Complex Oscillation Of The Linear Differential Equations With Entire Coefficients

In this paper, we mainly investigate the growth order of solutions of higher order linear differential equations with entire coefficients and the relations between solutions of a class of second-order linear differential equations and the small functions. It consists of four chapters.In chapter 1, we briefly introduce some background knowledge related to this paper and Nevanlinna theory, which is the basic tool used in this paper.In chapter 2, we investigate the growth of solutions of the differential equa-tions f(k)+Ak-1(z)f(k-1)+…+A0(z)f=0, where Aj(z)(j= 0,…, k-1) are entire functions. When there exists some coefficient As(z)(s ∈{1,…, k-1) being a nonzero solution of f"+P(z)f= 0, where P(z) is a polynomial with degree n(≥1), and the Taylor expansion of A0 (z) is Fabry gap, we obtain that every nonzero solution of such equations is of infinite order.In chapter 3, the growth and zeros of solutions of higher order linear differ-ential equations f(k)+Ak-1(z)f(k-1)+…A1(z)f’A0(z)f= F(z) with entire coefficients of finite iterated order are studied. Some estimations on the iterated order and the iterated convergence exponent of zero sequence are obtained, when there exists one dominant coefficient. The obtained results are extensions of some previous results.In chapter 4, we investigate the relations between the small functions ψ and the solutions, their 1st,2nd derivatives, differential polynomials to the differen-tial equations f"|A1eaznf’+A0ebznf=F, where A1{z),A0(z),F(z) are entire functions with orders less than n, a, b are nonzero complex numbers. Under cer-tain conditions, we obtain that the exponent of convergence of zero sequences of f-φ,f’-φ, f"-φ, d2f"+d1f’+dof-φ are all infinite.