For a given meromorphic function I(z) and an arbitrary value a,
Nevanlinna's value distribution theory, which can be derived from the
well known Poisson-Jensen for- mula, deals with relationships between
the growth of the function and quantitative estimations of the roots of
the equation: 1 (z) - a = O. In the 1920s as an application of the
celebrated Nevanlinna's value distribution theory of meromorphic
functions, R. Nevanlinna [188] himself proved that for two nonconstant
meromorphic func- tions I, 9 and five distinctive values ai (i =
1,2,3,4,5) in the extended plane, if 1 1- (ai) = g-l(ai) 1M (ignoring
multiplicities) for i = 1,2,3,4,5, then 1 = g. Fur- 1 thermore, if 1-
(ai) = g-l(ai) CM (counting multiplicities) for i = 1,2,3 and 4, then 1
= L(g), where L denotes a suitable Mobius transformation. Then in the
19708, F. Gross and C. C. Yang started to study the similar but more
general questions of two functions that share sets of values. For
instance, they proved that if 1 and 9 are two nonconstant entire
functions and 8, 82 and 83 are three distinctive finite sets such 1 1
that 1- (8 ) = g-1(8 ) CM for i = 1,2,3, then 1 = g.