Criteria of normality concerning the sequence of omitted functions
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In this paper, we research the normality of sequences of meromorphic functions concerning the sequence of omitted functions. The main result is listed below. Let $\{f_{n}(z)\}$ be a sequence of functions meromorphic in $D$, the multiplicities of whose poles and zeros are no less than $k+2,~k\in \mathbb N$. Let $\{b_{n}(z)\}$ be a sequence of functions meromorphic in $D$, the multiplicities of whose poles are no less than $ k+1$, such that $b_{n}(z)\overset\chi\Rightarrow b(z)$, where $b(z)(\neq 0)$ is meromorphic in $D$. If $f^{(k)}_{n}(z)\ne b_{n}(z)$, then $\{f_{n}(z)\}$ is normal in $D$. And we give some examples to indicate that there are essential differences between the normal family concerning the sequence of omitted functions and the normal family concerning the omitted function. Moreover, the conditions in our paper are best possible.
Abstract
In this paper, we research the normality of sequences of meromorphic functions concerning the sequence of omitted functions. The main result is listed below. Let $\{f_{n}(z)\}$ be a sequence of functions meromorphic in $D$, the multiplicities of whose poles and zeros are no less than $k+2,~k\in \mathbb N$. Let $\{b_{n}(z)\}$ be a sequence of functions meromorphic in $D$, the multiplicities of whose poles are no less than $ k+1$, such that $b_{n}(z)\overset\chi\Rightarrow b(z)$, where $b(z)(\neq 0)$ is meromorphic in $D$. If $f^{(k)}_{n}(z)\ne b_{n}(z)$, then $\{f_{n}(z)\}$ is normal in $D$. And we give some examples to indicate that there are essential differences between the normal family concerning the sequence of omitted functions and the normal family concerning the omitted function. Moreover, the conditions in our paper are best possible.
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