ANNALES POLONICI MATHEMATICI 110.3 (2014) A normality criterion for meromorphic functions having multiple zeros by Shanpeng Zeng (Hangzhou) and Indrajit Lahiri (Kalyani) Abstract. We prove a normality criterion for a family of meromorphic functions having multiple zeros which involves sharing of a non-zero value by the product of functions and their linear differential polynomials. 1. Introduction, definitions and results. Let f and g be two mero- morphic functions in the open complex plane C. If for some a ∈ C ∪ {∞} the functions f and g have the same set of a-points ignoring multiplicities, we say that f and g share the value a IM (ignoring multiplicities). In 1959 W. K. Hayman [6] proposed the following: Theorem A. If f is a transcendental meromorphic function in C, then f n f 0 assumes every finite non-zero complex value infinitely often for any positive integer n. Hayman [6] himself proved Theorem A for n ≥ 3, and n ≥ 2 if f is entire. Further it was proved by E. Mues [12] for n ≥ 2 and by J. Clunie [3] for n ≥ 1 if f is entire; also by W. Bergweiler and A. Eremenko [1] and by H. H. Chen and M. L. Fang [2] for n = 1. Thus Theorem A is completely established. In relation to Theorem A, Hayman [7] proposed the following conjecture on normal families. Theorem B (Hayman’s Conjecture). Let F be a family of meromorphic functions in a domain D ⊂ C, n be a positive integer and a be a non-zero finite complex number. If f n f 0 6= a in D for each f ∈ F, then F is normal. Theorem B was proved by L. Yang and G. Zhang [19, 20] (for n ≥ 5 and n ≥ 2 for a family of holomorphic functions), by Y. X. Gu [5] (for n =3, 4), by I. B. Oshkin [13] (for holomorphic functions and n = 1; see also [9]) and 2010 Mathematics Subject Classification : 30D35, 30D45. Key words and phrases : meromorphic function, shared value, normality. DOI: 10.4064/ap110-3-5 [283] c  Instytut Matematyczny PAN, 2014