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new criteria for meromorphic univalent functions - Project Euclid
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new criteria for meromorphic univalent functions - Project Euclid
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May 31, 1993 -
${\rm Re}\{\frac{Df(z)}{
D
^{n}f(z)}-2\}$. \langle. $-\frac{n+a}{n+1}$ .... $=kw(z_{O})$. ,. (2. 7) where $|w(z_{O})|$. $=1$ and $k\geq$.
$1
$ . From (2. 6).
Nihonkai Math. J. Vol.5 (1994), 1-11
NEW CRITERIA FOR MEROMORPHIC UNIVALENT FUNCTIONS
OF ORDER M.
ABSTRACT.
K.
Let
and
AOUF
$M_{n}(\alpha)$
be
$\alpha$
M.
H.
HOSSEN
class of
the
functions of
the
form $f(z)$
$=\frac{1}{z}+\sum_{k\approx 0}^{\infty}a_{k}z^{k}$
whi ch are regular in the punctured. disc
$U^{*}$
$=$
$\{z:0 \langle|z| \langle 1\}$
and satisfying $n+1$ ${\rm Re}\{\frac{Df(z)}{D^{n}f(z)}-2\}$
$n\in N_{O}=$
$\{0,1$ ,
...
$\}$
,
and
$D^{n}f(z)$
It is proved that
$ 0\leq$
\langle
$-\frac{n+a}{n+1}$
a
\langle 1 ,
$|z|$
\langle
where
$=\frac{1}{z}[\frac{z^{n+1}fz}{n!}()1(n)$
$M_{n+1}(^{\underline{\sim}})$
. Since
$\subset M_{n}(\underline{\sim})$
functions
in
$M_{n}(a)$
are univalent.
the integrals of functions in
KEY $WORDS-$ Univalent $\lambda MS$
(1991)
.
$M_{n}(a)$
meromorphic
,
Subject Classification.
–
1
–
is the class
$M_{o}(\sim\wedge)$
of meromorphical ly starl ike functions of order
all
1,
$\alpha$
,
$ 0\leq$
Further we
.
integrals.
$30C45-30C50$ .
a
$
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