An improvement of the Nevanlinna–Gundersen theorem

A well-known result of Nevanlinna states that for two nonconstant meromorphic functions f and g on the complex plane C and for four distinct values a j ∈ C ∪ { ∞ } , if ν f − a j = ν g − a j for all 1 ⩽ j ⩽ 4 , then g is a Möbius transformation of f. In 1983, Gundersen generalized the result of Nevanlinna to the case where the above condition is replaced by: min { ν f − a j , 1 } = min { ν g − a j , 1 } for j = 1 , 2 and ν f − a j = ν g − a j for j = 3 , 4 . In this paper, we prove that the theorem of Gundersen remains valid to the case where min { ν f − a j , 1 } = min { ν g − a j , 1 } for j = 1 , 2 , and min { ν f − a j , 2 } = min { ν g − a j , 2 } for j = 3 , 4 . Furthermore, we work on the case where { a j } are small functions.