Research Of Some Topics In The Theory Of Meromorphic Functions

In this dissertation we mainly study normal family of meromorphic functions, uniqueness of meromorphic functions, complex oscillation of differential equations in the complex domain and the value distrbution of quasimeromorphic mapping.Themain works are following:1. The normal family of meromorphic functions.Firstly, we study the normality of a family of meromorphic functions concerning differential polynomial and shared values and prove the followingTheorem 23.5 Let F be a family of meromorphic functions on the unite disc A, k , n , q be positive integers. P( ) = q + aq-1 (z)q-1 +...a1(z) be a polynomial , where aj(z) (j = 1,...,q-1) analytic in A and let H(f,f',...,f(k))be a differential polynomial each of whose monomials is of degree 1 at least , a , b be two distinct complex numbers , B 0 , and c be a nonzero complex number. If for every f F: the multiplicity of its zeros is at least k, andThen F is normal on for k 2, and for k = 1 so long as a (m + 1)b,(m = 1,2,...).Secondly, we study the normality of a family of meromorphic functions concerning multiplicity and shared values and prove the followingTheorem 2.4.2 Let F ={f(k)(z)],k N, where everyf(z) be meromorphic function in , and S = {a,b},a, b C,a b. If for every f(k) , g(k) Fwhere k , l be positive integers which satisfying the following conditionThen f is normal on A.2. The uniqueness of family of meromorphic functions.We mainly study the uniqueness of the familyR of meromorphic functionsand solve a more general problem concerning shared values and uniqueness in R . Theorem 3.1.5 Let . Ifand that .Then there exist at least n-1 functions of fj (z) (j' = 1, ...,n) be equal.3. The complex oscillation of linear differential equations.Firstly, we research the complex oscillation of a certain non-linear differential equations, and prove the followingTheorem 4.1.4 Let A.(z)(& 0)(j = 0,1),F(z)(& 0) be entire functions with cr(Aj) < 1 ,cr(F) < , cu bbe two complex constants and thatab 0 ,a = cb(c 1). Then all solutionsf(z)(x 0) of the equationsatisfyexcept at most an exceptional solution f0(z) with finite order.(ii) If there exists an exceptional solution/0(z) with finite order in ( i ) , then /0(z) satisfies tr(f0)1), Q(z) is a non-constant polynomial, or Q(z) = h(z)ebz, where h(z) is nonzero polynomial, F(Z) is an entire function with finite order. Then all solutions f(z)( 0) of the equationsatisfyexcept at most two exceptional complex a and an exceptional solutionf0(z) with finite order.Secondly, we study the growth of solutions of certain higher order linear differential equation and prove the followingTheorem 4.2.1 LetAj(z)( 0)be entire functions with cr(Aj) < l(j = 0,...,k-1,k 2),aj sC\{0}(y = 0,---,/t-l) with aj=Cjd., c} > l(j = \,---,k-1), and that ck_} > c := max c; .Then all solutions/(z)(x 0) of equationhave infinite order.Theorem 4.2.2 Let A; (z)(& 0), D} (z) be entire functions with cy(Aj) < 1, a(Dj )6C\{0}(y = 0,-,-l)withargflI arga0,a_, =ava,,aj >0(y = 2,---,A--l) or aj=cja0, 0< cj <1 (y = 1,..., k-1) . Then all solutions f(z)( 0) of the equationhave infinite order.Theorem 4.23 Let Aj(z)(j = 0,1,...,k-1) be nonzero polynomials,aj. eC\{0}(j = Q,---,k-\),ai -cja0,ci > 1, (j = \,...,k -l),and that there exist me (1,...,&-1} such that cm > c := max c; .Then all solutions/(z)( 0) of the equation (4.2.1)have infinite order and cr, (/) = 1, except at most two exceptional complex a0.At last, we research the growth of solutions of the second o...