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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 orderαin the open unit disc.
2000 Mathematics Subject Classification. 30C45.
1. Introduction. LetᏭ(p)denote the class of the functionsf (z)=zp+∞
n=p+1anzn which are analytic in the open unit discᏱ= {z:|z|<1}. A functionf (z)∈Ꮽ(p)is calledp-valently starlike if and only if the inequality
Re
zf(z) f (z)
>0 (1.1)
holds forz∈Ᏹ. A functionf (z)∈Ꮽ(p)is calledp-valently convex of orderα (0≤ α < p)if and only if the inequality
1+Re
zf(z) f(z)
> α (1.2)
holds forz∈Ᏹ. We denote byᏯ(p, α)the family of such functions. A functionf (z)∈ Ꮽ(p)is said to be strongly starlike of orderα (0< α≤1)if and only if the inequality
arg
zf(z) f (z)
<π
2α (1.3)
holds forz∈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 iff (z)∈ STS(p, α), then we have
Re
zf(z) f (z)
>0 inᏱ (1.4)
orf (z)isp-valently starlike inᏱand thereforef (z)isp-valent inᏱ(see [1, Lemma 7]).
Nunokawa [2,3] proved the following theorems.
Theorem1.1(see [2]). Iff (z)∈Ꮽ(p)satisfies 1+Re
zf(z) f(z)
< p+α
2, (1.5)
where0< α≤1, thenf (z)∈STS(p, α).
Theorem1.2(see [3]). Iff (z)∈Ꮽ(1)satisfies arg
1+zf(z) f(z)
<π
2α(β) inᏱ, (1.6)
then
arg
zf(z) f (z)
<π
2β inᏱ, (1.7)
where
α(β)=β+ 2 πtan−1
βq(β)sin(π /2)(1−β) p(β)+βq(β)cos(π /2)(1−β)
, p(β)=(1+β)(1+β)/2, q(β)=(1−β)(β−1)/2.
(1.8)
It is the purpose of the present paper to prove that iff (z)∈Ꮿ(1,1−(α/2)), then f (z)∈STS(1, α).
In this paper, we need the following lemma.
Lemma1.3. Letf (z)∈Ꮽ(1)be starlike with respect to the origin inᏱ. LetC(r , θ)= {f (teiθ): 0≤t≤r <1}andT (r , θ)be the total variation ofargf (teiθ)on C(r , θ), so that
T (r , θ)= r
0
∂
∂targ f
teiθ dt. (1.9)
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 orderαis the following.
Theorem2.1. Letf (z)∈Ꮽ(1)and 1+Re
zf(z) f(z)
>1−α
2 inᏱ, (2.1)
where0< α≤1. Then
arg
zf(z) f (z)
<π
2α inᏱ, (2.2)
orf (z)is strongly starlike of orderαinᏱ. Proof. We put
2 α
1+zf(z) f(z) −1+α
2
=zg(z)
g(z) , (2.3)
whereg(z)=z+∞
n=2bnzn. From assumption (2.1), we have Re
zg(z) g(z)
>0 inᏱ. (2.4)
This shows thatg(z)is starlike and univalent inᏱ. 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) f(z) dt=
1 0t−α/2
g tr eiθ g
r eiθ
α/2
dt, (2.7)
wherez=r eiθand 0< r <1. Sinceg(z)is starlike inᏱ, fromLemma 1.3, we have
−π <arg g
tr eiθ
−arg g
r eiθ
< π (2.8)
for 0< t≤1. Putting
ξ= g
tr eiθ g
r eiθα/2
, (2.9)
we have
args=α 2arg
g tr eiθ g
r eiθ
. (2.10)
From (2.8) and (2.10),slies in the convex sector
s:|args| ≤π 2α
(2.11) and the same is true of its integral mean of (2.7), (cf. [5, Lemma 1]). Therefore, we have
arg f (z)
zf(z) <π
2α inᏱ (2.12)
or
arg
zf(z) f (z)
<π
2α inᏱ. (2.13)
This shows that
Re
zf(z) f (z)
>0 inᏱ, (2.14)
which completes the proof of our main theorem.
Remark2.2. This result is sharp for the caseα→0 andα=1.
(a) For the caseα→0, putf (z)=z, thenf (z)is a convex function of order 1− (α/2)→1 andf (z)thenf (z)is a strongly starlike function of orderα→0.
(b) For the caseα=1, put
1+zf(z) f(z) = 1
1−z. (2.15)
Then we have
1+Re
zf(z) f(z)
>1
2 inᏱ, (2.16)
and thereforef (z)is a convex function of order 1/2. From (2.10), we easily have f(z)= 1
1−z, f (z)=log 1
1−z
. (2.17)
Putting|z| =1,z=eiθ, 0≤θ <2π, then it follows that z
1−z= −1
2+icos(θ/2) 2 sin(θ/2), log
1 1−z
=log 1
2+icos(θ/2) 2 sin(θ/2)
+iarg 1
2+icos(θ/2) 2 sin(θ/2)
.
θ→+0limarg
zf(z) f (z)
= lim
θ→+0arg
z/(1−z) log(1/(1−z))
= lim
θ→+0arg
−1
2+icos(θ/2) 2 sin(θ/2)
−lim
θ→+0arg
log 1
2+icos(θ/2) 2 sin(θ/2)
+iarg1
2+icos(θ/2) 2 sin(θ/2)
=π 2.
(2.18)
The above shows that the main theorem is sharp for the caseα→0 andα=1.
Applying the same method as above and [2], we can obtain the following result.
Theorem2.3. Iff (z)∈A(p)and satisfies
p−α
2<1+Re
zf(z) f(z)
inᏱ, (2.19)
where0< α≤1, thenf (z)∈STS(p, α).
References
[1] M. Nunokawa,On the theory of multivalent functions, Tsukuba J. Math.11(1987), no. 2, 273–286.MR 89d:30013. Zbl 639.30014.
[2] ,On certain multivalently starlike functions, Tsukuba J. Math.14(1990), no. 2, 275–
277.MR 92b:30016. Zbl 728.30014.
[3] ,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.
[4] M. Nunokawa and S. Owa,On certain subclass of analytic functions, Indian J. Pure Appl.
Math.19(1988), no. 1, 51–54.MR 89c:30029. Zbl 646.30020.
[5] C. Pommerenke,On close-to-convex analytic functions, Trans. Amer. Math. Soc.114(1965), 176–186.MR 30#4920. Zbl 132.30204.
[6] T. Sheil-Small,Some conformal mapping inequalities for starlike and convex functions, J.
London Math. Soc. (2)1(1969), 577–587.MR 40#2842. Zbl 201.40803.
Mamoru Nunokawa: Department of Mathematics, University of Gunma, Aramaki Maebashi Gunma,371-8510, Japan
E-mail address:[email protected]
Shigeyoshi Owa: Department of Mathematics, Kinki University, Higashi-Osaka, Osaka577-8502, Japan
E-mail address:[email protected]
Akira Ikeda: Department of Applied Mathematics, Fukuoka University, Nanakuma Jonan-ku Fukuoka,814-0180, Japan
E-mail address:[email protected]
