Normality Criteria of Meromorphic Functions Sharing a Holomorphic Function
Tóm tắt
Take three integers
$$m\ge 0,\,k\ge 1$$
, and
$$n\ge 2$$
. Let
$$a\ (\not \equiv 0)$$
be a holomorphic function in a domain
$$D$$
of
$$\mathbb {C}$$
such that multiplicities of zeros of
$$a$$
are at most
$$m$$
and divisible by
$$n+1$$
. In this paper, we mainly obtain the following normality criterion: Let
$${{{\fancyscript{F}}}}$$
be the family of meromorphic functions on
$$D$$
such that multiplicities of zeros of each
$$f\in {{\fancyscript{F}}}$$
are at least
$$k+m$$
and such that multiplicities of poles of
$$f$$
are at least
$$m+1$$
. If each pair
$$(f,g)$$
of
$${{\fancyscript{F}}}$$
satisfies that
$$f^{n}f^{(k)}$$
and
$$g^{n}g^{(k)}$$
share
$$a$$
(ignoring multiplicity), then
$${{\fancyscript{F}}}$$
is normal.
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