Conditions for Starlikeness and Convexity
Mamoru Nunokawa, Shigeyoshi Owa, Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
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SUFFICIENT CONDITIONS FOR STARLIKENESS AND CONVEXITY IN |z| <
12MAMORU NUNOKAWA
University of Gunma
798-8 Hoshikuki-machi, Chuo-ku, Chiba-shi Chiba 260-0808, Japan
EMail:mamoru_nuno@doctor.nifty.jp
SHIGEYOSHI OWA, YAYOI NAKAMURA AND TOSHIO HAYAMI
Department of Mathematics, Kinki University Higashi-Osaka, Osaka 577-8502, Japan
EMail:owa@math.kindai.ac.jp yayoi@math.kindai.ac.jp ha_ya_to112@hotmail.com Received: 18 February, 2008
Accepted: 04 June, 2008
Communicated by: A. Sofo 2000 AMS Sub. Class.: Primary 30C45.
Key words: Analytic, Starlike, Convex.
Abstract: For analytic functionsf(z)withf(0) =f0(0)−1 = 0in the open unit discE, T. H. MacGregor has considered some conditions forf(z)to be starlike or convex. The object of the present paper is to discuss some interesting problems forf(z)to be starlike or convex for|z|< 12. Acknowledgements: We would like to thank the referee for his very useful suggestions
which essentially improved this paper.
Conditions for Starlikeness and Convexity
Mamoru Nunokawa, Shigeyoshi Owa, Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
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Contents
1 Introduction 3
2 Starlikeness and Convexity 5
Conditions for Starlikeness and Convexity
Mamoru Nunokawa, Shigeyoshi Owa, Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
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1. Introduction
LetAdenote the class of functionsf(z)of the form f(z) =z+
∞
X
n=2
anzn
which are analytic in the open unit discE ={z ∈C : |z|< 1}. A functionf ∈ A is said to be starlike with respect to the origin inEif it satisfies
Re
zf0(z) f(z)
>0 (z∈E).
Also, a functionf ∈ Ais called as convex inEif it satisfies Re
1 + zf00(z) f0(z)
>0 (z ∈E).
MacGregor [2] has shown the following.
Theorem A. Iff ∈ Asatisfies
f(z)
z −1
<1 (z∈E),
then
zf0(z) f(z) −1
<1
|z|< 1 2
so that
Re
zf0(z) f(z)
>0
|z|< 1 2
. Therefore,f(z)is univalent and starlike for|z|< 12.
Conditions for Starlikeness and Convexity
Mamoru Nunokawa, Shigeyoshi Owa, Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
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Also, MacGregor [3] had given the following results.
Theorem B. Iff ∈ Asatisfies
|f0(z)−1|<1 (z ∈E), then
Re
1 + zf00(z) f0(z)
> 0 for|z|< 1 2. Therefore,f(z)is convex for|z|< 12.
Theorem C. Iff ∈ Asatisfies
|f0(z)−1|<1 (z ∈E), thenf(z)maps|z|< 2
√ 5
5 = 0.8944. . . onto a domain which is starlike with respect to the origin,
argzf0(z) f(z)
< π
2 for|z|< 2√ 5 5 or
Re zf0(z)
f(z) >0 for|z|< 2√ 5 5 .
The condition domains of Theorem A, Theorem B and Theorem C are some circular domains whose center is the pointz = 1.
It is the purpose of the present paper to obtain some sufficient conditions for starlikeness or convexity under the hypotheses whose condition domains are annular domains centered at the origin.
Conditions for Starlikeness and Convexity
Mamoru Nunokawa, Shigeyoshi Owa, Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
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2. Starlikeness and Convexity
We start with the following result for starlikeness of functionsf(z).
Theorem 2.1. Letf ∈ Aand suppose that 0.10583· · ·= exp
− π2 4 log 3
(2.1)
<
zf0(z) f(z)
<exp π2
4 log 3
= 9.44915. . . (z ∈E).
Thenf(z)is starlike for|z|< 12.
