SUFFICIENT CONDITIONS FOR STARLIKENESS
AND CONVEXITY IN |z| < 21
Conditions for Starlikeness
and Convexity
Mamoru Nunokawa, Shigeyoshi Owa,
Yayoi Nakamura and Toshio Hayami
MAMORU NUNOKAWA
University of Gunma
798-8 Hoshikuki-machi, Chuo-ku, Chiba-shi
Chiba 260-0808, Japan
EMail: mamoru_nuno@doctor.nifty.jp
vol. 9, iss. 2, art. 32, 2008
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
Title Page
Contents
ha_ya_to112@hotmail.com
JJ
II
04 June, 2008
J
I
Communicated by:
A. Sofo
Page 1 of 12
2000 AMS Sub. Class.:
Primary 30C45.
Key words:
Analytic, Starlike, Convex.
Abstract:
For analytic functions f (z) with f (0) = f 0 (0)−1 = 0 in the open unit
disc E, T. H. MacGregor has considered some conditions for f (z) to be
starlike or convex. The object of the present paper is to discuss some
interesting problems for f (z) to be starlike or convex for |z| < 21 .
Received:
18 February, 2008
Accepted:
Go Back
Full Screen
Acknowledgements:
We would like to thank the referee for his very useful suggestions
which essentially improved this paper.
Close
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
Title Page
Contents
JJ
II
J
I
Page 2 of 12
Go Back
Full Screen
Close
1.
Introduction
Let A denote the class of functions f (z) of the form
f (z) = z +
∞
X
an z n
n=2
which are analytic in the open unit disc E = {z ∈ C : |z| < 1}. A function f ∈ A
is said to be starlike with respect to the origin in E if it satisfies
0 zf (z)
Re
>0
(z ∈ E).
f (z)
Also, a function f ∈ A is called as convex in E if it satisfies
zf 00 (z)
Re 1 + 0
>0
(z ∈ E).
f (z)
MacGregor [2] has shown the following.
Theorem A. If f ∈ A satisfies
f (z)
z − 1 < 1
then
Conditions for Starlikeness
and Convexity
Mamoru Nunokawa, Shigeyoshi Owa,
Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
Title Page
Contents
JJ
II
J
I
Page 3 of 12
(z ∈ E),
Go Back
Full Screen
0
zf (z)
<1
−
1
f (z)
so that
Re
zf 0 (z)
f (z)
1
|z| <
2
|z| <
>0
1
2
Therefore, f (z) is univalent and starlike for |z| < 12 .
Close
.
Also, MacGregor [3] had given the following results.
Theorem B. If f ∈ A satisfies
|f 0 (z) − 1| < 1
then
zf 00 (z)
Re 1 + 0
f (z)
(z ∈ E),
> 0
1
for |z| < .
2
Conditions for Starlikeness
and Convexity
Mamoru Nunokawa, Shigeyoshi Owa,
Yayoi Nakamura and Toshio Hayami
Therefore, f (z) is convex for |z| < 12 .
vol. 9, iss. 2, art. 32, 2008
Theorem C. If f ∈ A satisfies
|f 0 (z) − 1| < 1
(z ∈ E),
Title Page
√
then f (z) maps |z| < 2 5 5 = 0.8944 . . . onto a domain which is starlike with respect
to the origin,
√
0
π
2
5
zf
(z)
<
arg
for |z| <
f (z)
2
5
or
0
Re
zf (z)
>0
f (z)
√
2 5
for |z| <
.
5
The condition domains of Theorem A, Theorem B and Theorem C are some
circular domains whose center is the point z = 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.
Contents
JJ
II
J
I
Page 4 of 12
Go Back
Full Screen
Close
2.
Starlikeness and Convexity
We start with the following result for starlikeness of functions f (z).
Theorem 2.1. Let f ∈ A and suppose that
π2
(2.1)
0.10583 · · · = exp −
4 log 3
0 zf (z) < f (z) 2 π
< exp
= 9.44915 . . .
4 log 3
Then f (z) is starlike for |z| <
Conditions for Starlikeness
and Convexity
Mamoru Nunokawa, Shigeyoshi Owa,
Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
(z ∈ E).
Title Page
1
.
