Uniqueness Theorems Of Meromorphic Functions And Their Derivatives

In 1920s, R. Nevanlinna introduced the characteristic functions of meromorphic functions and gave the famous Nevanlinna theory which is one of the greatest achievements in mathematics in the 20th century. This theory is considered to be the basis of modern meromorphic function theory, and it has a very important effect on the development and syncretic of many mathematical branches. Especially, Nevanlinna theory has been used as a very powerful tool and has made the research field more active and popular after it was successfully applied to the research of the global analytic solutions of complex differential equations. Taking advantage of the theory just made by himself, R. Nevanlinna[1] studied the conditions with which a meromorphic function can be determined and obtained two celebrated uniqueness theorems on meromorphic functions, which are called Nevanlinna's five-value theorem and four-value theorem. Since then, the research of meromorphic functions began. For over a half century, many foreign and domestic mathematicians have devoted themselves to the research and obtained lots of elegant results on the research of the uniqueness theory. In the past two decades, Professor Yi Hongxun[3][4] did much creative work on the uniqueness theory of meromorphic functions, and well improved the development of the uniqueness theory. In this paper, we will give some results on the uniqueness of memomorphic functions under the guidance of Professor Hu Pei-chu. This thesis consists of five chapters.In chapter 1, we will briefly introduce some fundamental results, definitions and some notations.In chapter 2, we study the uniqueness theorems of meromorphic functions sharing three values with the notion of weighted sharing which was introduced by I.Lahiri [21]. We relax the weight of one of the three shared values to 0 and improve the results given by M.Ozawa [37], I.Lahiri [23] [24] [25] [26] and others [67] . The main theorems are in the following.Theorem 1 Let / and g be two nonconstant meromorphic functions sharing (0,m),(∞,0)and (1,1), where m≥2. Ifthen either f≡g or fg≡1.In chapter 3, we study the uniqueness theorems of entire functions sharing two sets with their k-th derivatives. The results in this thesis improve the results given by Fang [9]. The main theorems are in the following.Theorem 2 Let n, k be positive integers, and let S₁={z:zⁿ=1},S₂= {a₁,a₂,…,a_m}, where a₁,a₂,…,a_m are distinct nonzero constants. If f and g satisfy one of the followings:then one of the following cases must occur: (1) f=tg,(a₁,a₂,…,a_m)=t(a₁,a₂,…,a_m), where t is a constant satisfying tⁿ = 1; (2) f(z)=de^cz,g(z)=(?),(a₁,a₂,…,a_m)= (-1)^kc^2kt((?)), where t, c, d are nonzero constants and tⁿ = 1.In chapter 4, we study the uniqueness theorems of transcendental entire functions sharing one value. The results in this thesis improve the results given by Fang [10], Xu and Qu [45]. The main theorems are in the following.Theorem 3 Let f(z) and g(z) be two transcendental entire functions, n, k two positive integers with n≥5k+8. If [fⁿ(z)]^k and [gⁿ(z)]^k share 1 IM. Then ei-ther f(z) = c₁e^cz, g(z) = c₂e^-cz, where c₁,c₂ and c are three constants satisfying (-1)^k(c₁c₂)ⁿ(nc)^2k=1 or f(z) = tg(z) for a constant t such that tⁿ = 1.Theorem 4 Let f(z) and g(z) be two transcendental entire functions, n, m, k be three positive integers with n+m>10k+14. If [fⁿ(z)(f(z)-1)^m]^k and [gⁿ(z)(g(z)-1)^m]^k share 1 IM. Then f and g satisfy the algebraic equation R(f, g)≡0, where R(ω₁,ω₂)=ω₁ⁿ(ω₁-1)^m-ω₂ⁿ(ω₂-1)^m.Theorem 5 Let f and g be two nonconstant meromorphic functions, n an integer and a∈C-{0} If f and g satisfy one of the followings: then either f = tg for some (n + l)th root of unity t or g(z)=c₁e^cz and f(z)=c₂e^-cz where c, c₁ and c₂ are constants and satisfy (c₁c₂)ⁿ⁺¹c²=-a².In this chapter, we also study the uniqueness problems on meromorphic function and its kth order derivative. The results in this paper improve the results Liu-Gu [35]. The main theorems are in the following.Theorem 6 Let f be a nonconstant meromorphic function, and let a be a small function of f such that a(z)(?)0,∞, If f and g satisfy one of the followings:In chapter 5, we study the uniqueness theorems of meromorphic functions con-cerning differential polynomial. The results in this thesis improve the results of Lin and Yi [34], Xiong-Lin-Mori [44]. The main theorems are in the following.Theorem 7 Let f and g be two transcendental meromorphic functions, n≥27 an integer. If fⁿ(f-1)f' and gⁿ(g-1)g' share z IM, then either f(z) = g(z) orTheorem 8 Let f and g be two transcendental meromorphic functions, n≥28 an integer. If fⁿ(f-1)²f' and gⁿ(g-1)²g' share z IM, then f (?)g.Theorem 9 Let f and g be two transcendental meromorphic functions,αbe a meromorphic function such that T(r,α)=o(T(r,f)+T(r,g) andα(?)0,∞. Let a be a nonzero constant. Suppose that m, n are positive integers. If f and g satisfy one of the followings:(1) n > m +10 andΨ'_f与Ψ'_g share (0,2); (2) 2n>3m+24 andΨ'_f与Ψ;_g share (0,1);(3) n > 4m+ 22 andΨ'_f与Ψ'_g share (0,0); (4) n > m+10 andΨ'_f与Ψ'_g share "(0,2)";(5) n > 2m+14 andΨ'_f与Ψ'_g share (0,2)^*; (6) n > m+10 and (?)(0,Ψ'_f)=(?)(0,Ψ'_g),then (i) if m≥2, then f(z)(?)g(z);(ii) if m = 1, either f(z) = g(z) or f and g satisfy the algebraic equation R(f,g) = 0, where R((?))=(n+1)((?))-(n+...