In this paper, we study the normality of families of meromorphic functions. We prove the result: Let α(z) be a holomorphic function and F{mathcal{F}} a family of meromorphic functions in a domain D, P(z) be a polynomial of degree at least 3. If P ○ f(z) and P ○ g(z) share α(z) IM for each pair f(z),g(z) Î F{f(z),g(z)in mathcal{F}} and one of the following conditions holds: (1) P(z) − α(z 0) has at least three distinct zeros for any z0 Î D{z_{0}in D}; (2) There exists z0 Î D{z_{0}in D} such that P(z) − α(z 0) has at most two distinct zeros and α(z) is nonconstant. Assume that β 0 is a zero of P(z) − α(z 0) with multiplicity p and that the multiplicities l and k of zeros of f(z) − β 0 and α(z) − α(z 0) at z 0, respectively, satisfy k ≠ lp, for all f(z) Î F{f(z)inmathcal{F}}. Then F{mathcal{F}} is normal in D. In particular, the result is a kind of generalization of the famous Montel criterion.
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