On the Uniqueness Results and Value Distribution of Meromorphic Mappings

This research concentrates on the analysis of meromorphic mappings. We derived several important results for value distribution of specific difference polynomials of meromorphic mappings, which generalize the work of Laine and Yang. In addition, we proved uniqueness theorems of meromorphic mappings. The difference polynomials of these functions have the same fixed points or share a nonzero value. This extends the research work of Qi, Yang and Liu, where they used the finite ordered meromorphic mappings.


Introduction
Let Ω be the set of finite linear measure of positive real numbers, which may not be the same at every occurrence.Assume T(q, α) denotes the Nevanlinna characteristic of a nonconstant meromorphic mapping α and S(q, α) represents any quantity fulfilling S(q, α) = o{T(q, α)}, as q → ∞ and q ∈ Ω.Consider a point c in the extended plane.Indicate two nonconstant meromorphic mappings by α and β.The mappings α and β share the value c IM, if they have the same c-points ignoring multiplicities [1].Also, c is called a small mapping of α, provided that c is a meromorphic mapping fulfilling T(q, c) = S(q, α) [1].All through the current paper, we consider meromorphic mappings in the complex plane and represent the order of α by ρ(α).Consider the following result which was proved by Clunie [2] and Hayman [3]: Theorem 1. Suppose k ≥ 1 be a positive integer.Let α(y) represents a transcendental entire mapping.Then there are infinitely many zeros of α k (y)α (y) − 1.
In this direction, Laine and Yang [4] derived the following result to deal with Problem 1: Theorem 2. Let λ = 0 be a complex number and α(y) be a finite order transcendental entire mapping.Then α(y) k α(y + λ) assumes every finite nonzero value c infinitely often for k ≥ 2.
We now give the following two examples, for details see [4,5].
Recently Liu and Yang proved the following result [5]: Theorem 3. Let k ≥ 2 be an integer and λ = 0 be a complex number.Assume that α(y) be a finite order transcendental entire mapping.Let P(y) ≡ 0 be a polynomial.Then there are infinitely many zeros of α k (y)α(y + λ) − P(y).
We recall the following two examples from [5].
In addition to Theorems 2 and 3 to deal with Problem 1 we will prove the following theorem: Theorem 4. Let k ≥ 7 be an integer.Suppose that the order of a transcendental meromorphic mapping α is given by ρ(α) = ρ < ∞.Let λ be a nonconstant complex number and ∆ λ α ≡ 0 .Assume P ≡ 0 be a polynomial.Then as q −→ ∞ and q ∈ Ω (k − 6)T(q, α) ≤ N q, 1 where Ω ⊂ (1, +∞) is a subset of finite logarithmic measure.
The following definition is borrowed from [6] which will be used in the forthcoming work of this article.
We define difference operators as The proof of Theorem 4 yields the following interesting result, which will be proved in Section 3.
Now consider the example given below, which indicates that the condition "ρ(α) < ∞" in Theorems 4 and 5 is necessary.
From Theorems 4 and 5 we can get the following results respectively.Corollary 1.Let k ≥ 7 be an integer.Suppose that the order of transcendental meromorphic mapping α is given by ρ(α) < ∞.Consider a nonconstant complex number such that ∆ λ α ≡ 0. Assume that P ≡ 0 be a polynomial.Then there are infinitely many zeros of α k ∆ λ α − P.
Corollary 2. Let k ≥ 3 be an integer.Suppose that the order of a transcendental entire mapping α is given by ρ(α) < ∞.Let λ be a nonconstant complex number such that ∆ λ α ≡ 0. Suppose that P ≡ 0 be a polynomial.Then there are infinitely many zeros of α k ∆ λ α − P.
He further studied the following result [7].
From Theorem 4 we will prove the following uniqueness results for meromorphic mappings associated to difference operators.Theorem 8. Suppose that k ≥ 12 be an integer and P ≡ 0 be a polynomial.Let the distinct transcendental meromorphic mappings α and β have finite orders.Assume that λ = 0 be a complex number such that ∆ λ α ≡ 0 and ∆ λ β ≡ 0. Suppose that α k ∆ λ α − P and β k ∆ λ β − P share 0 CM.Then Theorem 9. Let k ≥ 16 be an integer and λ = 0 be a complex number.Assume that the distinct nonconstant meromorphic mappings α and β have finite order.Suppose that α and β share 0, ∞ CM, α k ∆ λ α and then one of the two cases given below holds: Proving Theorem 9 in Section 3, we can obtain the following interesting uniqueness results.In the complex plane, the difference polynomials of the following meromorphic mappings have the same fixed points.Theorem 10.Suppose that k ≥ 16 be an integer and λ = 0 be a complex number.Let the distinct nonconstant meromorphic mappings α and β have finite orders.Suppose that α and β share 0, ∞ CM, α k (y)∆ λ α(y) − y and β k (y)∆ λ β(y) − y share 0 CM.If the inequality (4) holds, then one of the conclusions (i) and (ii) of Theorem 9 can occur.
In view of Theorem 5 and Lemma 2.9, we will derive the following results for entire mappings.
Theorem 11.Assume that k ≥ 5 be an integer, λ = 0 be a complex number and P ≡ 0 be a polynomial.Let the distinct transcendental meromorphic mappings α and β have finite orders.Suppose that ∆ λ α ≡ 0 and The above theorem gives us the following two uniqueness theorems of entire mappings.The difference polynomials of the mentioned mappings share a nonzero constant or have the same fixed points in the plane.
Theorem 12. Suppose that k ≥ 7 be an integer, λ be a nonzero complex number.Let the distinct nonconstant entire mappings α and β have finite order.Assume that α and β share 0, ∞ CM, α k ∆ λ α and β k ∆ λ β share 1 CM.Then one of the following arguments holds.
Theorem 13.Suppose that k ≥ 7 be an integer and λ be a nonzero complex number.Let the distinct nonconstant entire mappings α and β have finite orders.Assume that α and β share 0, ∞ CM, α k (y)∆ λ α(y) − y and β k (y)∆ λ β(y) − y share 0 CM.Then one of the conclusions (i) and (ii) of Theorem 12 holds.

