Abstract: Let F be a family of functions meromorphic in a domain D, let m, n_k, k be three positive integers and b be a finite nonzero complex number. Suppose that, (1) for each f∈F, all zeros of  f have multiplicities at least k ; (2) for each pair of functions f, g ∈F, P(f)H(f) and P(g)H(g) share b, where P(f) and H(f) were defined as (1.1) and (1.2) and n_k≥max_1≤i≤k-1{n_i}; (3) m ≥ 2 or n_k≥ 2, k ≥ 2, then F is normal in D.

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