Research On The Uniqueness Of Entire Functions Sharing Fixed Points And Growth Of Solutions Of A Kind Of Equations

In1920s, Finland mathematician R. Nevanlinna introduced the character-istic functions of mcrornorphic functions in the complex plane, and obtained the Nevanlinna value distribution theory([18]). Nevanlinna theory played ba-sic roles for researches of mcrornorphic functions. Nevanlinna theory keeps on developing. In spite of a certain completeness of the Nevanlinna value dis-tribution theory, extensive theory has been widely applied in other areas of mathematics, such as complex differential and difference equations, theory of functions of a complex variable, minimal surface, and so on.In this paper, the author researched the uniqueness problem on entire function sharing fixed points and the growth of some second order equations. The paper consists of three chapters.In chapter1. we simply introduce the background of this paper, which contains notations of Nevanlinna theory, basic concepts and results of mero-morphic functions.In chapter2, we investigate the uniqueness problem on entire function sharing fixed poinis, which extend the results of Zhang([27]). Qi-Yang([20.22]), and obtain the following main results:Theorem2.1:Let f(z) and g(z) be two transcendental entire functions, and let n, m and k be three positive integers with n>5k+4m*+7, λ and μ be constants that satisfy|λ|+|μ|≠0. If (fn(z)(λfm(z)+μt))(k) and (gn(z)(λgm(z)+μ))(k) share z IM, then the following conclusions hold:(i) If λμ≠0, then fd(z)=gd{z),d=GCD(n,m);especially, when d=1,f(z)≡9(z)；(ii)If λμ=0,theb.f=cg for a constant athat satisfies cn+m’=1:or k=1and f(z)=b1ebz2,g(z)=b2e-bz2for three constants b1,62and b that satisfy4(λ+μ)2(b1b2)n+m((n+m’)b)2=-1.In chapter3,we investigate the growth of somc second order equations, and obtain some results which supplement some results of Chiang-Feng([2]), and obtain the following main results:Theorem3.1:Let Pj(z),Qj(z)(j=1,2)be polynomials in z,If Then cach nontrivial entire solution f(z)of finite order of the equation(3.4) satisfies σ(f)≥2.Theorem3.2:Let Pj(z),Qj(z)(j=1,2)be polynomials in.,and A(z)=akzk+ak-1zk-1+…+.a0,(ak≠0)be a nonconstant polynomial.If Then each nontrivial entire solution,f(z) of finite order of the equation(3.5) satisfies σ(f)≥k+1.Theorem3.3:Let Pj(z),Qj(z)(j=1,2)be polynomials in z,and A(z) be a transcendental entire function. If Then cvcry solution of the cquation(3.5)is of infinite order and σ2(f(z))≥σ(A(z)). Let the expression(3.4)then takes the form then we have the following results.Theorem3.4:Let Pj(z),Qj(z)(j=1,2) be polynomials in z, If Then each nontrivial entire solution f(z) of finite order of the equation (3.6) satisfies a(f)≥2.Corollary3.1:Suppose that the assumptions of Theorem3.2arc satis-fied. Then each nontrivial entire solution f(z) of finite order of the equation satisfies a(f)>k+1.Corollary3.2:Suppose that the assumptions of Theorem3.3arc satis-fied.then every solution of the equation (3.7) is of infinite order and σ2(f(z))≥a(A(z)).