Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every f ∈ F, f and f (k) share 0, and |f(z)| ≥ M whenever f (k)(z) = h(z), then F is normal in D. The condition that f and f (k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f (k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f (k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc.
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