Vol 8, No 2 (2017)

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Available online throug

ISSN 2229 – 5046

International Journal of Mathematical Archive- 8(2), Feb. – 2017 129

UNIQUENESS OF ENTIRE FUNCTIONS CONCERNING q-SHIFT DIFFERENE POLYNOMIALS

HARINA P. WAGHAMORE*

1

_{, RAJESHWARI S.}

2

1,2

_{Department of Mathematics,}

Jnanabharathi Campus, Bangalore University, Bengaluru-560 056, India.

(Received On: 30-12-17; Revised & Accepted On: 09-02-17)

ABSTRACT

I

n this paper, we investigate the uniqueness problem of q-shift difference polynomials sharing a small functions. With the notion of weakly weighted sharing and relaxed weighted sharing we extend some well known previous results.

1. INTRODUCTION, DEFINITIONS AND RESULTS

By a meromorphic function we shall always mean a meromorphic function in the complex plane. Let k be a positive integer or infinity and 𝑎𝑎 ∈ 𝐶𝐶 ∪{∞}. Set 𝐸𝐸(𝑎𝑎,𝑓𝑓) = {𝑧𝑧:𝑓𝑓(𝑧𝑧)− 𝑎𝑎= 0}, where a zero point with multiplicity 𝑘𝑘 is counted k times in the set. If these zeros points are only counted once, then we denote the set by 𝐸𝐸�(𝑎𝑎,𝑓𝑓). Let 𝑓𝑓 and 𝑔𝑔 be two nonconstant meromorphic functions. If 𝐸𝐸(𝑎𝑎,𝑓𝑓) = 𝐸𝐸(𝑎𝑎,𝑔𝑔), then we say that 𝑓𝑓 and 𝑔𝑔 share the value a CM; if 𝐸𝐸�(𝑎𝑎, 𝑓𝑓) =𝐸𝐸�(𝑎𝑎, 𝑔𝑔), then we say that 𝑓𝑓 and 𝑔𝑔 share the value a IM. We denote by 𝐸𝐸_𝑘𝑘₎(𝑎𝑎,𝑓𝑓) the set of all a-points of 𝑓𝑓 with multiplicities not exceeding 𝑘𝑘, where an a-point is counted according to its multiplicity. Also we denote by 𝐸𝐸�𝑘𝑘)(𝑎𝑎, 𝑓𝑓) the set of distinct a-points of 𝑓𝑓 with multiplicities not greater than 𝑘𝑘. It is assumed that the reader is familiar with the notations of Nevanlinna theory such as 𝑇𝑇(𝑟𝑟,𝑓𝑓),𝑚𝑚(𝑟𝑟,𝑓𝑓),𝑁𝑁(𝑟𝑟,𝑓𝑓),𝑁𝑁�(𝑟𝑟,𝑓𝑓),𝑆𝑆(𝑟𝑟,𝑓𝑓) and so on, that can be found, for instance, in [4], [12]. We denote by 𝑁𝑁_𝑘𝑘₎(𝑟𝑟, 1

(𝑓𝑓−𝑎𝑎)) the counting function for zeros of 𝑓𝑓 − 𝑎𝑎 with multiplicity less or equal to 𝑘𝑘, and by𝑁𝑁�_𝑘𝑘₎(𝑟𝑟, 1

(𝑓𝑓−𝑎𝑎)) the corresponding one for which multiplicity is not counted. Let 𝑁𝑁(𝑘𝑘(𝑟𝑟, 1

(𝑓𝑓−𝑎𝑎)) be the counting function for zeros of 𝑓𝑓 − 𝑎𝑎 with multiplicity atleast 𝑘𝑘 and 𝑁𝑁�₍_𝑘𝑘(𝑟𝑟, 1

(𝑓𝑓−𝑎𝑎)) the corresponding one for which multiplicity is not counted.

Set

𝑁𝑁𝑘𝑘�𝑟𝑟,₍_{𝑓𝑓−𝑎𝑎}1 ₎�=𝑁𝑁� �𝑟𝑟,_{𝑓𝑓−𝑎𝑎}1 �+𝑁𝑁�(2�𝑟𝑟,_{𝑓𝑓−𝑎𝑎}1 �+⋯+𝑁𝑁�(𝑘𝑘�𝑟𝑟,_{𝑓𝑓−𝑎𝑎}1 �.

Let 𝑁𝑁𝐸𝐸(𝑟𝑟,𝑎𝑎; 𝑓𝑓,𝑔𝑔)�𝑁𝑁�𝐸𝐸(𝑟𝑟,𝑎𝑎; 𝑓𝑓,𝑔𝑔)�be the counting function (reduced counting function) of all common zeros of 𝑓𝑓 − 𝑎𝑎

and 𝑔𝑔 − 𝑎𝑎 with the same multiplicities and 𝑁𝑁₀(𝑟𝑟,𝑎𝑎; 𝑓𝑓,𝑔𝑔) (𝑁𝑁�₀(𝑟𝑟,𝑎𝑎; 𝑓𝑓,𝑔𝑔)) the counting function (reduced counting

function) of all common zeros of 𝑓𝑓 − 𝑎𝑎 and 𝑔𝑔 − 𝑎𝑎 ignoring multiplicities. If

𝑁𝑁� �𝑟𝑟,_{𝑓𝑓 − 𝑎𝑎�}1 +𝑁𝑁� �𝑟𝑟,_{𝑔𝑔 − 𝑎𝑎}1 � −2𝑁𝑁𝐸𝐸(𝑟𝑟,𝑎𝑎; 𝑓𝑓,𝑔𝑔) =𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔),

then we say that 𝑓𝑓 and 𝑔𝑔 share 𝑎𝑎"CM". On the other hand, if

𝑁𝑁� �𝑟𝑟,_{𝑓𝑓 − 𝑎𝑎�}1 +𝑁𝑁� �𝑟𝑟,_{𝑔𝑔 − 𝑎𝑎}1 � −2𝑁𝑁�0(𝑟𝑟,𝑎𝑎; 𝑓𝑓,𝑔𝑔) =𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔), then we say that 𝑓𝑓and 𝑔𝑔 share 𝑎𝑎 "𝐼𝐼𝐼𝐼".

