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Normal families of meromorphic functions concerning shared values. (English) Zbl 1145.30013
Let $ℱ$ be a family of meromorphic functions in a domain $D$. It is known that if every function in $ℱ$ omits three distinct values, then $ℱ$ is normal. W. Schwick [Arch. Math. 59, No. 1, 50–54 (1992; Zbl 0758.30028)] obtained a normality criteria from the point of view of value distribution theory, in particular, shared values. The author considers the sharing conditions with differential polynomials. Let $n$ be a positive integer, and $a$ be a nonzero constant. If $n\ge 4$ and for each pair of $f$ and $g$ in $ℱ$, ${f}^{\text{'}}-a{f}^{n}$ and ${g}^{\text{'}}-a{g}^{n}$ share a value $b$, then $ℱ$ is normal. The author also considers a family of entire functions. Examples are given which imply that results in this paper are sharp. The methods for the proofs are the value distribution theory and Zalcman’s lemma.

##### MSC:
 30D35 Distribution of values (one complex variable); Nevanlinna theory 30D45 Bloch functions, normal functions, normal families
##### Keywords:
differential polynomial; Zalcman’s lemma