Meromorphic functions sharing four values

On results of H. Ueda and G. Brosch concerning the unicity of meromorphic functions

“…For the remaining problem, still open at present, that whether the condition 1CM + 3IM = 4CM is true or not, he also did some partial but important work (see [3][4][5][6]). Another one should be mentioned is E. Mues, who, by introducing the concept of "CM" shared values, proved R. Nevanlinna's Four Value Theorem with much broader assumptions, which will be showed later (see [16]). Now, it is presumed that the reader is familiar with the fundamental results in Nevanlinna's value distribution theory of meromorphic functions of single complex variable in the open complex plane C, which is the main instrument used for studies in this paper, and hence by which the condition that any non-constant meromorphic function always means meromorphic in C is assumed throughout this paper, such as the First Main Theorem, the Second Main Theorem, and the lemma of logarithmic derivative, and the basic notations, such as the characteristic function T (r, f ), the proximity function m(r, f ), and the counting function N(r, f ) and the reduced counting functionN(r, f ) of poles.…”

Further results on meromorphic functions that share two values with their derivatives

“…For instance, it was shown by Lee-Yang [8] that if/ is entire and shares two finite values CM with / ' , then / = / ' . Since then, the subject of sharing values between meromorphic or entire functions and their derivatives or linear differential polynomials has been studied by many mathematicians, see, for example, [2,5,6,10]. In 1993, Riissmann [9] proved the following result: Let/ be a meromorphic function and where k > 2 and the aj 's are polynomials.…”

“…otherwise is the notation introduced by Mues in [5]. Here N 0 (r, l / ( / -a,)) denote the counting function of those a\-points off and L(f) of the same multiplicities but counted only once.…”

Shared Values and Borel Exceptional Values for High Order Difference Operators

“…If f (z) − a and g(z) − a have the same zeros with the same multiplicities, then we say that they share the value a CM (counting multiplicities). Classic Nevanlinna four values theorem says that if two nonconstant meromorphic functions f and g share four values CM, then f ≡ g or f is a Möbius transformation of g. The condition "4 CM" has been weakened to "2 CM+2 IM" by Gundersen [10], as well as by Mues [13]. But whether the condition can be weakened to "1 CM+3 IM" is still an open question.…”