2
Contents
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
0 0 0 Z
ζf (ζ) ζ + z
zf (z)
1
zf (z)
log
=
log dϕ + i arg
f (z)
2π |ζ|=R
f (ζ)
ζ −z
f (z) z=0
0 Z
zf (ζ) ζ + z
1
=
log dϕ
2π |ζ|=R
f (ζ) ζ − z
where |z| = r < |ζ| = R < 1, z = reiθ and ζ = Reiϕ .
JJ
II
J
I
Page 5 of 12
Go Back
Full Screen
Close
It follows that
0 0 Z
1
ζf (ζ) ζ
+
z
zf
(z)
=
arg
log
Im
dϕ
f (ζ) f (z) 2π |ζ|=R
ζ −z
Z 2π 0 ζf (ζ) 1
2Rr
sin(ϕ
−
θ)
log dϕ
≤
2π 0 f (ζ) R2 − 2Rr cos(ϕ − θ) + r2 Z 2π
π2
1
2Rr |sin(ϕ − θ)|
<
dϕ
2
4 log 3 2π 0 R − 2Rr cos(ϕ − θ) + r2
R+r
π2 2
=
log
.
4 log 3 π
R−r
Letting R → 1, we have
0
arg zf (z) < π log 1 + r
f (z) 2 log 3
1−r
π
log 3
<
2 log 3
1
π
=
|z| = r <
.
2
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
Title Page
Contents
JJ
II
J
I
Page 6 of 12
Go Back
Full Screen
Close
Theorem 2.2. Let f ∈ A and suppose that
3
0.472367 . . . = exp −
(2.2)
4
f (z) < z 3
< exp
= 2.177 . . .
4
Then we have
(z ∈ E).
Conditions for Starlikeness
and Convexity
Mamoru Nunokawa, Shigeyoshi Owa,
Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
0
zf (z)
<1
−
1
f (z)
1
|z| <
2
,
or f (z) is starlike for |z| < 12 .
Contents
Proof. From the assumption (2.2), we have
f (z) 6= 0
Title Page
(0 < |z| < 1).
JJ
II
J
I
Applying the harmonic function theory (cf. Duren [1]), we have
Z
f (ζ) ζ + z
f (z)
1
log
=
log dϕ,
z
2π |ζ|=R
ζ ζ −z
Page 7 of 12
where |z| = r < |ζ| = R < 1, z = reiθ and ζ = Reiϕ .
Then, it follows that
Z
f (ζ) zf 0 (z)
1
2ζz
−1=
log dϕ.
f (z)
2π |ζ|=R
ζ (ζ − z)2
Close
Go Back
Full Screen
This gives us
0
Z
zf (z)
2Rr
f (ζ) ≤ 1
−
1
dϕ
2
log 2π
f (z)
ζ R − 2Rr cos(ϕ − θ) + r2
|ζ|=R Z
3 1
2Rr
<
dϕ
4 2π |ζ|=R R2 − 2Rr cos(ϕ − θ) + r2
3 2Rr
=
.
4 R2 − r 2
Making R → 1, we have
0
3 2r
zf (z)
<
−
1
4 1 − r2 < 1
f (z)
Conditions for Starlikeness
and Convexity
Mamoru Nunokawa, Shigeyoshi Owa,
Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
|z| = r <
1
2
,
Title Page
which completes the proof of the theorem.
Contents
For convexity of functions f (z), we show the following corollary without the
proof.
Corollary 2.3. Let f ∈ A and suppose that
3
3
0
< |f (z)| < exp
= 2.117 . . .
(2.3) 0.472367 · · · = exp −
4
4
II
J
I
Page 8 of 12
(z ∈ E).
Go Back
Full Screen
Then f (z) is convex for |z| < 12 .
Close
Next our result for the convexity of functions f (z) is contained in
Theorem 2.4. Let f ∈ A and suppose that
(2.4)
0 zf (z) 1
1
0.778801 · · · = exp −
<
< exp
= 1.28403 . . .
4
f (z)
4
JJ
(z ∈ E).
Then f (z) is convex for |z| < 12 .
Proof. From the condition (2.4) of the theorem, we have
zf 0 (z)
6= 0
f (z)
in E.