Preliminaries
Building on the previous ideas of meromorphic mapping and Nevanlinna theory, this section contains the fundamental definitions, notions and results required for the further study of the subject.For more details on the concepts briefly discussed, readers are suggested to consult the papers [8][9][10][11][12][13][14].Let c ∈ C ∪ {∞}, p ∈ Z + and α be meromorphic mapping, which is not a constant.Then we give the following three definitions [15,16].Definition 2. The counting mapping of those c-points of α whose multiplicities are not greater than p is denoted N p) (q, 1/(α − c)).The corresponding reduced counting mapping (ignoring multiplicities) is indicated by N p) (q, 1/(α − c)).N (p (q, 1/(α − c)) represents the counting mapping of those c-points of α (counted with proper multiplicities) whose multiplicities are not less than p.By N (p (q, 1/(α − c)) we present the corresponding reduced counting mapping (ignoring multiplicities), where N Definition 3. Assume that k is a nonnegative integer.Let α be a meromorphic mapping, which is not constant.Suppose that c be any value in the extended complex plane.Then we set Definition 4. Let k ≥ 2 be an integer.Assume that α is a meromorphic mapping, which is not constant.The difference operators are defined by , where λ is a nonzero complex number.If λ = 1, we represent ∆ λ α(y) = ∆α(y).Also, Now we state some important lemmas.These lemmas will be used in the proof of our forthcoming results.The following first lemma is borrowed from [13] while second and third lemmas can be found in [17].Lemma 1.In the complex plane, consider a nonconstant meromorphic mapping α.Let c 0 , c 1 , • • • , c k−1 , c k be arbitrary constants and where c k = 0. Then m(q, P(α)) = km(q, α) + O(1).
for every q outside of a set Ω fulfilling lim sup q−→∞ Ω∩[1,q) dt/t log q = 0, i.e., outside of a set Ω of zero logarithmic density.If ρ 2 (α) = ρ 2 < 1 and ε > 0. Then for every q outside of a finite logarithmic measure m q, α(y where ε is a positive number. and δ ∈ (0, 1 − ζ), i.e., the hyper-order of T is strictly less than one.Then where outside of a set of finite logarithmic measure, q runs to infinity.
Lemma 4. Consider two meromorphic mappings F and G, which are nonconstant and G is a Möbius transformation of F. Assume that a subset I ⊂ R + with its linear measure mesI = +∞ exists and as q ∈ I and q −→ ∞, where λ < 1.If a point y 0 ∈ C exists in such a way that F(y 0 ) = G(y 0 ) = 1, then F = G or FG = 1.
Lemma 5. Consider two meromorphic mappings F and G, which are nonconstant.Let F and G share 1 CM.Assume that a subset I ⊂ R + with its linear measure mesI = ∞ exists and where µ < 1, T(q) = max{T(q, F), T(q, G)}.Then F = G or FG = 1.
Lemma 6.Consider two meromorphic mappings F and G, which are nonconstant.Let F and G share 1, ∞ CM.Assume that a subset I ⊂ R + with its linear measure mesI = +∞ exists and as q ∈ I and q −→ ∞, where λ < 1, T(q) = max{T(q, F), T(q, G)} and S(q) = o{T(q)}, as q ∈ I and q −→ ∞.Then F = G or FG = 1.
The following lemma can be found in [19].The following results will be utilized to prove Theorem 9.For its proof see [20].
Lemma 9. Let P(y) be a polynomial of degree k ≥ 1, and let ε > 0 be a given constant.Then we have (i) If δ(P, θ) > 0, then there exists an q(θ) > 0 such that for any q > q(θ), we have (ii) If δ(P, θ) < 0, then there exists an q(θ) < 0 such that for any q > q(θ), we have The proof of the following lemma can be found on page 177 of [21].
Lemma 10.Assume that α(y) be an analytic mapping of y = q iθ , regular in the region D between two straight lines making an angle π/a at the origin and on the lines themselves.Let | f (y)| ≤ M on the lines, where M > 0 be some constant, and that, as |y| → ∞, |α(y)| = O(e q β ), where b < a, uniformly in the angle.Then actually the inequality |α(y)| ≤ M holds throughout the region D.