We now explain in the following definition the notion of weakly weighted sharing which was introduced by Lin and Lin [8].

Definition 1 ([8]): Let 𝑓𝑓 and 𝑔𝑔 share 𝑎𝑎 "𝐼𝐼𝐼𝐼" and 𝑘𝑘 be a positive integer or ∞. 𝑁𝑁�_𝑘𝑘𝐸𝐸₎(𝑟𝑟,𝑎𝑎;𝑓𝑓,𝑔𝑔) denotes the reduced counting function of those 𝑎𝑎 −points of 𝑓𝑓 whose multiplicities are equal to the corresponging a-points of 𝑔𝑔, and both of their multiplicities are not greater than 𝑘𝑘.

Corresponding Author: Harina P. Waghamore*

1

_,

1,2

_{Department of Mathematics,}

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𝑁𝑁�(0𝑘𝑘(𝑟𝑟,𝑎𝑎;𝑓𝑓,𝑔𝑔) denotes the reduced counting function of those a-points of 𝑓𝑓 which are a-points of 𝑔𝑔, and both of their

multiplicities are not less than 𝑘𝑘.

Definition 2 ([8]): For 𝑎𝑎 ∈ 𝐶𝐶 ∪{∞}, if 𝑘𝑘 is a positive integer or ∞ and

𝑁𝑁�𝑘𝑘)�𝑟𝑟,₍_{𝑓𝑓 − 𝑎𝑎}1 ₎� − 𝑁𝑁�𝑘𝑘𝐸𝐸)(𝑟𝑟,𝑎𝑎;𝑓𝑓,𝑔𝑔) =𝑆𝑆(𝑟𝑟,𝑓𝑓),

𝑁𝑁�𝑘𝑘)�𝑟𝑟,₍_{𝑔𝑔 − 𝑎𝑎}1 ₎� − 𝑁𝑁�𝑘𝑘𝐸𝐸)(𝑟𝑟,𝑎𝑎;𝑓𝑓,𝑔𝑔) =𝑆𝑆(𝑟𝑟,𝑔𝑔),

𝑁𝑁�(𝑘𝑘+1�𝑟𝑟,₍_{𝑓𝑓 − 𝑎𝑎}1 ₎� − 𝑁𝑁�(0𝑘𝑘+1(𝑟𝑟,𝑎𝑎;𝑓𝑓,𝑔𝑔) =𝑆𝑆(𝑟𝑟,𝑓𝑓),

𝑁𝑁�(𝑘𝑘+1�𝑟𝑟,₍_{𝑔𝑔 − 𝑎𝑎}1 ₎� − 𝑁𝑁�(0𝑘𝑘+1(𝑟𝑟,𝑎𝑎;𝑓𝑓,𝑔𝑔) =𝑆𝑆(𝑟𝑟,𝑔𝑔),

or of 𝑘𝑘= 0 and 𝑁𝑁� �𝑟𝑟, 1

𝑓𝑓−𝑎𝑎� − 𝑁𝑁�0(𝑟𝑟,𝑎𝑎; 𝑓𝑓,𝑔𝑔) =𝑆𝑆(𝑟𝑟,𝑓𝑓), 𝑁𝑁� �𝑟𝑟, 1

𝑔𝑔−𝑎𝑎� − 𝑁𝑁�0(𝑟𝑟,𝑎𝑎; 𝑓𝑓,𝑔𝑔) =𝑆𝑆(𝑟𝑟,𝑔𝑔),

then we say 𝑓𝑓 and 𝑔𝑔 weakly share 𝑎𝑎 with weight 𝑘𝑘. Here we write 𝑓𝑓,𝑔𝑔 share "(a,k)" to mean that 𝑓𝑓,𝑔𝑔 weakly share 𝑎𝑎 with weight 𝑘𝑘.

Now it is clear from Definition 2 that weakly weighted sharing is a scaling between IM and CM.

Recently, A. Banerjee and S. Mukherjee [1] introduced another sharing notion which is also a scaling between IM and CM but weaker than weakly weighted sharing.

Definition 3 ([1]): We denote by 𝑁𝑁�(𝑟𝑟,𝑎𝑎;𝑓𝑓|=𝑝𝑝;𝑔𝑔| =𝑞𝑞) the reduced counting function of common a-points of 𝑓𝑓 and 𝑔𝑔 with multiplicities 𝑝𝑝 and 𝑞𝑞, respectively.

Definition 4 ([1]): Let 𝑓𝑓,𝑔𝑔 share 𝑎𝑎 "𝐼𝐼𝐼𝐼". Also let 𝑘𝑘 be a positive integer or ∞ and 𝑎𝑎 ∈ 𝐶𝐶 ∪{∞}. If � 𝑁𝑁�(𝑟𝑟,𝑎𝑎;𝑓𝑓|=𝑝𝑝;𝑔𝑔| =𝑞𝑞) =𝑆𝑆(𝑟𝑟),

𝑝𝑝,𝑞𝑞≤𝐾𝐾

then we say 𝑓𝑓 and 𝑔𝑔 share a with weight 𝑘𝑘 in a relaxed manner. Here we write 𝑓𝑓 and 𝑔𝑔 share (𝑎𝑎,𝑘𝑘)∗ to mean that 𝑓𝑓 and 𝑔𝑔 share a with weight 𝑘𝑘 in a relaxed manner.

