a positive integer, and π‘Ž, 𝑏 two finite complex numbers such that π‘Žβ‰ 0. If 𝑛β‰₯4(𝑛β‰₯2) and, for every pair of functions 𝑓 and 𝑔 in β„±, π‘“ξ…žβˆ’π‘Žπ‘“π‘› and π‘”ξ…žβˆ’π‘Žπ‘”π‘› share the value 𝑏, then β„± is normal in 𝐷.

Example 1.5 (see [11]). The family of meromorphic functions β„±={π‘“π‘—βˆš(𝑧)=1/(𝑗(π‘§βˆ’(1/𝑗)))βˆΆπ‘—=1,2,…,} is not normal in 𝐷={π‘§βˆΆ|𝑧|<1}. Obviously π‘“ξ…žπ‘—βˆ’π‘“3π‘—βˆš=βˆ’π‘§/(𝑗(π‘§βˆ’(1/𝑗))3). So for each pair π‘š,𝑗,π‘“ξ…žπ‘—βˆ’π‘“3𝑗 and π‘“ξ…žπ‘šβˆ’π‘“3π‘š share the value 0 in 𝐷, but β„± is not normal at the point 𝑧=0, since 𝑓#π‘—βˆš(0)=2(𝑗)3/(1+𝑗)β†’βˆž, as π‘—β†’βˆž.

Remark 1.6. Example 1.5 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. Let β„± be a family of meromorphic functions in 𝐷, 𝑛 a positive integer, and π‘Ž, 𝑏 two finite complex numbers such that π‘Žβ‰ 0. If 𝑛β‰₯2 and, for every pair of functions 𝑓 and 𝑔 in β„±, π‘“ξ…žβˆ’π‘Žπ‘“βˆ’π‘› and π‘”ξ…žβˆ’π‘Žπ‘”βˆ’π‘› share the value 𝑏, then β„± is normal in 𝐷.

Example 1.8. The family of holomorphic functions β„±={π‘“π‘—βˆš(𝑧)=𝑗(π‘§βˆ’(1/𝑗))βˆΆπ‘—=1,2,…,} is not normal in 𝐷={π‘§βˆΆ|𝑧|<1}. This is deduced by 𝑓#π‘—βˆš(0)=𝑗𝑗/(𝑗+1)β†’βˆž, as π‘—β†’βˆž and Marty's criterion [2], although, for any 𝑓𝑗(𝑧)βˆˆβ„±,π‘“ξ…žπ‘—+π‘“π‘—βˆ’1√=𝑗𝑗𝑧/(π‘—π‘§βˆ’1).

Remark 1.9. Example 1.8 shows that the condition that added 𝑛β‰₯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. Let β„± be a family of meromorphic functions in 𝐷, 𝑛β‰₯2, and let π‘Ž be a nonzero finite complex number. If, for every pair of functions 𝑓 and 𝑔 in β„±, π‘“π‘›π‘“ξ…ž and π‘”π‘›π‘”ξ…ž share the value π‘Ž, then β„± is normal in 𝐷.

A natural problem is what conditions are added such that Theorem 1.7 holds when 𝑛=1. Next we give an answer.

Theorem 1.11. Let β„± be a family of meromorphic functions in 𝐷, and let π‘Ž and 𝑏 be two finite complex numbers such that π‘Žβ‰ 0. Suppose that all of zeros are multiple for each 𝑓(𝑧)βˆˆβ„±. If, for every pair of functions 𝑓 and 𝑔 in β„±, π‘“ξ…žβˆ’π‘Žπ‘“βˆ’1 and π‘”ξ…žβˆ’π‘Žπ‘”βˆ’1 share the value 𝑏, then β„± is normal in 𝐷.

Remark 1.12. Example 1.8 shows that the condition that all of zeros are multiple for each 𝑓(𝑧)βˆˆβ„± added in Theorem 1.7 is best possible. In Theorem 1.11 taking 𝑏=0 we get Corollary 1.13.