Then, it follows that
0
(2.5)
log
1
zf (z)
=
f (z)
2π
Z
0
log
|ζ|=R
ζf (ζ)
f (ζ)
ζ +z
dϕ,
ζ −z
Conditions for Starlikeness
and Convexity
Mamoru Nunokawa, Shigeyoshi Owa,
Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
where |z| = r < |ζ| = R < 1, z = reiθ and ζ = Reiϕ .
Differentiating (2.5) and multiplying by z, we obtain that
0 Z
ζf (ζ) zf 0 (z)
1
2ζz
zf 00 (z)
1+ 0
=
+
log dϕ.
f (z)
f (z)
2π |ζ|=R
f (ζ)
(ζ − z)2
In view of Theorem 2.1, f (z) is starlike for |z| < 21 and therefore, we have
1−r
1
zf 0 (z)
Re
≥
|z| = r <
.
f (z)
1+r
2
Then, we have
0 Z
ζf (ζ) zf 00 (z)
zf 0 (z)
1
2ζz
1 + Re 0
= Re
+
log Re
dϕ
f (z)
f (z)
2π |ζ|=R
f (ζ) (ζ − z)2
Z
1−r
1
1 2Rr
>
−
dϕ
1 + r 2π |ζ|=R 4 |ζ − z|2
1 − r 1 2Rr
−
.
=
1 + r 4 R2 − r 2
Title Page
Contents
JJ
II
J
I
Page 9 of 12
Go Back
Full Screen
Close
Letting R → 1, we see that
1 + Re
zf 00 (z)
1 − r 1 2r
>
−
0
f (z)
1 + r 4 1 − r2
1 1 4
= − ·
3 4 3
=0
1
|z| = r <
2
Conditions for Starlikeness
and Convexity
Mamoru Nunokawa, Shigeyoshi Owa,
,
which completes the proof of our theorem.
Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
Finally, we prove
Theorem 2.5. Let f ∈ A and suppose that
π2
0.10583 . . . = exp −
4 log 3
0 2 zf (z) π
<
< exp
= 9.44915 . . .
f (z)
4 log 3
Title Page
Contents
(z ∈ E).
Then f (z) is convex in |z| < r0 where r0 is the root of the equation
(4 log 3)r2 − 2(4 log 3 + π 2 )r + 4 log 3 = 0,
p
π 2 − 4 log 3 − π π 2 + 8 log 3
r0 =
= 0.15787 . . . .
4 log 3
Proof. Applying the same method as the proof of Theorem 2.5, we have
0 Z
ζf (ζ) zf 00 (z)
zf 0 (z)
1
2ζz
1 + Re 0
= Re
+
log Re
dϕ
f (z)
f (z)
2π
f (ζ) (ζ − z)2
|ζ|=R
2
>
1−r
π
2Rr
−
2
1 + r 4 log 3 R − r2
JJ
II
J
I
Page 10 of 12
Go Back
Full Screen
Close
where |z| = r < |ζ| = R < 1, z = reiθ and ζ = Reiϕ .
Putting R → 1, we have
1 + Re
zf 00 (z)
1−r
π2
2r
>
−
0
f (z)
1 + r 4 log 3 1 − r2
n
o
1
2
2
(4 log 3)r − 2(4 log 3 + π )r + 4 log 3
=
(1 − r2 )4 log 3
>0
(|z| < r0 ).
Conditions for Starlikeness
and Convexity
Mamoru Nunokawa, Shigeyoshi Owa,
Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
Remark 1. The condition in Theorem A by MacGregor [2] implies that
f (z)
0 < Re
<2
(z ∈ E).
z
However, the condition in Theorem 2.2 implies that
f (z)
−2.117 · · · < Re
< 2.117 . . .
z
(z ∈ E).
Furthermore, the condition in Theorem B by MacGregor [3] implies that
0 < Re f 0 (z) < 2
Contents
JJ
II
J
I
Page 11 of 12
Go Back
Full Screen
(z ∈ E).
Close
However, the condition in Corollary 2.3 implies that
−2.117 · · · < Re f 0 (z) < 2.117 . . .
Title Page
(z ∈ E).
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.
Conditions for Starlikeness
and Convexity
Mamoru Nunokawa, Shigeyoshi Owa,
Yayoi Nakamura and Toshio Hayami
vol. 9, iss. 2, art. 32, 2008
Title Page
Contents
JJ
II
J
I
Page 12 of 12
Go Back
Full Screen
Close