W.K Hayman proposed the following well-known conjecture in [5].

Hayman’s conjecture: If an entire function 𝑓𝑓 satisfies 𝑓𝑓𝑛𝑛𝑓𝑓′ ≠1 for all positive integers 𝑛𝑛 ∈ 𝑁𝑁, then 𝑓𝑓 is a constant.

It has been verified by Hayman himself in [6] for the case 𝑛𝑛> 1 and Clunie in [3] for the case 𝑛𝑛 ≥1, respectively.

It is well-known that if 𝑓𝑓 and 𝑔𝑔 share four distinct values CM, then 𝑓𝑓 is Mobius transformation of 𝑔𝑔. In 2011, Liu and Cao [10], have obtained results on the uniqueness and value distribution of q-shift difference polynomials. Some of them are stated below.

Theorem A. [10, Theorem 1.1]: Let 𝑓𝑓(𝑧𝑧) be a transcendental meromorphic (resp. entire) function with zero order, and let 𝑚𝑚,𝑛𝑛 be positive integers and 𝑎𝑎,𝑞𝑞 be non-zero complex constants. If 𝑛𝑛 ≥6(𝑟𝑟𝑟𝑟𝑟𝑟𝑝𝑝.𝑛𝑛 ≥2), then 𝑓𝑓𝑛𝑛₍_𝑧𝑧₎₍_𝑓𝑓𝑚𝑚₍_𝑧𝑧₎_{− 𝑎𝑎}₎_𝑓𝑓₍_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎_{− 𝛼𝛼}₍_𝑧𝑧₎_{has infinitely many zeros, where}_𝛼𝛼₍_𝑧𝑧₎_{is a non-zero small function with respect}

to 𝑓𝑓. In particular, if 𝑓𝑓(𝑧𝑧) is a transcendental entire function and 𝛼𝛼(𝑧𝑧) is a non-zero rational function, then 𝑚𝑚 and 𝑛𝑛 can be any positive integers.

Theorem B [10, Theorem 1.5]: Let 𝑓𝑓(𝑧𝑧) and 𝑔𝑔(𝑧𝑧) be a transcendental entire functions with zero order. If 𝑛𝑛 ≥ 𝑚𝑚+ 5, and 𝑓𝑓𝑛𝑛(𝑧𝑧)(𝑓𝑓𝑚𝑚(𝑧𝑧)− 𝑎𝑎)𝑓𝑓(𝑞𝑞𝑧𝑧+𝑐𝑐)and 𝑔𝑔𝑛𝑛(𝑧𝑧)(𝑔𝑔𝑚𝑚(𝑧𝑧)− 𝑎𝑎)𝑔𝑔(𝑞𝑞𝑧𝑧+𝑐𝑐) share a non-zero polynomial 𝑝𝑝(𝑧𝑧) CM, then 𝑓𝑓(𝑧𝑧)≡ 𝑔𝑔(𝑧𝑧).

In 2015, on the basis of Theorems A and B, Q. Zhao and J. Zhang [14] study the k-th derivative of q-shift difference polynomials and proved the following results.

Theorem C: Let 𝑓𝑓(𝑧𝑧) be a transcendental meromorphic function with zero order, and let 𝑛𝑛,𝑘𝑘 be positive integers. If 𝑛𝑛>𝑘𝑘+ 5, then (𝑓𝑓𝑛𝑛(𝑧𝑧)𝑓𝑓(𝑞𝑞𝑧𝑧+𝑐𝑐))(𝑘𝑘)−1 has infinitely many zeros.

Theorem D: Let 𝑓𝑓(𝑧𝑧) be a transcendental entire function with zero order, and let 𝑛𝑛,𝑘𝑘 be positive integers, then

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Theorem E: Let 𝑓𝑓(𝑧𝑧) be a transcendental entire functions with zero order, and let 𝑛𝑛,𝑘𝑘 be positive integers. If 𝑛𝑛> 2𝑘𝑘+ 5, and (𝑓𝑓𝑛𝑛(𝑧𝑧)𝑓𝑓(𝑞𝑞𝑧𝑧+𝑐𝑐))(𝑘𝑘) and 𝑔𝑔𝑛𝑛(𝑧𝑧)𝑔𝑔(𝑞𝑞𝑧𝑧+𝑐𝑐))(𝑘𝑘) share z CM, then 𝑓𝑓=𝑡𝑡𝑔𝑔 for a constant 𝑡𝑡 with 𝑡𝑡𝑛𝑛+1_{= 1.}

Theorem F: Let 𝑓𝑓(𝑧𝑧) be a transcendental entire functions with zero order, and let 𝑛𝑛,𝑘𝑘 be positive integers. If 𝑛𝑛> 2𝑘𝑘+ 5, and (𝑓𝑓𝑛𝑛(𝑧𝑧)𝑓𝑓(𝑞𝑞𝑧𝑧+𝑐𝑐))(𝑘𝑘) and 𝑔𝑔𝑛𝑛(𝑧𝑧)𝑔𝑔(𝑞𝑞𝑧𝑧+𝑐𝑐))(𝑘𝑘) share 1 CM, then 𝑓𝑓=𝑡𝑡𝑔𝑔 for a constant 𝑡𝑡 with 𝑡𝑡𝑛𝑛+1_{= 1.}

When sharing a single value IM, and obtain the following theorems.

