On the Cardinality of Unique Range Sets with Weight One

Two meromorphic functions f and g are said to share a set S subset of C boolean OR[{infinity} with weight l is an element of N boolean OR[{0}boolean OR[{infinity} if E-f (S, l) = E-g(S, l), where E-f (S, l) = boolean OR(a is an element of S) ((z, t) is an element of C x N | f(z) = a with multiplicity p}, provided that t = p for p <= l and t = p + 1 for p > l. We improve and supplement the result by L. W. Liao and C. C. Yang [Indian J. Pure Appl. Math., 31, No. 4, 431-440 (2000)] by showing that there exists a finite set S with 13 elements such that E-f (S, 1) = E-g(S, 1) implies that f equivalent to g.