Corollary 1.13. Let β„± be a family of meromorphic functions in 𝐷, and let π‘Ž be a nonzero finite complex number. Suppose that all of zeros are multiple for each 𝑓(𝑧)βˆˆβ„±. If, for every pair of functions 𝑓 and 𝑔 in β„±, π‘“π‘“ξ…ž and π‘”π‘”ξ…ž share the value π‘Ž, then β„± is normal in 𝐷.

From the proof of Theorem 1.7 we know that the following corollary holds.

Corollary 1.14. Let β„± be a family of meromorphic functions in 𝐷, 𝑛 be a positive integer and π‘Ž, 𝑏 be two finite complex numbers such that π‘Žβ‰ 0. If for each function 𝑓 in β„±, π‘“ξ…žβˆ’π‘Žπ‘“βˆ’π‘›β‰ π‘, then β„± 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]). Let β„± be a family of meromorphic functions on the unit disc satisfying all zeros of functions in β„± that have multiplicity β‰₯𝑝 and all poles of functions in β„± that have multiplicity β‰₯π‘ž. Let 𝛼 be a real number satisfying βˆ’π‘ž<𝛼<𝑝. Then β„± is not normal at 0 if and only if there exist
(a)anumber0<π‘Ÿ<1; (b)points𝑧𝑛with|𝑧𝑛|<π‘Ÿ; (c)functionsπ‘“π‘›βˆˆβ„±; (d)positivenumbersπœŒπ‘›β†’0such that 𝑔𝑛(𝜁)∢=πœŒβˆ’π›Όπ‘“π‘›(𝑧𝑛+πœŒπ‘›πœ) converges spherically uniformly on each compact subset of β„‚ to a nonconstant meromorphic function 𝑔(𝜁), whose all zeros have multiplicity β‰₯𝑝 and all poles have multiplicity β‰₯π‘ž and order is at most 2.

Remark 2.2. If β„± 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 of 𝑔 is defined by using Nevanlinna's characteristic function 𝑇(π‘Ÿ,𝑔): 𝜌(𝑔)=limπ‘Ÿβ†’βˆžsuplog𝑇(π‘Ÿ,𝑔).logπ‘Ÿ(2.1)

Lemma 2.3 (see [25] or [26]). Let 𝑓(𝑧) be a meromorphic function and π‘βˆˆβ„‚β§΅{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]). Let 𝑓(𝑧) be a transcendental meromorphic function of finite order in β„‚ and have no simple zero, then 𝑓′(𝑧) assumes every nonzero finite value infinitely often.