Theorem G: Let 𝑓𝑓(𝑧𝑧) and 𝑔𝑔(𝑧𝑧) be transcendental entire functions with zero order, and let 𝑛𝑛,𝑘𝑘 be positive integer. If 𝑛𝑛> 5𝑘𝑘+ 11, and (𝑓𝑓𝑛𝑛(𝑧𝑧)𝑓𝑓(𝑞𝑞𝑧𝑧+𝑐𝑐))(𝑘𝑘) and 𝑔𝑔𝑛𝑛(𝑧𝑧)𝑔𝑔(𝑞𝑞𝑧𝑧+𝑐𝑐))(𝑘𝑘) share z IM, then 𝑓𝑓=𝑡𝑡𝑔𝑔 for a constant 𝑡𝑡 with 𝑡𝑡𝑛𝑛+1_{= 1.}

Theorem H: Let 𝑓𝑓(𝑧𝑧) and 𝑔𝑔(𝑧𝑧) be transcendental entire functions with zero order, and let 𝑛𝑛,𝑘𝑘 be positive integer. If 𝑛𝑛> 5𝑘𝑘+ 11, and (𝑓𝑓𝑛𝑛(𝑧𝑧)𝑓𝑓(𝑞𝑞𝑧𝑧+𝑐𝑐))(𝑘𝑘) and 𝑔𝑔𝑛𝑛(𝑧𝑧)𝑔𝑔(𝑞𝑞𝑧𝑧+𝑐𝑐))(𝑘𝑘) share 1 IM, then 𝑓𝑓=𝑡𝑡𝑔𝑔 for a constant 𝑡𝑡 with 𝑡𝑡𝑛𝑛+1_{= 1.}

In this paper by introducing the small function 𝛼𝛼(𝑧𝑧), we prove the following results.

Theorem 1: Let 𝑓𝑓(𝑧𝑧) and 𝑔𝑔(𝑧𝑧) be a transcendental entire functions of finite order, and𝛼𝛼(𝑧𝑧) be a small function with respect to both 𝑓𝑓(𝑧𝑧) and 𝑔𝑔(𝑧𝑧). Suppose that 𝑐𝑐 is a non-zero complex constant and 𝑛𝑛 ≥4𝑘𝑘+𝑚𝑚+ 6 is an integer. If

(𝑓𝑓𝑛𝑛₍_𝑧𝑧₎₍_𝑓𝑓𝑚𝑚₍_𝑧𝑧₎₋₁₎_𝑓𝑓₍_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎₎(𝑘𝑘)_and₍_𝑔𝑔𝑛𝑛₍_𝑧𝑧₎₍_𝑔𝑔𝑚𝑚₍_𝑧𝑧₎₋₁₎_𝑔𝑔₍_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎₎(𝑘𝑘)_share_"(_𝛼𝛼₍_𝑧𝑧_{), 2)"}_{, then}_𝑓𝑓₍_𝑧𝑧₎

_≡

_𝑔𝑔₍_𝑧𝑧_).

Theorem 2: Let 𝑓𝑓(𝑧𝑧) and 𝑔𝑔(𝑧𝑧) be a transcendental entire functions of finite order, and𝛼𝛼(𝑧𝑧) be a small function with respect to both 𝑓𝑓(𝑧𝑧) and 𝑔𝑔(𝑧𝑧). Suppose that 𝑐𝑐 is a non-zero complex constant and 𝑛𝑛> 6𝑘𝑘+ 3𝑚𝑚+ 8 is an integer. If

(𝑓𝑓𝑛𝑛₍_𝑧𝑧₎₍_𝑓𝑓𝑚𝑚₍_𝑧𝑧₎₋₁₎_𝑓𝑓₍_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎₎(𝑘𝑘)_and₍_𝑔𝑔𝑛𝑛₍_𝑧𝑧₎₍_𝑔𝑔𝑚𝑚₍_𝑧𝑧₎₋₁₎_𝑔𝑔₍_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎₎(𝑘𝑘)_share₍_𝛼𝛼₍_𝑧𝑧_{), 2)}∗_{, then}_𝑓𝑓₍_𝑧𝑧₎

_≡

_𝑔𝑔₍_𝑧𝑧_).

Without the notions of weakly weighted sharing and relaxed weighted sharing we prove the following theorem.

Theorem 3: Let 𝑓𝑓(𝑧𝑧) and 𝑔𝑔(𝑧𝑧) be a transcendental entire functions of finite order, and𝛼𝛼(𝑧𝑧) be a small function with respect to both 𝑓𝑓(𝑧𝑧) and 𝑔𝑔(𝑧𝑧). Suppose that 𝑐𝑐 is a non-zero complex constant and 𝑛𝑛> 10𝑘𝑘+ 5𝑚𝑚+ 12 is an integer. If

E�₂₎�𝛼𝛼(𝑧𝑧),𝑓𝑓𝑛𝑛₍_𝑧𝑧₎₍_𝑓𝑓𝑚𝑚₍_𝑧𝑧₎₋₁₎_𝑓𝑓₍_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎_�₌_E_�

2)(𝛼𝛼(𝑧𝑧),𝑔𝑔𝑛𝑛(𝑧𝑧)(𝑔𝑔𝑚𝑚(𝑧𝑧)−1)𝑔𝑔(𝑞𝑞𝑧𝑧+𝑐𝑐)) then 𝑓𝑓(𝑧𝑧)≡ 𝑔𝑔(𝑧𝑧).

