Composite meromorphic functions and normal families

In this paper, we study the normality of families of meromorphic functions. We prove the result: Let alpha(z) be a holomorphic function and F a family of meromorphic functions in a domain D, P(z) be a polynomial of degree at least 3. If P circle f(z) and P circle g(z) share alpha(z) IM for each pair f(z), g(z) is an element of F and one of the following conditions holds: (1) P(z) - alpha(z(0)) has at least three distinct zeros for any z(0) is an element of D; (2) There exists z(0) is an element of D such that P(z) - alpha(z(0)) has at most two distinct zeros and alpha(z) is nonconstant. Assume that beta(0) is a zero of P(z) - alpha(z(0)) with multiplicity p and that the multiplicities l and k of zeros of f(z) - beta(0) and alpha(z) - alpha(z(0)) at z(0), respectively, satisfy k not equal lp, for all f(z) is an element of F. Then F is normal in D. In particular, the result is a kind of generalization of the famous Montel criterion.