3. Proof of the Results

Proof of Theorem 1.7. Suppose that β„± is not normal in 𝐷. Then there exists at least one point 𝑧0 such that β„± 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 πœŒπ‘—β†’0, and functions π‘“π‘—βˆˆβ„± such that 𝑔𝑗(πœ‰)=πœŒπ‘—βˆ’1/(𝑛+1)𝑓𝑗𝑧𝑗+πœŒπ‘—πœ‰ξ€ΈβŸΉπ‘”(πœ‰)(3.1) locally uniformly with respect to the spherical metric, where 𝑔 is a nonconstant meromorphic function in β„‚. Moreover, the order of 𝑔 is ≀2.
From (3.1) we know π‘”ξ…žπ‘—(πœ‰)=πœŒπ‘—π‘›/(𝑛+1)π‘“ξ…žπ‘—ξ€·π‘§π‘—+πœŒπ‘—πœ‰ξ€ΈβŸΉπ‘”ξ…žπœŒ(πœ‰),𝑗𝑛/(𝑛+1)𝑓𝑗′𝑧𝑗+πœŒπ‘—πœ‰ξ€Έβˆ’π‘Žπ‘“π‘—βˆ’π‘›ξ€·π‘§π‘—+πœŒπ‘—πœ‰ξ€Έξ€Έβˆ’π‘=𝑔𝑗′(πœ‰)βˆ’π‘Žπ‘”π‘—βˆ’π‘›(πœ‰)βˆ’πœŒπ‘—π‘›/(𝑛+1)π‘βŸΉπ‘”ξ…ž(πœ‰)βˆ’π‘Žπ‘”βˆ’π‘›(πœ‰)(3.2) in ℂ⧡𝐒 locally uniformly with respect to the spherical metric, where 𝐒 is the set of all poles of 𝑔(πœ‰).
If π‘”β€²π‘”π‘›βˆ’π‘Žβ‰‘0, then βˆ’1/(𝑛+1)𝑔𝑛+1β‰‘π‘Žπœ‰+𝑐, where 𝑐 is a constant. This contradicts with 𝑔 being a meromorphic function. So π‘”β€²π‘”π‘›βˆ’π‘Žβ‰’0.
If π‘”β€²π‘”π‘›βˆ’π‘Žβ‰ 0, by Lemma 2.3, then 𝑔 is also a constant which is a contradiction with 𝑔 being a nonconstant. Hence, π‘”β€²π‘”π‘›βˆ’π‘Ž is a nonconstant meromorphic function and has at least one zero.
Next we prove that π‘”β€²π‘”π‘›βˆ’π‘Ž has just a unique zero. On the contrary, let πœ‰0 and πœ‰βˆ—0 be two distinct zeros of π‘”β€²π‘”π‘›βˆ’π‘Ž, and choose 𝛿(>0) small enough such that 𝐷(πœ‰0,𝛿)∩𝐷(πœ‰βˆ—0,𝛿)=πœ™, where 𝐷(πœ‰0,𝛿)={πœ‰βˆΆ|πœ‰βˆ’πœ‰0|<𝛿} and 𝐷(πœ‰βˆ—0,𝛿)={πœ‰βˆΆ|πœ‰βˆ’πœ‰βˆ—0|<𝛿}. From (3.2), by π»π‘’π‘Ÿπ‘€π‘–π‘‘π‘§β€²π‘  theorem, there exist points πœ‰π‘—βˆˆπ·(πœ‰0,𝛿), πœ‰βˆ—π‘—βˆˆπ·(πœ‰βˆ—0,𝛿) such that for sufficiently large π‘—π‘“ξ…žπ‘—ξ€·π‘§π‘—+πœŒπ‘—πœ‰π‘—ξ€Έβˆ’π‘Žπ‘“π‘—βˆ’π‘›ξ€·π‘§π‘—+πœŒπ‘—πœ‰π‘—ξ€Έπ‘“βˆ’π‘=0,ξ…žπ‘—ξ€·π‘§π‘—+πœŒπ‘—πœ‰βˆ—π‘—ξ€Έβˆ’π‘Žπ‘“π‘—βˆ’π‘›ξ€·π‘§π‘—+πœŒπ‘—πœ‰βˆ—π‘—ξ€Έβˆ’π‘=0.(3.3)
