The Uniqueness Of Meromorphic Functions Involving Shared Values

This thesis mainly studies the functional relations of meromorphic functions with three distinct shared values IM and a shared value CM. The sufficient conditions of the meromorphic functions shared four values CM are obtained. In special situation, it improves G.G.Gundersen related result (4/5-value theorem), and the result is extended to the condition of the existence of a real number λ>2/3. Secondly, the degree relations of rational functions with three shared values IM is also invesigated. We obtain three main results.If f(z) and g(z) are two distinct nonconstant meromorphic functions that share three values 0,1, c IM and a fourth value ∞ CM, c∈{-1,1/2,2}. Suppose that there exists some real constant λ>2/3 and some set Ic that has infinite linear measure such that N(r, f)≥λT(r,f) for all r∈Ic. Then f(z) and g(z) share all four values 0,1,c,∞ CM.If f(z) and g(z) are two nonconstant meromorphic functions that share three values 0,1,-1 IM and a fourth value ∞ CM. Suppose that there exists some real constant ε> 0 and some set Iε that has infinite linear measure such that N(r,f)≥N(r,1/f)+εT(r,f) for all r∈Iε. Then(z)= g(z).For any three distinct values a1,a2, a3 in the extended complex plane and a natural number n, there exsist two nonconstant rational functions that share values a1,a2, a3 IM and deg(f)=n, deg(g)=2n.