IJMMS 28:1 (2001) 51–55
PII. S0161171201006202
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON THE STRONGLY STARLIKENESS OF MULTIVALENTLY
CONVEX FUNCTIONS OF ORDER α
MAMORU NUNOKAWA, SHIGEYOSHI OWA, and AKIRA IKEDA
(Received 16 November 2000)
Abstract. The object of the present paper is to derive some sufficient conditions for
strongly starlikeness of multivalently convex functions of order α in the open unit disc.
2000 Mathematics Subject Classification. 30C45.
∞
1. Introduction. Let Ꮽ(p) denote the class of the functions f (z) = zp + n=p+1 anzn
which are analytic in the open unit disc Ᏹ = {z : |z| < 1}. A function f (z) ∈ Ꮽ(p) is
called p-valently starlike if and only if the inequality
Re
zf (z)
f (z)
>0
(1.1)
holds for z ∈ Ᏹ. A function f (z) ∈ Ꮽ(p) is called p-valently convex of order α (0 ≤
α < p) if and only if the inequality
1 + Re
zf (z)
f (z)
>α
(1.2)
holds for z ∈ Ᏹ. We denote by Ꮿ(p, α) the family of such functions. A function f (z) ∈
Ꮽ(p) is said to be strongly starlike of order α (0 < α ≤ 1) if and only if the inequality
arg zf (z) < π α
f (z) 2
(1.3)
holds for z ∈ Ᏹ. We also denote by STS(p, α) the family of functions which satisfy
the above inequality for the argument. From the definition, it follows that if f (z) ∈
STS(p, α), then we have
Re
zf (z)
f (z)
>0
in Ᏹ
(1.4)
or f (z) is p-valently starlike in Ᏹ and therefore f (z) is p-valent in Ᏹ (see [1, Lemma 7]).
Nunokawa [2, 3] proved the following theorems.
Theorem 1.1 (see [2]). If f (z) ∈ Ꮽ(p) satisfies
1 + Re
zf (z)
f (z)
where 0 < α ≤ 1, then f (z) ∈ STS(p, α).
< p+
α
,
2
(1.5)
52
MAMORU NUNOKAWA ET AL.
Theorem 1.2 (see [3]). If f (z) ∈ Ꮽ(1) satisfies
arg 1 + zf (z) < π α(β) in Ᏹ,
f (z) 2
(1.6)
then
arg zf (z) < π β
f (z) 2
in Ᏹ,
(1.7)
where
α(β) = β +
βq(β) sin(π /2)(1 − β)
2
tan−1
,
π
p(β) + βq(β) cos(π /2)(1 − β)
p(β) = (1 + β)(1+β)/2 ,
(1.8)
q(β) = (1 − β)(β−1)/2 .
It is the purpose of the present paper to prove that if f (z) ∈ Ꮿ(1, 1 − (α/2)), then
f (z) ∈ STS(1, α).
In this paper, we need the following lemma.
Lemma 1.3. Let f (z) ∈ Ꮽ(1) be starlike with respect to the origin in Ᏹ. Let C(r , θ) =
{f (teiθ ) : 0 ≤ t ≤ r < 1} and T (r , θ) be the total variation of arg f (teiθ ) on C(r , θ),
so that
r ∂
arg f teiθ dt.
T (r , θ) =
(1.9)
∂t
0
Then
T (r , θ) < π .
(1.10)
We owe this lemma to Sheil-Small [6, Theorem 1].
2. Main theorem. Our main theorem for the starlikeness of multivalently convex
functions of order α is the following.
Theorem 2.1. Let f (z) ∈ Ꮽ(1) and
1 + Re
zf (z)
f (z)
> 1−
α
2
in Ᏹ,
(2.1)
where 0 < α ≤ 1. Then
arg zf (z) < π α
f (z) 2
in Ᏹ,
(2.2)
or f (z) is strongly starlike of order α in Ᏹ.
Proof. We put
zf (z)
α
zg (z)
2
1+ −1+
=
,
α
f (z)
2
g(z)
(2.3)
ON THE STRONGLY STARLIKENESS . . .
where g(z) = z +
∞
n=2 bn z
n
53
. From assumption (2.1), we have
Re
zg (z)
g(z)
>0
in Ᏹ.
(2.4)
This shows that g(z) is starlike and univalent in Ᏹ. With an easy calculation (cf. [4]),
(2.3) gives us that
f (z) =
g(z)
z
α/2
.