By the hypothesis that, for each pair of functions 𝑓 and 𝑔 in β„±, π‘“β€²βˆ’π‘Žπ‘“βˆ’π‘› and π‘”β€²βˆ’π‘Žπ‘”βˆ’π‘› share 𝑏 in 𝐷, we know that for any positive integerπ‘šπ‘“ξ…žπ‘šξ€·π‘§π‘—+πœŒπ‘—πœ‰π‘—ξ€Έβˆ’π‘Žπ‘“π‘šβˆ’π‘›ξ€·π‘§π‘—+πœŒπ‘—πœ‰π‘—ξ€Έπ‘“βˆ’π‘=0,ξ…žπ‘šξ€·π‘§π‘—+πœŒπ‘—πœ‰βˆ—π‘—ξ€Έβˆ’π‘Žπ‘“π‘šβˆ’π‘›ξ€·π‘§π‘—+πœŒπ‘—πœ‰βˆ—π‘—ξ€Έβˆ’π‘=0.(3.4)
Fix π‘š, take π‘—β†’βˆž, and note 𝑧𝑗+πœŒπ‘—πœ‰π‘—β†’0, 𝑧𝑗+πœŒπ‘—πœ‰βˆ—π‘—β†’0, then π‘“π‘šβ€²(0)βˆ’π‘Žπ‘“π‘šβˆ’π‘›(0)βˆ’π‘=0. Since the zeros of π‘“π‘šβ€²βˆ’π‘Žπ‘“π‘šβˆ’π‘›βˆ’π‘ have no accumulation point, so 𝑧𝑗+πœŒπ‘—πœ‰π‘—=0,𝑧𝑗+πœŒπ‘—πœ‰βˆ—π‘—=0.(3.5) Hence, πœ‰π‘—=βˆ’π‘§π‘—/πœŒπ‘—,πœ‰βˆ—π‘—=βˆ’π‘§π‘—/πœŒπ‘—. This contradicts with πœ‰π‘—βˆˆπ·(πœ‰0,𝛿),πœ‰βˆ—π‘—βˆˆπ·(πœ‰βˆ—0,𝛿), and 𝐷(πœ‰0,𝛿)∩𝐷(πœ‰βˆ—0,𝛿)=πœ™. So π‘”β€²π‘”π‘›βˆ’π‘Ž has just a unique zero, which can be denoted by πœ‰0. By Lemma 2.4, 𝑔 is not any transcendental function.
If 𝑔 is a nonconstant polynomial, then π‘”β€²π‘”π‘›βˆ’π‘Ž=𝐴(πœ‰βˆ’πœ‰0)𝑙, where A is a nonzero constant, 𝑙 is a positive integer, because 𝑙β‰₯𝑛β‰₯3. Set πœ™=(1/(𝑛+1))𝑔𝑛+1, then πœ™ξ…ž=𝐴(πœ‰βˆ’πœ‰0)𝑙+π‘Ž and πœ™ξ…žξ…ž=𝐴𝑙(πœ‰βˆ’πœ‰0)π‘™βˆ’1. Note that the zeros of πœ™ are of multiplicity β‰₯4. But πœ™ξ…žξ…ž has only one zero πœ‰0, so πœ™ has only the same zero πœ‰0 too. Hence, πœ™ξ…ž(πœ‰0)=0 which contradicts with πœ™ξ…ž(πœ‰0)=π‘Žβ‰ 0. Therefore, 𝑔 and πœ™ are rational functions which are not polynomials, and πœ™ξ…žβˆ’π‘Ž has just a unique zero πœ‰0.
Next we prove that there exists no rational function such as πœ™. Noting that πœ™=(1/(𝑛+1))𝑔𝑛+1, we can set ξ€·πœ™(πœ‰)=π΄πœ‰βˆ’πœ‰1ξ€Έπ‘š1ξ€·πœ‰βˆ’πœ‰2ξ€Έπ‘š2β‹―ξ€·πœ‰βˆ’πœ‰π‘ ξ€Έπ‘šπ‘ ξ€·πœ‰βˆ’πœ‚1𝑛1ξ€·πœ‰βˆ’πœ‚2𝑛2β‹―ξ€·πœ‰βˆ’πœ‚π‘‘ξ€Έπ‘›π‘‘,(3.6) where 𝐴 is a nonzero constant, 𝑠β‰₯1,𝑑β‰₯1,π‘šπ‘–β‰₯𝑛+1β‰₯3(𝑖=1,2,…,𝑠),𝑛𝑗β‰₯𝑛+1β‰₯3(𝑗=1,2,…,𝑑). For stating briefly, denote π‘š=π‘š1+π‘š2+β‹―+π‘šπ‘ β‰₯3𝑠,𝑁=𝑛1+𝑛2+β‹―+𝑛𝑑β‰₯3𝑑.