Complex Oscillation Theory Of Differential Euqations And Uniqueness Theory Of Functions

In 1920s, R. Nevanlinna introduced the characteristic function of meromorphic functions and gave two fundamental theorems, which construct Nevanlinna theory of meromorphic functions. For over half a century, Nevanlinna theory has been well developed and widely applied in the research of complex differential equations and the uniqueness of meromorphic functions. The complex oscillation theory of differential equations is the borderline field and intersected subject research, which studies differential equations by the theory and method of complex analysis. It became popular after that S. Bank and I Laine made some original work in 1980s. Many mathematicians have studied deeply and paid close attention to it, in which Prof. Shi-An Gao and Prof. Zong-Xuan Chen made contributions to the introduction and development of the theory in China. The research on the uniqueness theory of meromorphic functions is a very active international subject in recent decades. The research on the uniqueness theory of meromorphic functions dealing with shared values originates from some works of R. Nevanlinna, which lay the foundation of uniqueness theory. After that, a lot of elegant results were given by many mathematicians. For two recent decades, Prof. Hong-Xun Yi have made a lot of remarkable works and contributions to the development of the uniqueness theory of functions in China.In this thesis, we introduce the researches on the complex oscillation theory of differential equations and the uniqueness theory of functions under the guidance of Prof. Hong-Xun Yi. It consists of five chapters.In Chapter 1, we will briefly introduce the basic results of the Nevanlinna theory (see [35], [75], [81], [47]) and Wiman-Valiron theory (see [40], [47]), which are the powerful tools in the field of complex differential equations and uniqueness theory of meromorphic functions.In Chapter 2, we investigate the complex oscillation theory of linear differential equations with iterated order meromorphic coefficients in the plane C, which extend and improve some results of [24],[45],[20],[12]. Now we show three of our main results as followsTheorem 0.0.1. Let B₀,... ,B_k-i be meromorphic functions such thatThen every meromorphic solutions f(?) 0 of the equationsatisfies i(f) = p + 1 andσ_p+1(f) =σ_p{B₀). Theorem 0.0.2. Let B₀,..., B_k-1 be meromorphic functions such thati(B_s) = p(0 p(B_j): j≠s} <σ_p{B_s) =σandThen every transcendental meromorphic solution f of the equationsatisfies p≤i(f)≤p+1 andσ_p+1(f)≤σ_p(B_s)≤σ_p(f). Furthermore, if all solutions f(z) of (0.0.2) are meromorphic functions, then there is at least one meromorphic solution f₁ which satisfies i(f₁) = p + 1 andσ_p+1(f₁) =σ_p(B_s).Theorem 0.0.3. Let B₀,B₁,…, B_k-1 be meromorphic functions. There exists one B_s,(0≤s≤k-1) such thatThen all transcendental meromorphic solutions f of (0.0.2) satisfy i(f) = 2 andσ₂(f) =σ(B_s), and every nontranscendental meromorphic solutions f of (0.0.2) is a polynomial with degree deg(f)≤s - 1.In Chapter 3, we consider the complex oscillation theory of linear differential equations in the unit disc. In the first section, we investigate the complex oscillation of the differential equation f" + A(z)f = 0 where A(z) is analytic in the unit disc. In the second section, we introduce some simple results on the solutions of finite order of differential equations in the unit disc. In the third section, We investigate the complex osculation of a class of higher order differential equations in the unit disc. At the last section, the coefficients of iterated order of higher order differential equations in the unit disc are considered. Here we show the main results of the first section in the second chapter.Theorem 0.0.4. Let A(z) be a nonadmissible analytic function in the unit disc. Assume that f₁ and f₂ are two linearly independent solutions ofand set E = f₁f₂, then Theorem 0.0.5. Let A(z) be an admissible analytic function in the unit disc. Then all nonzero solutions f of (0.0.3) are of infinite order and satisfyσ(A)≤σ₂(f) =σ_M(A).Theorem 0.0.6. Let A(z) be an admissible analytic function in the unit disc, and let f\ and f₂ are two linearly independent solutions of (0.0.3), and set E = f₁f₂, thenIfλ(E) <∞, thenλ(f) = 00 holds for all solutions of type f = c₁f₁- c₂f₂, where c₁≠0 and C₂≠0.Theorem 0.0.7. Let A(z) be an admissible analytic function in the unit disc. Ifλ|-(A) <σ(A), then all nonzero solutions f of (0.0.3) satisfyσ(A)≤λ|-(f).In Chapter 4, we investigate the multiple values and uniqueness of meromorphic functions sharing small functions as targets and obtain a more general result as follows, which improves and extends strongly the results of R. Nevanlinna [54], Li-Qiao [50], Yao [77], Yi [79] [80], and Thai-Tan [62].Theorem 0.0.8. Let f₁ and f₂ be two nonconstant meromorphic functions on C, a_j(j = 1,2,..., q) be q distinct meromorphic functions in R(f₁)∩R(f₂), and k_j(j = 1,2,..., q) be positive integers or 00 such that where m and n are positive integers in {1,2,... ,q} and a is an arbitrary meromorphic function in R(f_i) (t = 1,2). IfThen f₁(z) = f₂(z).In Chapter 5, we introduce some results of the uniqueness theory of algebroid functions in the plane, which improve and summarize the results obtained by G. Valiron [69], Y.-Z. He [37], [38], [39], H.-X. Yi [78], Y.-Y. Lin [53], Sun-Gao [60], Xuan-Gao [73] and some results in [81]. Here we show the main results as follows.Theorem 0.0.9. Let W(z) and M(z) be k—valued and s—valued algebroid functions, respectively. Let a_j (j = 1,2,..., q) be q distinct complex numbers, and let t_j (j = 1,2,..., q) be q positive integers or∞such thatThen W(z) = M(z).Theorem 0.0.10. Let W₁(z) and W₂(z) be k-valued and s-valued algebroid functions, respectively. Let a_j (j = 1,2,...,q)beq distinct complex numbers, and let t_j {j = 1,2,...,q) be q positive integers or∞satisfying (0.0.4). Set If for any j = 1,2,..., q,andmin{A_i,A₂}≥0, max{A₁,A₂} > 0.Then W₁{z) = W₂(z).