Proof. From the assumption (2.1), we get
f(z)6= 0 (0<|z|<1).
From the harmonic function theory (cf. Duren [1]), we have log
zf0(z) f(z)
= 1 2π
Z
|ζ|=R
log
ζf0(ζ) f(ζ)
ζ+z
ζ−zdϕ+iarg
zf0(z) f(z)
z=0
= 1 2π
Z
|ζ|=R
log
zf0(ζ) f(ζ)
ζ+z ζ−zdϕ where|z|=r <|ζ|=R <1,z =reiθ andζ =Reiϕ.
Conditions for Starlikeness and Convexity
Mamoru Nunokawa, Shigeyoshi Owa, Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
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It follows that
arg
zf0(z) f(z)
=
1 2π
Z
|ζ|=R
log
ζf0(ζ) f(ζ)
Imζ+z ζ−z
dϕ
≤ 1 2π
Z 2π
0
log
ζf0(ζ) f(ζ)
2Rrsin(ϕ−θ) R2−2Rrcos(ϕ−θ) +r2
dϕ
< π2 4 log 3
1 2π
Z 2π
0
2Rr|sin(ϕ−θ)|
R2−2Rrcos(ϕ−θ) +r2dϕ
= π2 4 log 3
2
πlog R+r R−r. LettingR →1, we have
argzf0(z) f(z)
< π
2 log 3log 1 +r 1−r
< π
2 log 3log 3
= π 2
|z|=r < 1 2
. This completes the proof of the theorem.
Next we derive the following
Conditions for Starlikeness and Convexity
Mamoru Nunokawa, Shigeyoshi Owa, Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
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Theorem 2.2. Letf ∈ Aand suppose that 0.472367. . .= exp
−3 4
(2.2)
<
f(z) z
<exp 3
4
= 2.177. . . (z ∈E).
Then we have
zf0(z) f(z) −1
<1
|z|< 1 2
, orf(z)is starlike for|z|< 12.
Proof. From the assumption (2.2), we have
f(z)6= 0 (0<|z|<1).
Applying the harmonic function theory (cf. Duren [1]), we have log
f(z) z
= 1 2π
Z
|ζ|=R
log
f(ζ) ζ
ζ+z ζ−zdϕ, where|z|=r <|ζ|=R <1,z =reiθ andζ =Reiϕ.
Then, it follows that zf0(z)
f(z) −1 = 1 2π
Z
|ζ|=R
log
f(ζ) ζ
2ζz (ζ−z)2dϕ.
Conditions for Starlikeness and Convexity
Mamoru Nunokawa, Shigeyoshi Owa, Yayoi Nakamura and Toshio Hayami
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This gives us
zf0(z) f(z) −1
≤ 1 2π
Z
|ζ|=R
log
f(ζ) ζ
2Rr
R2 −2Rrcos(ϕ−θ) +r2dϕ
< 3 4
1 2π
Z
|ζ|=R
2Rr
R2−2Rrcos(ϕ−θ) +r2dϕ
= 3 4
2Rr R2−r2. MakingR →1, we have
zf0(z) f(z) −1
< 3 4
2r 1−r2 <1
|z|=r < 1 2
, which completes the proof of the theorem.
For convexity of functions f(z), we show the following corollary without the proof.
Corollary 2.3. Letf ∈ Aand suppose that (2.3) 0.472367· · ·= exp
−3 4
<|f0(z)|<exp 3
4
= 2.117. . . (z ∈E).
Thenf(z)is convex for|z|< 12.
Next our result for the convexity of functionsf(z)is contained in Theorem 2.4. Letf ∈ Aand suppose that
(2.4)
0.778801· · ·= exp
−1 4
<
zf0(z) f(z)
<exp 1
4
= 1.28403. . . (z ∈E).
Conditions for Starlikeness and Convexity
Mamoru Nunokawa, Shigeyoshi Owa, Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
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Thenf(z)is convex for|z|< 12.