(3.7) From (3.6), πœ™ξ…žπ΄ξ€·(πœ‰)=πœ‰βˆ’πœ‰1ξ€Έπ‘š1βˆ’1ξ€·πœ‰βˆ’πœ‰2ξ€Έπ‘š2βˆ’1β‹―ξ€·πœ‰βˆ’πœ‰π‘ ξ€Έπ‘šπ‘ βˆ’1β„Ž(πœ‰)ξ€·πœ‰βˆ’πœ‚1𝑛1+1ξ€·πœ‰βˆ’πœ‚2𝑛2+1β‹―ξ€·πœ‰βˆ’πœ‚π‘‘ξ€Έπ‘›π‘‘+1=𝑝1(πœ‰)π‘ž1(πœ‰),(3.8) where β„Ž(πœ‰)=(π‘šβˆ’π‘βˆ’π‘‘)πœ‰π‘ +π‘‘βˆ’1+π‘Žπ‘ +π‘‘βˆ’2πœ‰π‘ +π‘‘βˆ’2+β‹―+π‘Ž0,𝑝1(ξ€·πœ‰)=π΄πœ‰βˆ’πœ‰1ξ€Έπ‘š1βˆ’1ξ€·πœ‰βˆ’πœ‰2ξ€Έπ‘š2βˆ’1β‹―ξ€·πœ‰βˆ’πœ‰π‘ ξ€Έπ‘šπ‘ βˆ’1π‘žβ„Ž(πœ‰),1ξ€·(πœ‰)=πœ‰βˆ’πœ‚1𝑛1+1ξ€·πœ‰βˆ’πœ‚2𝑛2+1β‹―ξ€·πœ‰βˆ’πœ‚π‘‘ξ€Έπ‘›π‘‘+1(3.9) are polynomials. Since πœ™ξ…ž(πœ‰)+π‘Ž has only a unique zero πœ‰0, set πœ™ξ…žπ΅ξ€·(πœ‰)+π‘Ž=πœ‰βˆ’πœ‰0ξ€Έπ‘™ξ€·πœ‰βˆ’πœ‚1𝑛1+1ξ€·πœ‰βˆ’πœ‚2𝑛2+1β‹―ξ€·πœ‰βˆ’πœ‚π‘‘ξ€Έπ‘›π‘‘+1,(3.10) where 𝐡 is a nonzero constant, so πœ™ξ…žξ…žξ€·(πœ‰)=πœ‰βˆ’πœ‰0ξ€Έπ‘™βˆ’1𝑝2(πœ‰)ξ€·πœ‰βˆ’πœ‚1𝑛1+2ξ€·πœ‰βˆ’πœ‚2𝑛2+2β‹―ξ€·πœ‰βˆ’πœ‚π‘‘ξ€Έπ‘›π‘‘+2,(3.11) where 𝑝2(πœ‰)=𝐡(π‘™βˆ’π‘βˆ’2𝑑)πœ‰π‘‘+π‘π‘‘βˆ’1πœ‰π‘‘βˆ’1+β‹―+𝑏0 is a polynomial. From (3.8) we also have πœ™ξ…žξ…žξ€·(πœ‰)=πœ‰βˆ’πœ‰1ξ€Έπ‘š1βˆ’2ξ€·πœ‰βˆ’πœ‰2ξ€Έπ‘š2βˆ’2β‹―ξ€·πœ‰βˆ’πœ‰π‘ ξ€Έπ‘šπ‘ βˆ’2𝑝3(πœ‰)ξ€·πœ‰βˆ’πœ‚1𝑛1+2ξ€·πœ‰βˆ’πœ‚2𝑛2+2β‹―ξ€·πœ‰βˆ’πœ‚π‘‘ξ€Έπ‘›π‘‘+2,(3.12) where 𝑝3(πœ‰) is also a polynomial.
Let deg(𝑝) denote the degree of a polynomial 𝑝(πœ‰).
From (3.8) and (3.9), 𝑝deg(β„Ž)≀𝑠+π‘‘βˆ’1,deg1ξ€Έξ€·π‘žβ‰€π‘š+π‘‘βˆ’1,deg1ξ€Έ=𝑁+𝑑.(3.13) Similarly from (3.11), (3.12) and noting (3.13), 𝑝deg2≀𝑑,(3.14)𝑝deg3𝑝≀deg1ξ€Έ+π‘‘βˆ’1βˆ’(π‘šβˆ’2𝑠)≀2𝑑+2π‘ βˆ’2.(3.15)
Note that π‘šπ‘–β‰₯3(𝑖=1,2,…,𝑠), it follows from (3.8) and (3.10) that πœ™ξ…ž(πœ‰π‘–)=0(𝑖=1,2,…,𝑠) and πœ™ξ…ž(πœ‰0)=π‘Žβ‰ 0. Thus, πœ‰0β‰ πœ‰π‘–(𝑖=1,2,…,𝑠), and then (πœ‰βˆ’πœ‰0)π‘™βˆ’1 is a factor of 𝑝3(πœ‰). Hence, we get that π‘™βˆ’1≀deg(𝑝3). Combining (3.11) and (3.12) we also have π‘šβˆ’2𝑠=deg(𝑝2)+π‘™βˆ’1βˆ’deg(𝑝3)≀deg(𝑝2). By (3.14) we obtain ξ€·π‘π‘šβˆ’2𝑠≀deg2≀𝑑.(3.16)
Since π‘šβ‰₯3𝑠, we know by (3.16) that 𝑠≀𝑑.(3.17)
If 𝑙β‰₯𝑁+𝑑, by (3.15), then 𝑝4π‘‘βˆ’1≀𝑁+π‘‘βˆ’1β‰€π‘™βˆ’1≀deg3≀2𝑑+2π‘ βˆ’2.(3.18) Noting (3.17), we obtain 1≀0; a contradiction.
If 𝑙<𝑁+𝑑, from (3.8) and (3.10), then deg(𝑝1)=deg(π‘ž1). Noting that deg(β„Ž)≀𝑠+π‘‘βˆ’1, deg(𝑝1)β‰€π‘š+π‘‘βˆ’1, and deg(π‘ž1)=𝑁+𝑑, hence π‘šβ‰₯𝑁+1β‰₯3𝑑+1. By (3.16), 2𝑑+1≀2𝑠. From (3.17), we obtain 1≀0; a contradiction.
The proof of Theorem 1.7 is complete.

