IJMMS 26:5 (2001) 317–319
PII. S0161171201004550
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
SUFFICIENT CONDITIONS FOR MEROMORPHIC STARLIKENESS
AND CLOSE-TO-CONVEXITY OF ORDER α
NAK EUN CHO and SHIGEYOSHI OWA
(Received 7 February 2000)
Abstract. The object of the present paper is to derive a property of certain meromorphic functions in the punctured unit disk. Our main theorem contains certain sufficient
conditions for starlikeness and close-to-convexity of order α of meromorphic functions.
2000 Mathematics Subject Classification. 30C45.
1. Introduction. Let
denote the class of functions of the form
f (z) =
∞
1 +
an zn
z n=1
(1.1)
which are analytic in the punctured unit disk Ᏸ = {z : 0 < |z| < 1}. A function f ∈
is said to be meromorphic starlike of order α if it satisfies
− Re
zf (z)
f (z)
>α
z ∈ ᐁ = Ᏸ − {0}
(1.2)
∗
(α) the class of all meromorphic starlike
for some α (0 ≤ α < 1). We denote by
functions of order α.
Let MC(α) be the subclass of
consisting of functions f which satisfy
− Re z2 f (z) > α
(z ∈ ᐁ)
(1.3)
for some α (0 ≤ α < 1). A function f in MC(α) is meromorphic close-to-convex of
order α in Ᏸ (see [1]).
2. Main result. In proving our main theorem, we need the following lemma due to
Owa, Nunokawa, Saitoh, and Fukui [2].
Lemma 2.1. Let p be analytic in ᐁ with p(0) = 1. Suppose that there exists a point
z0 ∈ ᐁ such that Re p(z) > 0 (|z| < |z0 |), Re p(z0 ) = 0, and p(z) = 0. Then we have
p(z) = ia(a = 0) and
z0 p z0
1
k
=i
a+
,
(2.1)
p z0
2
a
where k is a real number with k ≥ 1.
With the aid of Lemma 2.1, we derive the following theorem.
318
N. E. CHO AND S. OWA
satisfies f (z)f (z) = 0 in Ᏸ and
zf (z) zf (z)
− < 2(2 − α) − β (z ∈ ᐁ),
Re α
f (z)
f (z)
Theorem 2.2. If f ∈
then
− Re
z2−α f (z)
f α (z)
>
1
1 + 2(2 − α) − 2β
(z ∈ ᐁ),
(2.2)
(2.3)
where α ≤ 2 and (2(2 − α) − 1)/2 ≤ β < 2 − α.
Proof. We define the function p in ᐁ by
−
z2−α f (z)
= γ + (1 − γ)p(z)
f α (z)
(2.4)
with γ = 1/(1 + 2(2 − α) − 2β). Then p is analytic in ᐁ with p(0) = 1 and
α
(1 − γ)zp (z)
zf (z) zf (z)
− = 2−α−
.
f (z)
f (z)
γ + (1 − γ)p(z)
Suppose that there exists a point z0 ∈ ᐁ such that
Re p(z) > 0 |z| < |z0 | , Re p z0 = 0,
p(z) = 0.
Then, applying Lemma 2.1, we have p(z) = ia(a = 0) and
z0 p z0
1
k
=i
a+
(k ≥ 1).
p z0
2
a
It follows from this that
z f z
(1 − γ)z0 p z0
k(1 − γ) 1 + a2
z0 f z0
− 0 0 = 2 − α −
= 2−α+ .
α
f z0
f z0
γ + (1 − γ)p z0
2 γ + i(1 − γ)a
Therefore, we have
z0 f z0
z0 f z0
k(1 − γ) 1 + a2
Re α
−
= 2−α+
f z0
f z0
2 γ 2 + (1 − γ)2 a2
≥ 2−α+
k(1 − γ)
≥ 2(2 − α) − β.
2γ
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
This contradicts our assumption. Thus, we conclude that Re p(z) > 0 for all z ∈ ᐁ,
that is,
z2−α f (z)
1
− Re
>γ=
(z ∈ ᐁ).
(2.10)
f α (z)
1 + 2(2 − α) − 2β
Putting β = (2(2 − α) − 1)/2 in Theorem 2.2, we have
Corollary 2.3. If f ∈ satisfies f (z)f (z) = 0 in Ᏸ and
3
zf (z) zf (z)
− < − α (z ∈ ᐁ),
Re α
f (z)
f (z)
2
then
− Re
where α ≤ 2.
z2−α f (z)
f α (z)
>
1
2
(z ∈ ᐁ),
(2.11)
(2.12)
SUFFICIENT CONDITIONS FOR MEROMORPHIC STARLIKENESS . . .
319
Taking α = 1 in Theorem 2.2, we have the following corollary.
Corollary 2.4. If f ∈ satisfies f (z)f (z) = 0 in Ᏸ and
Re
zf (z) zf (z)
− f (z)
f (z)
then
− Re
that is, f ∈
∗
zf (z)
f (z)
>
< 2−β
1
3 − 2β
(z ∈ ᐁ),
(z ∈ ᐁ),
(2.13)
(2.14)
(1/(3 − 2β)), where 1/2 ≤ β < 1.
Further, letting α = 0 in Theorem 2.2, we have the following corollary.
Corollary 2.5. If f ∈ satisfies f (z)f (z) = 0 in Ᏸ and
then
zf (z)
f (z)
< 4−β
(z ∈ ᐁ),
(2.15)
− Re z2 f (z) > 5 − 2β
(z ∈ ᐁ),
(2.16)
− Re
that is, f ∈ MC(1/(5 − 2β)), where 3/2 ≤ β < 2.
Acknowledgement. This work was supported by the Korea Research Foundation
Grant (KRF-99-015-DP0019).
References
[1]
[2]
M. D. Ganigi and B. A. Uralegaddi, Subclasses of meromorphic close-to-convex functions,
Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 33(81) (1989), no. 2, 105–109.
MR 91b:30036. Zbl 685.30004.
S. Owa, M. Nunokawa, H. Saitoh, and S. Fukui, Starlikeness and close-to-convexity of certain
analytic functions, Far East J. Math. Sci. 2 (1994), no. 2, 143–148. MR 97d:30014.
Zbl 933.30008.
Nak Eun Cho: Department of Applied Mathematics, Pukyong National University,
Pusan 605-737, Korea
E-mail address: necho@dolphin.pknu.ac.kr
Shigeyoshi Owa: Department of Mathematics, Kinki University, Higashi-Osaka,
Osaka 577-8502, Japan
E-mail address: owa@math.kindai.ac.jp