Some Results Of Meromorphic Functions Involve Sharing Values And Sharing Value Sets

This thesis mainly studies two aspects of the uniqueness of meromorphic functions.On the one hand,we study the uniqueness of analytic functions in a certain angular domain involving shared values.We get the following conclusions:Let ε0∈(0,π/4),f(z)be an analytic function in Ωε0(0,π/2)∪{z||z|<ε0}with L:argz=π/2 as a Julia direction,and there exist positive numbers G,t,δand η,such that for the points z*and w*in Ωε0(0,π/2),as long as |f(z*)|>G and |z*-w*|<δ,there must be t|z*-w*|η≤|f(z*)-f’(ω*)|.Then●If f(z)and f’(z)have four distinct finitely IM sharing values on Ωε0(0,π/2),then f(z)≡f’(z).●If αk(k=1,2,3)be three distinct finite complex numbers,f(z)and f’(z)share a1 as CM in Ωε0(0,π/2)and share every one of ak(k=2,3)as IM inΩε0(0,π/2),and cross ratio(α1,α2,α3,∞)is not equal to-1,1/2 and 2,then f(z)≡f’(z).On the other hand,the uniqueness of meromorphic functions on complex planes involving shared value sets is investigated.The following conclusions are obtained:Let Sn={w/wn=1}(n=1,2,…),f(z)and g(z)are non-constant meromorphic functions in complex plane.Then●If Θ(∞,f)>1/2,(?)(∞,g)>1/2,S6 be a CM sharing value set of f(z)and g(z),then T(r,f)～T(r,g).(r(?)E,r→∞,mesE<+∞)●Le Θ(∞,f)>1/2,Θ(∞,g)>1/2,n≥7,and Sn be a CM sharing value set off(z)and g(z),then g(z)must be a fractional linear transformation of f(z).