Researches On Problems Related To Properties Of Zeros Of Polynomials And Transcendental Meromorphic Functions

The dissertation focuses on two important conjectures on the properties of zeros of transcendental meromorphic functions and polynomials, and the dynamical properties of the special transcendental functions related to the first conjecture in complex dynamical system.In 2001,German scholar W.Bergweiler raised a conjecture on the zeros of transcendental meromorphic functions in the process of research for normal rules:Let f be a transcendental meromorphic function in C If f’(z)≠ 1 for all z ∈ C, then the set composed of values of the first order derivative of all the zeros f’(f-1 (0)) is unbounded.BL.Sendov raised a conjecture on the zeros of polynomial and its zeros of the first order derivative:If all zeros of a polynomial v=1 are all in the unit closed disk D(0; 1). Then for each zero zv(n = 1,2,3, …, n), the closed disk D(zv; 1) contains at least one zero of p’(z).The main contents of the dissertation are as follows:In Chapter 1, it is introduced that the origin, development, relationship and relevant achievements of value distribution and normal family, as well as the background and devel-opment of the theory of iteration of dynamical system. The main results of the dissertation are listed.In Chapter 2, all the knowledge prepared is introduced. We introduce the context of Berg-weiler conjecture by the theory of value distribution and normal family and the classic results and development of iteration of entire functions.In Chapter 3, we give a new proof for Bergweiler conjecture, moreover, the conjecture is expanded in certain extent, the hypothesis the first order derivative omit 1 can be changed by the first order derivative take 1 at most finitely times.In Chapter 4, we research some problems related to Bergweiler’s conjecture:suppose a transcendental meromorphic function has infinitely many zeros and its first order derivative take 0 at most finitely times, then whether the set composed of the values of the first order derivative of all the zeros is unbounded? we apply the theory of normal family to proof that if a meromorphic function takes a non-zero at most finitely times and its order is greater than 2, then the set composed of the values of the first order derivative of all the zeros is unbounded. For the entire functions whose order is greater than 1, if it takes a non-zero value at most finitely times, then the set composed of the values of the first order derivative of all the zeros is unbounded; here there exists example to show that the hypothesis the order is greater than 1 is sharp.In Chapter 5, dynamical properties of a kind of functions related to Bergweiler conjecture is researched, especially the properties of buried points of the finite type entire function just like with p(z) is real coefficient polynomial whose leading coefficient is positive, suppose the zeros of nd if they are real, we prove when J(fμ(z))≠C and parameter μ satisfy some hypothesis, then the unbounded positive real interval belong to the set of buried points of J(fμ) withIn Chapter 6, for the polynomial whose zeros are all in closed unit disk, we study the distance between the zeros of the polynomial and the zeros of it’s first order derivative, that is the conjecture from BL.Sendov. we have:let p(z) be a polynomial with degree n, all the zeros of p(z) are in closed unit disk, assume a is a zero of p(z) with |α| ≤ sin π/n, then the closed disk take a as center with radius 1 contains at least one zero of p’(z). The result partly confirms the conjecture.