2. LEMMAS

In this section, we present some lemmas which play an important role in the proof of the main results. We will denote by 𝐻𝐻 the following function;

𝐻𝐻=�𝐹𝐹_𝐹𝐹"_′−_{𝐹𝐹 −}2𝐹𝐹′₁� − �𝐺𝐺_𝐺𝐺"_′ −_{𝐺𝐺 −}2𝐺𝐺′₁�

Lemma 1 ([1]):𝐻𝐻 be defined as above. If 𝐹𝐹 and 𝐺𝐺 share "(1,2)" and 𝐻𝐻 ≢0, then

𝑇𝑇(𝑟𝑟,𝐹𝐹)≤ 𝑁𝑁2�𝑟𝑟,_𝐹𝐹�1 +𝑁𝑁2�𝑟𝑟,_𝐺𝐺�1 +𝑁𝑁2(𝑟𝑟,𝐹𝐹) +𝑁𝑁2(𝑟𝑟,𝐺𝐺)− � 𝑁𝑁�(𝑝𝑝�𝑟𝑟,_𝐺𝐺𝐺𝐺_′�+𝑆𝑆(𝑟𝑟,𝐹𝐹) +𝑆𝑆(𝑟𝑟,𝐺𝐺), ∞

𝑝𝑝=3 and the same inequality holds for 𝑇𝑇(𝑟𝑟,𝐺𝐺).

Lemma 2 ([1]): Let 𝐻𝐻 be defined as above. If 𝐹𝐹 and 𝐺𝐺 share (1,2)∗ and 𝐻𝐻 ≢0, then

𝑇𝑇(𝑟𝑟,𝐹𝐹)≤ 𝑁𝑁2�𝑟𝑟,_𝐹𝐹1�+𝑁𝑁2�𝑟𝑟,_𝐺𝐺1�+𝑁𝑁2(𝑟𝑟,𝐹𝐹) +𝑁𝑁2(𝑟𝑟,𝐺𝐺) +𝑁𝑁� �𝑟𝑟,1_𝐹𝐹�+𝑁𝑁�(𝑟𝑟,𝐹𝐹)− 𝑚𝑚 �𝑟𝑟,_{𝐺𝐺 −}1 ₁�+𝑆𝑆(𝑟𝑟,𝐹𝐹) +𝑆𝑆(𝑟𝑟,𝐺𝐺),

and the same inequality holds for 𝑇𝑇(𝑟𝑟,𝐺𝐺).

Lemma 3 ([13]): Let 𝐻𝐻 be defined as above. If 𝐻𝐻 ≡0 and

limsup 𝑟𝑟→∞

𝑁𝑁� �𝑟𝑟, 1_𝐹𝐹�+𝑁𝑁�(𝑟𝑟,𝐹𝐹) +𝑁𝑁� �𝑟𝑟, 1_𝐺𝐺�+𝑁𝑁�(𝑟𝑟,𝐺𝐺)

𝑇𝑇(𝑟𝑟) < 1,𝑟𝑟 ∈ 𝐼𝐼,

where 𝑇𝑇(𝑟𝑟) = max⁡{𝑇𝑇(𝑟𝑟,𝐹𝐹),𝑇𝑇(𝑟𝑟,𝐺𝐺)} and 𝐼𝐼 is a set with infinite linear measure then 𝐹𝐹 ≡ 𝐺𝐺 or 𝐹𝐹𝐺𝐺 ≡1.

Lemma 4 ([2]): Let 𝑓𝑓(𝑧𝑧) be a meromorphic function in the complex plane of finite order 𝜎𝜎(𝑓𝑓), and let 𝜂𝜂 be a fixed non-zero complex number. Then for each 𝜖𝜖> 0, one had

(4)

Lemma 5 ([11]): Let 𝑓𝑓(𝑧𝑧) be an entire function of finite order 𝜎𝜎(𝑓𝑓), c is a fixed non-zero complex number, and 𝑃𝑃(𝑧𝑧) =𝑎𝑎𝑛𝑛𝑓𝑓𝑛𝑛(𝑧𝑧) +𝑎𝑎𝑛𝑛−1𝑓𝑓𝑛𝑛−1(𝑧𝑧) +⋯+𝑎𝑎1𝑓𝑓(𝑧𝑧) +𝑎𝑎0

where 𝑎𝑎_𝑗𝑗(𝑗𝑗= 0,1, … ,𝑛𝑛) are constants. If 𝐹𝐹(𝑧𝑧) =𝑃𝑃(𝑧𝑧)𝑓𝑓(𝑧𝑧+𝑐𝑐), then 𝑇𝑇(𝑟𝑟,𝐹𝐹) = (𝑛𝑛+ 1)𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑂𝑂�𝑟𝑟𝜎𝜎(𝑓𝑓)−1+𝜖𝜖_�₊_𝑂𝑂_(log_𝑟𝑟_).

Lemma 6 ([9]): Let 𝐹𝐹 and 𝐺𝐺 be two nonconstant entire functions, and 𝑝𝑝 ≥2 an integer. If 𝐸𝐸�_𝑝𝑝₎(1,𝐹𝐹) =𝐸𝐸�_𝑝𝑝₎(1,𝐺𝐺) and 𝐻𝐻 ≢0, then

𝑇𝑇(𝑟𝑟,𝐹𝐹) =𝑁𝑁2�𝑟𝑟,_𝐹𝐹�1 +𝑁𝑁2�𝑟𝑟,_𝐺𝐺�1 + 2𝑁𝑁� �𝑟𝑟,_𝐹𝐹�1 +𝑁𝑁� �𝑟𝑟,_𝐺𝐺�1 +𝑆𝑆(𝑟𝑟,𝐹𝐹) +𝑆𝑆(𝑟𝑟,𝐺𝐺).