(2.5)
Since
f (z) ≠ 0,
0 < |z| < 1,
(2.6)
we easily have
f (z)
=
zf (z)
1
0
f (tz)
dt =
f (z)
1
t
−α/2
0
α/2
  g tr eiθ 
dt,
 g r eiθ 
(2.7)
where z = r eiθ and 0 < r < 1. Since g(z) is starlike in Ᏹ, from Lemma 1.3, we have
(2.8)
−π < arg g tr eiθ − arg g r eiθ < π
for 0 < t ≤ 1. Putting
ξ=
g tr eiθ α/2
,
g r eiθ
(2.9)
we have
arg s =
g tr eiθ
α
.
arg
2
g r eiθ
From (2.8) and (2.10), s lies in the convex sector
π
s : | arg s| ≤ α
2
(2.10)
(2.11)
and the same is true of its integral mean of (2.7), (cf. [5, Lemma 1]). Therefore, we have
arg f (z) < π α in Ᏹ
(2.12)
zf (z) 2
or
arg zf (z) < π α
f (z) 2
in Ᏹ.
(2.13)
This shows that
Re
zf (z)
f (z)
>0
which completes the proof of our main theorem.
in Ᏹ,
(2.14)
54
MAMORU NUNOKAWA ET AL.
Remark 2.2. This result is sharp for the case α → 0 and α = 1.
(a) For the case α → 0, put f (z) = z, then f (z) is a convex function of order 1 −
(α/2) → 1 and f (z) then f (z) is a strongly starlike function of order α → 0.
(b) For the case α = 1, put
zf (z)
1
=
.
f (z)
1−z
1+
(2.15)
Then we have
1 + Re
zf (z)
f (z)
>
1
2
in Ᏹ,
(2.16)
and therefore f (z) is a convex function of order 1/2. From (2.10), we easily have
1
1
f (z) =
,
f (z) = log
.
(2.17)
1−z
1−z
Putting |z| = 1, z = eiθ , 0 ≤ θ < 2π , then it follows that
z
1
cos (θ/2)
= − +i
,
1−z
2
2 sin (θ/2)
1
1
cos(θ/2) + i arg 1 + i cos(θ/2) .
log
= log +i
1−z
2
2 sin(θ/2)
2
2 sin(θ/2)
zf (z)
z/(1 − z)
lim arg
= lim arg
f (z)
log(1/(1 − z))
θ→+0
θ→+0
(2.18)
cos(θ/2)
1
= lim arg − + i
2
2 sin(θ/2)
θ→+0
1
cos(θ/2) + i arg 1 + i cos(θ/2)
− lim arg log
+i
2
2 sin(θ/2)
2
2 sin(θ/2)
θ→+0
=
π
.
2
The above shows that the main theorem is sharp for the case α → 0 and α = 1.
Applying the same method as above and [2], we can obtain the following result.
Theorem 2.3. If f (z) ∈ A(p) and satisfies
p−
α
zf (z)
< 1 + Re
2
f (z)
in Ᏹ,
(2.19)
where 0 < α ≤ 1, then f (z) ∈ STS(p, α).
References
[1]
[2]
[3]
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, On certain multivalently starlike functions, Tsukuba J. Math. 14 (1990), no. 2, 275–
277. MR 92b:30016. Zbl 728.30014.
, On the order of strongly starlikeness of strongly convex functions, Proc. Japan Acad.
Ser. A Math. Sci. 69 (1993), no. 7, 234–237. MR 95f:30019. Zbl 793.30007.
ON THE STRONGLY STARLIKENESS . . .
[4]
[5]
[6]
55
M. Nunokawa and S. Owa, On certain subclass of analytic functions, Indian J. Pure Appl.
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Mamoru Nunokawa: Department of Mathematics, University of Gunma, Aramaki
Maebashi Gunma, 371-8510, Japan
E-mail address: nunokawa@edu.gunma-u.ac.jp
Shigeyoshi Owa: Department of Mathematics, Kinki University, Higashi-Osaka,
Osaka 577-8502, Japan
E-mail address: owa@math.kindai.ac.jp
Akira Ikeda: Department of Applied Mathematics, Fukuoka University, Nanakuma
Jonan-ku Fukuoka, 814-0180, Japan
E-mail address: aikeda@sf.sm.fukuoka-u.ac.jp