new criteria for meromorphic univalent functions - Project Euclid

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

$