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WANG Xue, LIU Xiao-jun, CHEN Qiao-yu. Normal criterion concerning differential polynomials and omitted functions[J]. Journal of East China Normal University (Natural Sciences), 2012, (3): 61-70.
Citation:
WANG Xue, LIU Xiao-jun, CHEN Qiao-yu. Normal criterion concerning differential polynomials and omitted functions[J]. Journal of East China Normal University (Natural Sciences), 2012, (3): 61-70.
WANG Xue, LIU Xiao-jun, CHEN Qiao-yu. Normal criterion concerning differential polynomials and omitted functions[J]. Journal of East China Normal University (Natural Sciences), 2012, (3): 61-70.
Citation:
WANG Xue, LIU Xiao-jun, CHEN Qiao-yu. Normal criterion concerning differential polynomials and omitted functions[J]. Journal of East China Normal University (Natural Sciences), 2012, (3): 61-70.
In this paper, we proved: Let $k\geqslant 2$ be a positive integer, $\mathcal{F}$ be a family of holomorphic functions, all of whose zeros have multiplicities at least $k$, and let $h(z)$, $a_1(z)$, $a_2(z)$, $\cdots$, $a_k(z)$ are all nonequivalent to $0$ on $D$. If for any $f\in\mathcal{F}$, the following two conditions are satisfied: (a)~$f(z)=0\Rightarrow |f^{(k)}(z)+a_1(z)f^{(k-1)}(z)+\cdots+a_k(z)f(z)||h(z)|$; (b)~$f^{(k)}(z)+a_1(z)f^{(k-1)}(z)+\cdots+a_k(z)f(z)\neq h(z),$~ where ~$a_1(z), a_2(z),\cdots ,a_k(z)$ and $f$ have no common zeros, then $\mathcal{F}$ is normal on $D$.
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