Research Of Some Problems In The Uniqueness And Value Distribution Theory Of Meromorphic Functions

This thesis studies some problems in the uniqueness and value distribution theory of meromorphic functions.It contains the following parts.In Chapter 1,we introduce the research background and significance of value distribution theory of meromorphic functions and give some preliminary knowledge.In Chapter 2,we study the uniqueness of entire functions concerning deficient value and exponent of convergence and mainly prove the following theorem.Let S={1,ω,ω2,…,ωn-1},where ωn=1,n(≥1)is an integer,let k be a positive integer,and let f be a nonconstant entire function such that λ(f)<ρ(f)<∞.If f(z)andΔηkf(z)share S IM,where η is a nonzero complex number,then f(z)=eaz+b,where a(≠0)and b are two finite complex numbers.In Chapter 3,we study the uniqueness question of meromorphic functions concerning fixed points and mainly prove the following theorem.Let f and g be two nonconstant meromorphic functions,let n,k be two positive integers with n>3k+10.5-Θmin(k+6.5)if Θmin≥2.5/(k+6.5),otherwise n>3k+8,and let(fn(z))(k)and(gn(z))(k)share z CM,f(z)and g(z)share ∞ IM,then one of the following two cases holds:If k=1,then either f(z)=c1ecz2,g(z)=c2e-cz2,where c1,c2 and c are three constants satisfying 4n2(c1c2)nc2=-1,or f≡tg for a constant t such that tn=1;if k≥2,then f≡tg for a constant t such that tn=1.In Chapter 4,we study value distribution of differential polynomials and mainly prove the following theorem.Let P be a polynomial with deg P≥3,let f be a transcendental meromorphic function,and let α be a small funtion of f.If α is a constant,we also require that there exists a constant A≠α such that P(z)-A has a zero of multiplicity at least 3.Then for any 0<ε<1,we have where if P’(z)has only one zero,then k=1/(deg P-2);if P’(z)has two distinct zeros a,b with P(a)≠P(b),and α is nonconstant,then k=1/(1-ε);otherwise k=1.In Chapter 5,we summarize the main contents of this thesis and pose some problems for further research.