Proof of Results
In this section, we provide the proof of theorems, stated in first section.
Subcase 2.2.Suppose that In view of the hypothesis that α and β share 0, ∞ CM we get where α is an entire function.Noting that ρ(α) = ρ(β) := ρ < ∞, we can get from (50) that ρ(e P 1 ) ≤ ρ, and so P 1 is a polynomial with degree ≤ ρ.Suppose that P 1 is some constant, then e P 1 is some nonzero constant, say e P 1 = c 2 .Thus from (49) and (50) we get If ∆ λ β = 0, then we can get the conclusion (ii) from (49).Next we suppose that ∆ λ β ≡ 0, and so we have from (51) that c k+1 2 = 1, which together with (50) and e P 1 = c 2 reveals the conclusion (ii) of Theorem 9. Suppose that P 1 is a nonzero polynomial.Then P 1 (y) = (a 1 + ib 1 )y k + • • • be a polynomial of degree k 1 ≥ 1, where a 1 and b 1 are real numbers such that a 1 + ib 1 = 0.By (49) and (50) we have Combining (66) with the assumption that α and β are entire functions sharing 0 CM, we have where f and g are nonconstant polynomials.By substituting (67) into (66) we have e k f (y)+kg(y) [e f (y+λ) − e f (y) ][e g(y+λ) − e g(y) ] = 1 (68) for all y ∈ C. By (68) we have e f (y+λ) − e f (y) = e γ(y) (69) for all y ∈ C, where γ is a polynomial.By (69) and Lemma 7 we can find that e f (y+η)− f (y) = 1 is a constant.Similarly e g(y+λ)−g(y) = 1 is also a constant.Set where for all y ∈ C. From (73) we can find that at least one of e f (y+λ)− f (y) − 1 and e g(y+λ)−g(y) − 1, say e f (y+λ)− f (y) − 1 has a zero of y = 0, and so e f (y+λ)− f (y) is transcendental entire mapping, which implies that e f (y+λ)− f (y) − 1 has infinitely many zeros in the complex plane.But, from (73) we can find that e f (y+λ)− f (y) − 1 at most has one zero of y = 0, this is a contradiction.Thus the proof stands completed.

Conclusions
In the present article, we have proved several important results for value distribution of meromorphic mappings.We hope the techniques used in the present paper will play a key role to provide a framework for the concepts briefly discussed.

Problem 1 .Problem 2 .
It has been shown that the difference polynomials of the mentioned mappings have the same fixed points or share nonzero values.We have provided examples that the previous work of Laine and Yang need generalization.The results have been derived in more general domains.Several uniqueness results of meromorphic mapping have been explored.The research work of Qi, Yang and Liu has been generalized.The current work opens several new research directions.For instance, from Corollary 1, Corollary 2 and Example 3 we give the following problem: What can be said about the conclusion of Corollary 1, if 2 ≤ n ≤ 6?From Theorems 12 and 13 we pose the following problem.What can be said about the conclusions of Theorems 12 and 13, if 2 ≤ n ≤ 6?
Consider two transcendental meromorphic mappings α and β.Then, from Theorem 8 we have