Proof. From the condition (2.4) of the theorem, we have zf0(z)
f(z) 6= 0 inE. Then, it follows that
(2.5) log zf0(z) f(z) = 1
2π Z
|ζ|=R
log ζf0(ζ) f(ζ)
ζ+z ζ−zdϕ, where|z|=r <|ζ|=R <1,z =reiθ andζ =Reiϕ.
Differentiating (2.5) and multiplying byz, we obtain that 1 + zf00(z)
f0(z) = zf0(z) f(z) + 1
2π Z
|ζ|=R
log
ζf0(ζ) f(ζ)
2ζz (ζ−z)2dϕ.
In view of Theorem2.1,f(z)is starlike for|z|< 12 and therefore, we have Rezf0(z)
f(z) ≥ 1−r 1 +r
|z|=r < 1 2
. Then, we have
1 + Rezf00(z)
f0(z) = Rezf0(z) f(z) + 1
2π Z
|ζ|=R
log
ζf0(ζ) f(ζ)
Re 2ζz (ζ−z)2
dϕ
> 1−r 1 +r − 1
2π Z
|ζ|=R
1 4
2Rr
|ζ−z|2dϕ
= 1−r 1 +r − 1
4
2Rr R2−r2.
Conditions for Starlikeness and Convexity
Mamoru Nunokawa, Shigeyoshi Owa, Yayoi Nakamura and Toshio Hayami
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LettingR →1, we see that
1 + Rezf00(z)
f0(z) > 1−r 1 +r − 1
4 2r 1−r2
= 1 3 −1
4 · 4 3
= 0
|z|=r < 1 2
, which completes the proof of our theorem.
Finally, we prove
Theorem 2.5. Letf ∈ Aand suppose that 0.10583. . .= exp
− π2 4 log 3
<
zf0(z) f(z)
<exp π2
4 log 3
= 9.44915. . . (z ∈E).
Thenf(z)is convex in|z|< r0 wherer0 is the root of the equation (4 log 3)r2−2(4 log 3 +π2)r+ 4 log 3 = 0, r0 = π2−4 log 3−πp
π2+ 8 log 3
4 log 3 = 0.15787. . . . Proof. Applying the same method as the proof of Theorem2.5, we have
1 + Rezf00(z)
f0(z) = Rezf0(z) f(z) + 1
2π Z
|ζ|=R
log
ζf0(ζ) f(ζ)
Re 2ζz (ζ−z)2
dϕ
> 1−r
1 +r − π2 4 log 3
2Rr R2−r2
Conditions for Starlikeness and Convexity
Mamoru Nunokawa, Shigeyoshi Owa, Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
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where|z|=r <|ζ|=R <1,z =reiθ andζ =Reiϕ. PuttingR →1, we have
1 + Rezf00(z)
f0(z) > 1−r
1 +r − π2 4 log 3
2r 1−r2
= 1
(1−r2)4 log 3 n
(4 log 3)r2−2(4 log 3 +π2)r+ 4 log 3o
>0 (|z|< r0).
Remark 1. The condition in TheoremAby MacGregor [2] implies that 0<Re
f(z) z
<2 (z ∈E).
However, the condition in Theorem2.2implies that
−2.117· · ·<Re
f(z) z
<2.117. . . (z ∈E).
Furthermore, the condition in TheoremBby MacGregor [3] implies that 0<Ref0(z)<2 (z ∈E).
However, the condition in Corollary2.3implies that
−2.117· · ·<Ref0(z)<2.117. . . (z ∈E).
Conditions for Starlikeness and Convexity
Mamoru Nunokawa, Shigeyoshi Owa, Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
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References
[1] P. DUREN, Harmonic mappings in the plane, Cambridge Tracts in Mathematics 156, Cambridge Univ. Press, 2004.
[2] T.H. MacGREGOR, The radius of univalence of certain analytic functions. II, Proc. Amer. Math. Soc., 14(3) (1963), 521–524.
[3] T.H. MacGREGOR, A class of univalent functions, Proc. Amer. Math. Soc., 15 (1964), 311–317.