Lemma 7 ([7]): Let 𝑓𝑓(𝑧𝑧) be a nonconstant meromorphic function, and let s, k be two positive integers. Then

𝑁𝑁𝑟𝑟�𝑟𝑟,_𝑓𝑓1(𝑘𝑘)� ≤ 𝑇𝑇�𝑟𝑟,𝑓𝑓(𝑘𝑘)� − 𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑁𝑁𝑟𝑟+𝑘𝑘�𝑟𝑟,1_𝑓𝑓�+𝑆𝑆(𝑟𝑟,𝑓𝑓),

𝑁𝑁𝑟𝑟�𝑟𝑟,_𝑓𝑓1₍_𝑘𝑘₎� ≤ 𝑘𝑘𝑁𝑁�(𝑟𝑟,𝑓𝑓) +𝑁𝑁𝑟𝑟+𝑘𝑘�𝑟𝑟,1_𝑓𝑓�+𝑆𝑆(𝑟𝑟,𝑓𝑓).

Clearly, 𝑁𝑁� �𝑟𝑟, 1

𝑓𝑓(𝑘𝑘)�=𝑁𝑁1�𝑟𝑟,_𝑓𝑓(1𝑘𝑘)�.

3. PROOF OF THEOREM 1

Let 𝐹𝐹(𝑧𝑧) =[𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚−1)𝑓𝑓(𝑞𝑞𝑧𝑧+𝑐𝑐)](𝑘𝑘)

𝛼𝛼(𝑧𝑧) , 𝐺𝐺(𝑧𝑧) =

[𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚−1)𝑔𝑔(𝑞𝑞𝑧𝑧+𝑐𝑐)](𝑘𝑘)

𝛼𝛼(𝑧𝑧) , Then 𝐹𝐹(𝑧𝑧) and 𝐺𝐺(𝑧𝑧) share "(1,2)" except the zeros or poles of 𝛼𝛼(𝑧𝑧). By Lemma 5, we have

𝑇𝑇�𝑟𝑟,𝐹𝐹(𝑧𝑧)�=𝑇𝑇�𝑟𝑟,𝑓𝑓𝑛𝑛₍_𝑓𝑓𝑚𝑚₋₁₎_𝑓𝑓₍_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎_�₊_{𝑘𝑘𝑁𝑁�}₍_𝑟𝑟_,_𝑓𝑓_{) +}_𝑆𝑆₍_𝑟𝑟_,_𝑓𝑓_).₍₁₎

𝑇𝑇�𝑟𝑟,𝐺𝐺(𝑧𝑧)�=𝑇𝑇�𝑟𝑟,𝑔𝑔𝑛𝑛₍_𝑔𝑔𝑚𝑚 ₋₁₎_𝑔𝑔₍_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎_�₊_{𝑘𝑘𝑁𝑁�}₍_𝑟𝑟_,_𝑔𝑔_{) +}_𝑆𝑆₍_𝑟𝑟_,_𝑔𝑔_).₍₂₎

Also from Lemma 7, we obtain

𝑁𝑁2�𝑟𝑟,1_𝐹𝐹� ≤ 𝑁𝑁𝑘𝑘+2�𝑟𝑟,_𝑓𝑓𝑛𝑛₍_𝑓𝑓𝑚𝑚₋₁₎1_𝑓𝑓₍_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎�+𝑆𝑆(𝑟𝑟,𝑓𝑓) ≤(𝑘𝑘+ 2)𝑁𝑁 �𝑟𝑟,_𝑓𝑓1�+𝑁𝑁 �𝑟𝑟,_𝑓𝑓𝑚𝑚1₋₁�+𝑁𝑁 �𝑟𝑟,

1

𝑓𝑓(𝑞𝑞𝑧𝑧+𝑐𝑐)�+𝑘𝑘𝑁𝑁�(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑓𝑓) (3) ≤(2𝑘𝑘+𝑚𝑚+ 3)𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑓𝑓)

and

𝑁𝑁2�𝑟𝑟,1_𝐺𝐺� ≤(2𝑘𝑘+𝑚𝑚+ 3)𝑇𝑇(𝑟𝑟,𝑔𝑔) +𝑆𝑆(𝑟𝑟,𝑔𝑔) (4)

Suppose 𝐻𝐻 ≢0, then by Lemma 1 and Lemma 4, we have

𝑇𝑇(𝑟𝑟,𝐹𝐹) +𝑇𝑇(𝑟𝑟,𝐺𝐺)≤2𝑁𝑁2�𝑟𝑟,_𝐹𝐹1�+ 2𝑁𝑁2�𝑟𝑟,_𝐺𝐺1�+𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔)

(𝑛𝑛+𝑚𝑚+ 1)[𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑇𝑇(𝑟𝑟,𝑔𝑔)]≤(4𝑘𝑘+ 2𝑚𝑚+ 6)[𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑇𝑇(𝑟𝑟,𝑔𝑔)] +𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔)

(𝑛𝑛 −4𝑘𝑘 − 𝑚𝑚 −5)[𝑇𝑇(𝑟𝑟,𝑓𝑓) +𝑇𝑇(𝑟𝑟,𝑔𝑔)]≤ 𝑂𝑂�𝑟𝑟𝜎𝜎(𝑓𝑓)−1+𝜖𝜖_�₊_{𝑂𝑂�𝑟𝑟}𝜎𝜎(𝑔𝑔)−1+𝜖𝜖_�₊_𝑆𝑆₍_𝑟𝑟_,_𝑓𝑓_{) +}_𝑆𝑆₍_𝑟𝑟_,_𝑔𝑔₎₍₅₎

which contradicts with 𝑛𝑛> 4𝑘𝑘+𝑚𝑚+ 6. Thus we have 𝐻𝐻 ≡0. Note that

𝑁𝑁� �𝑟𝑟,1_𝐹𝐹�+𝑁𝑁� �𝑟𝑟,_𝐺𝐺1� ≤(2𝑘𝑘+𝑚𝑚+ 2)𝑇𝑇(𝑟𝑟,𝑓𝑓) + (2𝑘𝑘+𝑚𝑚+ 2)𝑇𝑇(𝑟𝑟,𝑔𝑔) +𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔) ≤ 𝑇𝑇(𝑟𝑟).

