On The Shared Values Of Meromorphic Functions

Concerning the theory of normal families, we can get the following result:Let T be a family of meromorphic functions in unit disc, if for any f∈F, we have E_f(0) = E_f'(0), and E_f'(0)(1) (?)E_f(1), then T is a normal family in unit disc.Concerning value distribution theory, Q.C.Zhang got the following result:Let f(z) be a non-constant meromorphic function in C, b is a finite non-zero complex number, if f and f' CM share 0 and IM share b, then:(i) f ≡ f', or(ii) f = (2b)/(1-ce-2z) here c is any non-zero constant. Noticing the similarity between the conditions of the two results above (Q.C.Zhang's condition is more strong), and guided by the Bloch Conjecture, we guess whether we can weaken the condition " f and f' IM share b" in Q.C.Zhang's result to " f'(z) = b => f(z) = b". Following this, and using the analysing methods in value distribution theory and the theory of normal families, we give affirmative answer to theguess, and get the following result:Let f be a non-constant meromorphic function, if f(z), f'(z) CM share 0 and f'(z) = 1 => f(z) = 1, then either f(z) ≡ f'(z) or f(z)=2/(1-ce-2z) , here c is any non-zero constant.