Uniqueness Theorem Of Certain Entire Functions

In this paper, we consider two problems in the theory of meromorphic functions. First, we study the uniqueness theorem of entire functions. We consider certain entire functions of small order and get:Theoreml Let f and g be two transcendental entire functions, and g have two finite IM sharing values, f ≡ g .Theorem2 Let f and g be two non-constant entire functions, ρ_f <1/4 .If f and g have two finite IM sharing values, f ≡ g .Second, we study a conjecture which belongs to W.Bergweiler in document[3] and get:Theorem3 Let f(z) be a transcendental meromorphic function in the finiteplane and f(z) has finite poles in number at the most. If f'(z)≠1the setis not bounded.Corollary. Let g(z) be a transcendental meromorphic function in the finiteplane and g(z) has finite poles in number at the most. If g' has no zeros in , exist pointsz_n(n =1,2,3,-)suppose that g(z_n) = z_n(n =1,2,3, ...) and lim|g'(z_n)|= +∞.