Normal family of meromorphic functions sharing holomorphic functions and the converse of the bloch principle

In 1996, C. C. Yang and P. C. Hu 8] showed that: Let f be a transcendental meromorphic function on the complex plane, and a ≠ 0 be a complex number; then assume that n ≥ 2, n₁, …,n_k are nonnegative integers such that $n_1+?+n_k\ge 1;$ thus

$f^n{\left(f^{\prime }\right)}^{n_1}?{\left(f^{\left(k\right)}\right)}^{n_k}?a$

has infinitely zeros. The aim of this article is to study the value distribution of differential polynomial, which is an extension of the result of Yang and Hu for small function and all zeros of f having multiplicity at least k ≥ 2. Namely, we prove that

$f^n{\left(f^{\prime }\right)}^{n_1}?{\left(f^{\left(k\right)}\right)}^{n_k}?a\left(z\right)$

has infinitely zeros, where f is a transcendental meromorphic function on the complex plane whose all zeros have multiplicity at least k ≥ 2, and a(z) ≡ 0 is a small function of f and n ≥ 2, n₁,…,n_k are nonnegative integers satisfying n₁+…+ n_k ≥ 1. Using it, we establish some normality criterias for a family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. The results of this article are supplement of some problems studied by J. Yunbo and G. Zongsheng 6], and extension of some problems studied X. Wu and Y. Xu 10]. The main result of this article also leads to a counterexample to the converse of Bloch's principle.