Proof of Theorem 1.11. The proof of this theorem is the same as the proof of Theorem 1.7, some different places are stated as follows.
The zeros of 𝑔 are multiple; 𝑙β‰₯2𝑛+1=3.(3.19)
The zeros of πœ™ are of multiplicity β‰₯4: π‘šπ‘–β‰₯2(𝑛+1)=4(𝑖=1,2,…,𝑠),𝑛𝑗β‰₯𝑛+1=2(𝑗=1,2,…,𝑑);(3.20)π‘š=π‘š1+π‘š2+β‹―+π‘šπ‘ β‰₯4𝑠,𝑁=𝑛1+𝑛2+β‹―+𝑛𝑑β‰₯2𝑑.((3.7)ξ…ž)
Noting π‘šβ‰₯4𝑠, by (3.16) we have 2𝑠≀𝑑.((3.17)ξ…ž)
If 𝑙β‰₯𝑁+𝑑, by (3.15), then 𝑝3π‘‘βˆ’1≀𝑁+π‘‘βˆ’1β‰€π‘™βˆ’1≀deg3≀2𝑑+2π‘ βˆ’2.(3.21) Noting (3.17), we obtain 1≀0; a contradiction.
If 𝑙<𝑁+𝑑, from (3.8) and (3.10), then deg(𝑝1)=deg(π‘ž1). Noting that deg(β„Ž)≀𝑠+π‘‘βˆ’1, deg(𝑝1)β‰€π‘š+π‘‘βˆ’1, and deg(π‘ž1)=𝑁+𝑑, hence π‘šβ‰₯𝑁+1β‰₯2𝑑+1. By (3.16), 2𝑑+1≀2𝑠+𝑑. From ((3.17)ξ…ž), we obtain 1≀0; a contradiction.
The proof of Theorem 1.11 is complete.

Acknowledgment

The authors would like to express their hearty thanks to Professor Qingcai Zhang for supplying them his helpful reprint. They wish to thank the managing editor and referees for their very helpful comments and useful suggestions. This work was completed of the support with the NSF of China (10771220) and Doctorial Point Fund of National Education Ministry of China (200810780002).

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