Where 𝑇𝑇(𝑟𝑟) = max{𝑇𝑇(𝑟𝑟,𝐹𝐹),𝑇𝑇(𝑟𝑟,𝐺𝐺)}. By Lemma 3, we deduce that either 𝐹𝐹 ≡ 𝐺𝐺 or 𝐹𝐹𝐺𝐺 ≡1. Next we will consider the following two cases, respectively.

Case-1: 𝐹𝐹 ≡ 𝐺𝐺, thus 𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚−1)𝑓𝑓(𝑞𝑞𝑧𝑧+𝑐𝑐)≡ 𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚−1)𝑔𝑔(𝑞𝑞𝑧𝑧+𝑐𝑐). Let 𝜑𝜑(𝑧𝑧) =𝑓𝑓(𝑧𝑧)

𝑔𝑔(𝑧𝑧). If 𝜑𝜑𝑛𝑛+𝑚𝑚(𝑧𝑧)(𝑞𝑞𝑧𝑧+𝑐𝑐)≢1, we have

𝑔𝑔𝑚𝑚₍_𝑧𝑧_{) =} 𝜑𝜑𝑛𝑛(𝑧𝑧)𝜑𝜑(𝑞𝑞𝑧𝑧+𝑐𝑐)−1

(5)

© 2017, IJMA. All Rights Reserved 133 Then 𝜑𝜑(𝑧𝑧) is a transcendental meromorphic function of finite order since 𝑔𝑔(𝑧𝑧) is transcendental. By Lemma 4, we have 𝑇𝑇�𝑟𝑟,𝜑𝜑(𝑞𝑞𝑧𝑧+𝑧𝑧)�=𝑇𝑇�𝑟𝑟,𝜑𝜑(𝑧𝑧)�+𝑆𝑆(𝑟𝑟,𝜑𝜑). (7)

If 𝜑𝜑𝑛𝑛+𝑚𝑚(𝑧𝑧)𝜑𝜑(𝑧𝑧+𝑐𝑐) =𝑘𝑘(≠1), where 𝑘𝑘 is a constant, the Lemma 4 and (7) imply that

(𝑛𝑛+𝑚𝑚)𝑇𝑇�𝑟𝑟,𝜑𝜑(𝑧𝑧)�=𝑇𝑇�𝑟𝑟,𝜑𝜑(𝑧𝑧+𝑐𝑐)�+𝑂𝑂(1) =𝑇𝑇(𝑟𝑟,𝜑𝜑(𝑧𝑧))+𝑂𝑂�𝑟𝑟𝜎𝜎�𝜑𝜑(𝑧𝑧)�−1+𝜖𝜖�+𝑂𝑂(𝑙𝑙𝑙𝑙𝑔𝑔𝑟𝑟)

which contradicts with 𝑛𝑛 ≥4𝑘𝑘+𝑚𝑚+ 6. Thus 𝜑𝜑𝑛𝑛+𝑚𝑚(𝑧𝑧)𝜑𝜑(𝑞𝑞𝑧𝑧+𝑐𝑐) is not a constant.

Suppose that there exists a point 𝑧𝑧₀ such that 𝜑𝜑𝑛𝑛+𝑚𝑚(𝑧𝑧₀)𝜑𝜑(𝑞𝑞𝑧𝑧₀+𝑐𝑐) = 1. Then 𝜑𝜑𝑛𝑛(𝑧𝑧₀)𝜑𝜑(𝑞𝑞𝑧𝑧₀+𝑐𝑐) = 1 since 𝑔𝑔(𝑧𝑧) is an entire functions. Hence 𝜑𝜑𝑚𝑚(𝑧𝑧₀) = 1 and

𝑁𝑁� �𝑟𝑟,_𝜑𝜑_𝑛𝑛₊_𝑚𝑚₍_𝑧𝑧₎_𝜑𝜑1₍_𝑧𝑧₊_𝑐𝑐₎₋₁� ≤ 𝑁𝑁� �𝑟𝑟,_𝜑𝜑_𝑚𝑚₍_𝑧𝑧1₎₋₁� ≤ 𝑚𝑚𝑇𝑇�𝑟𝑟,𝜑𝜑(𝑧𝑧)�+𝑂𝑂(1).

We apply the second Nevanlinna fundamental theorem to 𝜑𝜑𝑛𝑛+𝑚𝑚(𝑧𝑧)𝜑𝜑(𝑞𝑞𝑧𝑧+𝑐𝑐):

𝑇𝑇�𝑟𝑟,𝜑𝜑𝑛𝑛+𝑚𝑚₍_𝑧𝑧₎_𝜑𝜑₍_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎_{� ≤ 𝑁𝑁��𝑟𝑟}_,_𝜑𝜑𝑛𝑛+𝑚𝑚₍_𝑧𝑧₎_𝜑𝜑₍_𝑧𝑧₊_𝑐𝑐₎_�₊_{𝑁𝑁� �𝑟𝑟}_, 1

𝜑𝜑𝑛𝑛+𝑚𝑚₍_𝑧𝑧₎_𝜑𝜑₍_𝑧𝑧₊_𝑐𝑐₎� +𝑁𝑁� �𝑟𝑟,_𝜑𝜑_𝑛𝑛₊_𝑚𝑚₍_𝑧𝑧₎_𝜑𝜑₍1_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎₋₁�+𝑆𝑆(𝑟𝑟,𝜑𝜑).

