Normality of Composite Analytic Functions and Sharing an Analytic Function

A result of Hinchliffe 2003 is extended to transcendental entire function, and an alternative proof is given in this paper. Our main result is as follows: let α z be an analytic function, F a family of analytic functions in a domain D, and H z a transcendental entire function. If H ◦ f z and H ◦ g z share α z IM for each pair f z , g z ∈ F, and one of the following conditions holds: 1 H z −α z0 has at least two distinct zeros for any z0 ∈ D; 2 α z is nonconstant, and there exists z0 ∈ D such that H z − α z0 : z − β0 Q z has only one distinct zero β0, and suppose that the multiplicities l and k of zeros of f z −β0 and α z −α z0 at z0, respectively, satisfy k / lp, for each f z ∈ F, where Q β0 / 0; 3 there exists a z0 ∈ D such thatH z − α z0 has no zero, and α z is nonconstant, then F is normal in D.