Value Distributions For Solution Of Complex Differential Equation And Differential Polynomials

The globe theory of complex differential equations, studied from the point of view of Nevanlinna theory, has been attracting much attention since 1982 when the article by Bank and Laine [5] appeared in Tran. Amer. Math. Soc. They examined second-order complex differential equations of the formwhere A(z) is an entire function. Up to now, a considerable number of research papers has been written in this special area of differential equations.The recent investigations of the equations mainly concentrate on three general types of problems. The first one involves determining the frequency of zeros of a non-trivial solution which is usually measured by the exponent of convergence(see.[5, 6, 8]); The second type involves studying the distribution and asymptotic behavior of the zeros of solutions(see. [4, 41, 52]); The third type involves considering the growth of the solutions(see. [34, 35]).Concerning the problem for the distribution of the zeros of solutions to (0.1), the situation when A(z) is a polynomial is fairly clear. The case for two linearly independent solutions was first considered by Bank and Laine in [5] from which it can be inferred that at least one of any two linearly independent solutions has infinitely many zeros in some sector(s). When A(z) is transcendental in (0.1), however, the situation becomes much more complicated. The main problem in this case is the Bank-Laine conjecture: Let f₁, f₂ be any two linearly independent solutions. Then max{λ(f₁),λ(f₂)} =∞whenever the order of A(z) is finite and non-integer. This conjecture still remains open in general and, in fact, most of the researches on (0.1) has been made more or less towards proving this conjecture, see, e.g., [47], Chapter 5. Furthermore, concerning the distribution and asymptotic behavior of the zeros of solutions to (0.1), it also seems more difficult in this case. No general results has been obtained in this aspect, except for some special cases of (0.1), in which cases the zero-rich and zero-scare regions can be determined explicitly, see, e.g., [4, 41, 52].After some well-known preliminaries in Chapter 1, we will investigate the solutions of the high order linear differential equation in some angle in Chapter 2.where A_j(j = 0,1,..., k-1) are analytic on (?) (α,β). In the first section, we study the growth order and the asymptotic behavior of the zeros of solutions of (0.2).Theorem 0.1. Let A_j(z)(j = 0,1,..., k-1) be analytic on (?)(α,β)(0<β-α≤2Ï€), if for any K > 0 theθ's which satisfyα≤θ≤βandform a set of positive measure. Then for every solution f (?) 0 of (0.2) we have _pαβ(f)=+∞.Theorem 0.2. Let A_j(z)(j = 0,1,...,k-1) be analytic on (?)(α,β)(0<β-α< 2Ï€)such that for any arbitrary given constant K>0, if z∈Ω(α,β)Then if f (?) 0 is a solution of (0.2) with _pαβ(f) <∞, then there exists a constant b_8-1≠0 such that for everyε(0 <ε< (β-α)/2), we have f^(m)(z)→0 (m≥s) as z→∞in any subangleΩ(α+ε,β-ε) ofΩ(α,β).In the second section, the distribution of the zeros of solutions of (1.2) in some angle is investigated.Theorem 0.3. Let A_j(j= 0,1,..., k-2) be an entire function of finite order, if there exists one coefficient _p0,θ(A_j) =∞for (0.2), then the ray arg z =θis zero rich.Meanwhile, we also investigate a certain class differential equation which has the exponent type coefficient. Theorem 0.4.Let P(z) and Q(z) be two polynomials of degrees n≥1 and m≥0. respectively. Let f₁, f₂,...,f_k be any k linearly independent solutions ofand set E = f₁f₂...f_k.Then for anyθsatisfyingθ(P,θ) < 0, we haveλ_θ(E) <∞.Our results generalized some results which were proved by S. J. Wu [57. 58]. I. Laine and S. J. Wu [48] and S. P. Wang[56].Chapter 3 is devoted to study the frequency of zeros of solutions in the special case ofwhere k≥3, P(z) is a polynomial of degree n≥1, Q₁,..., Q_k-2 are polynomials, while Q₀ is a transcendental entire function of order < n. For such equations, we investigate the condition which will guarantee that all non-trivial solutions of (0.3) satisfyλ=∞. The second case first studied by Bank and Langley [5]. then the results improved by Y. Chiang, I. Laine and S. P. Wang in [23].Theorem 0.5. Suppose that k≥3. P(z) is a polynomial of degree n≥1. R(z),Q₀are non-zero polynomials. Q₁,..., Q₍k-2 are not all polynomials, andσ(Q_j) < n j = 0,1,..., k-2. Then any non-trivial solution f ofsatisfiesλ(f) =∞.Note thatδ(0, e^P) =1 in Theorem 3.D, we here wish to generalize e^P(z) to A(z) satisfyingδ(0,A) = 1.Theorem 0.6. Let A be an entire function of finite orderλsuch that (?)(Ï„,(1/A)) = S(Ï„,A), and let Q_j, j = 0,1,..., k-2 be an entire function such that T(Ï„, Q_j) = S(Ï„,A),j = 0,1,..., k - 2. Suppose that k≥3,admits a non-trivial solution f such that N(Ï„,(1/f))=S(Ï„,A). Then A(z) has no zeros, and so there is a polynomial P(z) of degree A such that A(z)=exp P(z). Moreover, f has no zeros.Chapter 4 investigate the growth of solutions of the high order linear differential equations with meromorphic function coefficients. In the most cases, we focus on our attention to investigate the equation with the entire function coefficients. Since the main tool is the Wiman-Valiron theory which, in general, adopts for the transcendent . entire function. For the transcendent meromorphic functions, we use the special form of Wiman-Valiron theory which were proved by Wang and Yi (see.[55]) to investigate the solutions with the infinite order for the meromorphic function coefficients.Theorem 0.7. Let H be a set of complex numbers satisfying (?){|z| : z∈H} > 0, and let A₀(z),A₁(z),..., A_k-1(z), A_k(z) be entire functions with A₀(z) (?) 0 such thatand for some constants 0≤β<αand for anyε> 0 sufficiently small, we haveas z→∞for z∈H. Then every meromorphic solution f (?) 0 ofsatisfiesσ(f) =∞andσ₂(f) =σ(A₀).Chapter 5 investigate the value distributions of the differential polynomials. Our work mainly concentrated on giving some quantitative estimation on some certain differentialpolynomials f²f^(k)-1. As we all known, the second fundamental theorem in Nevanlinna's theory of value distribution use the reduced counting function to estimate the Nevanlinna characteristic function. Naturally, we can pose the following important question: whether one can give some quantitative estimates on the differential polynomialsby the reduced counting function? In this chapter, we give some estimations. At the end of the chapter, we also give some precise estimations on the differential polynomial f²-f^(k)-1. Our result improved the result of H. Chen, X. Hua and Y. Xu. Examples show our result in the last section is precise in some sense.