Uniqueness Of Meromorphic Functions With Sharing Values

The uniqueness theory of meromorphic functions mainly studies conditions underwhich there is only one function. We know that the conditions used to determinea transcendental meromorphic function is different from that of a polynomial. Therefore, complicated and interesting. In recent decades, many mathematicians paid close attention to it, which makes it become a very active subject all around the world. Many new results appear as time goes on. For this new subject, the value distribution theory founded by R.Nevanlinna ([1])in 1920s has become the mail tool. Nevanlinna himself gave 5IM theorem and 4CM theorem which are known as classic results in this field. The uniqueness theory of meromorphic functions dealing with shared values originated from some works in R.Nevanlinna. These works lay the foundation of the research on the uniqueness theory. A lot of deep results were obtainedby Xiong Qinglai([3]), Yang Le([1],[26],[31]), Yi Hongxun([1],[2],[29],[30],[31]), Hu Peichu([6]), Yang Lianzhong([27],[28]), Zhang Qingcai([33])and so on. Many other mathematicians such as F.Gross([8]), H.Ueda([7]), R.Bruck([14]), E.Mues([25]), A.Banerjee([10],[11]), I.Lahiri([19],[20],[21],[22],[23]), W.Hayman([9]), A.Sarkar([20]), G.Gundersen([16],[17]) also obtained a lot of elegant results on the research of the uniqueness theory.In this paper, we will give some results on the uniqueness of meromorphic functionswith sharing values under the guidance of Professor Hu Peichu. It consists of three chapters.In chapter 1, we briefly introduce the background of this thesis, which contains some fundamental results and notations of Nevanlinna theory.In chapter 2, we studied uniqueness theorem of meromorphic functions of finitenon-integer(lower) order, which is a generalization and improvement of H.X.Yi ,Weichuan Lin , Lin Yan etc. Main result is stated as follows:Theorem 1. f(z) and g(z) be non-constant meromorphic functions, such that the order X(g) of g(z) is finite and non-integer. Assume that f(z) and g(z) are sharing 0,∞CM, if a₁(z) and a₂(2) arc two distinct small functions of f(z) and g(z) such that a₁(z), a₂(z)(?)0,∞. k₁(≥1) and k₂(≥2) are two integers, if (?)(a_j,f)(?)(a_j,g)(j=1,2) and (?)(∞,f)＞(?), then f(z)≡g(z).Corollary 1. f(z) and g(z) be non-constant entire functions, such that the orderλ(g) of g(z) is finite and non-integer. Assume that f(z) and g(z) are sharing0CM, if a₁(z) and a₂(z) are two distinct small functions of f(z) and g(z) such that a₁(z), a₂(z)(?)0,∞. k₁(≥1) and k₂(≥2) are two integers, such that (?)(a_j,f)(?)(a_j,g)(j=1,2),thenf(z)≡g(z).Corollary 2. f(z) and g(z) be non-constant meromorphic functions, such that the orderλ(g))of g(z) is finite and non-integer. Assume that f(z) and g(z) are sharing 0,∞CM, if there are two distinct finite non-zero complex numbers a₁ and a₂ and two integers k₁(≥1) and k₂(≥2), such that (?)(a_j,f)(?)(a_j,g)(j= 1,2), (k₁ = k₂ =∞) and (?)(∞,f)＞0, then f(z)≡g(z).In chapter 3, we deal with the uniqueness problems of meromorphic functions that share a small function with its differential polynomials. and improve some resultsof Liu, Gu, Lahiri, Zhang, and A.Banerjee, and also answer some questions of Kit-Wing Yu. Main result is stated as follows:Theorem 3. f(z) be a non-constant meromorphic function, k(≥1) and l(≥0) be integers, and a≡a(z) be a non-constant meromorphic small function. Suppose that f -a and L(f) -a share (0,l). Then f≡L(f) if one of the following assump- tions holds, (i) l≥2 and(ii) l=1 and (iii) l=0 andCorollary 3. f(z) be a non-constant entire function, and a≡a(z) be a non-constant meromorphic small function. If f - a and L(f) - a share (0,1), thenδ_2+k(0,f)＞3/5 or if f-a and L(f)-a share (0,0) andδ_2+k(0,f)＞4/5-1/5[2(?)(0,f)+δ₂(0,f) +δ_1+k(0, f) -δ_2+k(0,f)] then f≡L(f).Sinceδ₂(0,f)≥δ_1+k(0,f)≥δ_2+k(0,f)≥(?)(0,f), we have the following corollary.Corollary 4. f(z) be a non-constant meromorphic function, k(≥1) and l(≥0) be integers. and a≡a(z) be a non-constant meromorphic small function. Suppose that f-a and L(f) - a share (0,l). Then f≡L(f) if one of the following assumptionsholds, (i) l≥2 and(ii) l=1 and(iii) l=0 and...