This paper is a continuation of work of the author in [Sci. China, Ser. A 32, No.7, 782-791 (1989;

Zbl 0687.30023)] which makes use of L. Zalcman’s heuristic principle for normality: a family of holomorphic functions which have a common property P in a domain D is (apt to be) a normal family in D if P cannot be possessed by a non-constant entire function in the finite plane. The author uses a modified version of Zalcman’s principle to establish the normality of the family of meromorphic functions

$f\left(z\right)$ satisfying

${f}^{\text{'}}-a{f}^{n}\ne b$,

$n\ge 4$,

$a\ne 0$,

$\infty $,

$b\ne \infty $ in a domain D. This conjecture of

*W. K. Hayman* [Research problems in function theory (1967;

Zbl 0158.063)] was proved for

$n\ge 5$ by

*L. Xianjin* [Sci. Sin. Ser. A 28 (1985)].