In this article, we prove some normality criteria for a family of meromorphic functions, which involves sharing of a nonzero value by certain differential monomials generated by the members of the family. These results generalize some of the results of Schwick. | Turk J Math (2016) 40: 1258 – 1273 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Some normality criteria Gopal DATT1,∗, Sanjay KUMAR2 Department of Mathematics, University of Delhi, Delhi, India 2 Department of Mathematics, Deen Dayal Upadhyaya College, University of Delhi, Delhi, India 1 Received: • Accepted/Published Online: • Final Version: Abstract: In this article, we prove some normality criteria for a family of meromorphic functions, which involves sharing of a nonzero value by certain differential monomials generated by the members of the family. These results generalize some of the results of Schwick. Key words: Meromorphic functions, holomorphic functions, shared values, normal families 1. Introduction and main results The notion of normal families was introduced by Montel in 1907. Let us begin by recalling the definition. A family of meromorphic functions defined on a domain D ⊂ C is said to be normal in the domain if every sequence in the family has a subsequence that converges spherically uniformly on compact subsets of D to a meromorphic function or to ∞ (see [1, 6, 9, 14]). One important aspect of the theory of complex analytic functions is to find normality criteria for families of meromorphic functions. Montel obtained a normality criterion, now known as the fundamental normality test, which says that a family of meromorphic functions in a domain is normal if it omits three distinct complex numbers. This result has undergone various extensions. In 1975, Zalcman [15] proved a remarkable result, now known as Zalcman’s lemma, for families of meromorphic functions that are not normal in a domain. Roughly speaking, it says that a nonnormal family can be rescaled at small scale to obtain a nonconstant meromorphic function in the limit. This result of Zalcman gave birth to many new normality criteria. These normality criteria have been
