ANNALES POLONICI MATHEMATICI 80 (2003) Normal families and shared values of meromorphic functions by Mingliang Fang (Nanjing) and Lawrence Zalcman (Ramat-Gan) To Professor J´ ozef Siciak, with admiration and friendship Abstract. Let F be a family of meromorphic functions on a plane domain D, all of whose zeros are of multiplicity at least k ≥ 2. Let a, b, c, and d be complex numbers such that d = b, 0 and c = a. If, for each f ∈F ,f (z)= a ⇔ f (k) (z)= b, and f (k) (z)= d ⇒ f (z)= c, then F is a normal family on D. The same result holds for k = 1 so long as b =(m + 1)d, m =1, 2, .... 1. Introduction. Let f and g be meromorphic functions on a domain D in C, and let a and b be complex numbers. If g(z )= b whenever f (z )= a, we write f (z )= a ⇒ g(z )= b. If f (z )= a ⇒ g(z )= b and g(z )= b ⇒ f (z )= a, we write f (z )= a ⇔ g(z )= b. If f (z )= a ⇔ g(z )= a, then we say that f and g share a in D. Mues and Steinmetz [11] proved Theorem A. Let f be a nonconstant meromorphic function , and let a 1 , a 2 , and a 3 be distinct complex numbers. If f and f ′ share a 1 , a 2 , and a 3 , then f ≡ f ′ . Schwick [15] discovered a connection between normality criteria and shared values. He proved Theorem B. Let F be a family of meromorphic functions in a domain D, and let a 1 , a 2 , and a 3 be distinct complex numbers. If , for each f ∈F , f and f ′ share a 1 , a 2 , and a 3 in D, then F is normal in D. 2000 Mathematics Subject Classification : Primary 30D45. Key words and phrases : normal families, shared values. Research of the first author supported by the National Natural Science Foundation of China (grant no. 10071038) and by the Fred and Barbara Kort Sino-Israel Post Doctoral Fellowship Program at Bar-Ilan University. Research of the second author supported by the German-Israeli Foundation for Sci- entific Research and Development, G.I.F. grant no. G-643-117.6/1999. [133]