≤(𝑚𝑚+ 5)𝑇𝑇�𝑟𝑟,𝜑𝜑(𝑧𝑧)�+𝑆𝑆(𝑟𝑟,𝜑𝜑).

By Lemma 5 we deduce

(𝑛𝑛 − 𝑚𝑚 −4)𝑇𝑇�𝑟𝑟,𝜑𝜑(𝑧𝑧)� ≤ 𝑂𝑂�𝑟𝑟𝜎𝜎�𝜑𝜑(𝑧𝑧)�−1+𝜖𝜖_�₊_𝑆𝑆₍_𝑟𝑟_,_𝜑𝜑_),₍₈₎

which contradicts with 𝑛𝑛 ≥4𝑘𝑘+𝑚𝑚+ 6. So 𝜑𝜑𝑛𝑛+𝑚𝑚(𝑧𝑧)𝜑𝜑(𝑞𝑞𝑧𝑧+𝑐𝑐)≡1. Thus 𝜑𝜑(𝑧𝑧)≡1, that is 𝑓𝑓(𝑧𝑧)≡ 𝑔𝑔(𝑧𝑧).

Case-2: 𝐹𝐹(𝑧𝑧)𝐺𝐺(𝑧𝑧)≡1, that is

𝑓𝑓𝑛𝑛₍_𝑓𝑓𝑚𝑚₋₁₎_𝑓𝑓₍_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎_𝑔𝑔𝑛𝑛₍_𝑔𝑔𝑚𝑚₋₁₎_𝑔𝑔₍_{𝑞𝑞𝑧𝑧}₊_𝑐𝑐₎_{≡ 𝛼𝛼}2₍_𝑧𝑧_).₍₉₎ Since 𝑓𝑓 and 𝑔𝑔 are transcendental entire functions, we can deduce from (9) that 𝑁𝑁 �𝑟𝑟,1

𝑓𝑓�=𝑆𝑆(𝑟𝑟,𝑓𝑓),𝑁𝑁(𝑟𝑟,𝑓𝑓) =𝑆𝑆(𝑟𝑟,𝑓𝑓)

and 𝑁𝑁 �𝑟𝑟, 1

𝑓𝑓−1�=𝑆𝑆(𝑟𝑟,𝑓𝑓). Then 𝛿𝛿(0,𝑓𝑓) +𝛿𝛿(∞,𝑓𝑓) +𝛿𝛿(1,𝑓𝑓) = 3, which contradicts the deficiency relation. This completes the proof of Theorem 1.

[𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚−1)𝑔𝑔(𝑞𝑞𝑧𝑧+𝑐𝑐)](𝑘𝑘) 𝛼𝛼(𝑧𝑧) ,

Then 𝐹𝐹(𝑧𝑧) and 𝐺𝐺(𝑧𝑧) share (1,2)∗ except the zeros or poles of 𝛼𝛼(𝑧𝑧). Obviously

2𝑁𝑁2�𝑟𝑟,_𝐹𝐹1�+ 2𝑁𝑁2�𝑟𝑟,_𝐺𝐺1�+𝑁𝑁� �𝑟𝑟,_𝐹𝐹1�+𝑁𝑁� �𝑟𝑟,_𝐺𝐺1�+𝑆𝑆(𝑟𝑟,𝐹𝐹) +𝑆𝑆(𝑟𝑟,𝐺𝐺)

≤(6𝑘𝑘+ 3𝑚𝑚+ 8)𝑇𝑇(𝑟𝑟,𝑓𝑓) + (6𝑘𝑘+ 3𝑚𝑚+ 8)𝑇𝑇(𝑟𝑟,𝑔𝑔) +𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔). (10)

According to (10) and Lemma 2, we can prove Theorem 2 in a similar way as in Section 3.

[𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚−1)𝑔𝑔(𝑞𝑞𝑧𝑧+𝑐𝑐)](𝑘𝑘) 𝛼𝛼(𝑧𝑧) ,

Then E�₂₎�𝛼𝛼(𝑧𝑧), [𝑓𝑓𝑛𝑛(𝑓𝑓𝑚𝑚−1)𝑓𝑓(𝑞𝑞𝑧𝑧+𝑐𝑐)](𝑘𝑘)�= E�₂₎(𝛼𝛼(𝑧𝑧), [𝑔𝑔𝑛𝑛(𝑔𝑔𝑚𝑚−1)𝑔𝑔(𝑞𝑞𝑧𝑧+𝑐𝑐)](𝑘𝑘)) except the zeros or poles of 𝛼𝛼(𝑧𝑧). Obviously

2𝑁𝑁2�𝑟𝑟,_𝐹𝐹�1 + 2𝑁𝑁2�𝑟𝑟,_𝐺𝐺�1 + 3𝑁𝑁� �𝑟𝑟,_𝐹𝐹�1 + 3𝑁𝑁� �𝑟𝑟,_𝐺𝐺�1 +𝑆𝑆(𝑟𝑟,𝐹𝐹) +𝑆𝑆(𝑟𝑟,𝐺𝐺)

≤(10𝑘𝑘+ 5𝑚𝑚+ 12)𝑇𝑇(𝑟𝑟,𝑓𝑓) + (10𝑘𝑘+ 5𝑚𝑚+ 12)𝑇𝑇(𝑟𝑟,𝑔𝑔) +𝑆𝑆(𝑟𝑟,𝑓𝑓) +𝑆𝑆(𝑟𝑟,𝑔𝑔). (11)

(6)

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