Journal of Inequalities in Pure and
Applied Mathematics
COEFFICIENT INEQUALITIES FOR CLASSES OF UNIFORMLY
STARLIKE AND CONVEX FUNCTIONS
volume 7, issue 5, article 160,
2006.
SHIGEYOSHI OWA, YAŞAR POLATOĞLU AND EMEL YAVUZ
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502
Japan
EMail: owa@math.kindai.ac.jp
Department of Mathematics and Computer Science
İstanbul Kültür University
Bakırköy 34156, İstanbul, Turkey
EMail: y.polatoglu@iku.edu.tr
EMail: e.yavuz@iku.edu.tr
Received 28 September, 2006;
accepted 06 November, 2006.
Communicated by: N.E. Cho
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In view of classes of uniformly starlike and convex functions in the open unit
disc U which was considered by S. Shams, S.R. Kulkarni and J.M. Jahangiri,
some coefficient inequalities for functions are discussed.
2000 Mathematics Subject Classification: Primary 30C45.
Key words: Uniformly starlike, Uniformly convex.
Coefficient Inequalities for
Classes of Uniformly Starlike
and Convex Functions
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Coefficient Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
Shigeyoshi Owa, Yaşar Polatoğlu
and Emel Yavuz
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1.
Introduction
Let A be the class of functions f (z) of the form
(1.1)
f (z) = z +
∞
X
an z n
n=2
which are analytic in the open unit disc U = {z ∈ C | |z| < 1}.
Let S ∗ (β) denote the subclass of A consisting of functions f (z) which satisfy
0 zf (z)
(1.2)
Re
>β
(z ∈ U)
f (z)
for some β (0 5 β < 1). A function f (z) ∈ S ∗ (β) is said to be starlike of
order β in U. Also let K(β) be the subclass of A consisting of all functions
f (z) which satisfy
zf 00 (z)
(1.3)
Re 1 + 0
>β
(z ∈ U)
f (z)
for some β (0 5 β < 1). A function f (z) in K(β) is said to be convex of order
β in U. In view of the class S ∗ (β), Shams, Kulkarni and Jahangiri [3] have
introduced the subclass SD(α, β) of A consisting of functions f (z) satisfying
0
0 zf (z)
zf (z)
> α − 1 + β
(z ∈ U)
(1.4)
Re
f (z)
f (z)
Coefficient Inequalities for
Classes of Uniformly Starlike
and Convex Functions
Shigeyoshi Owa, Yaşar Polatoğlu
and Emel Yavuz
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for some α = 0 and β (0 5 β < 1). We also denote by KD(α, β) the subclass
of A consisting of all functions f (z) which satisfy
00 zf (z) zf 00 (z)
+β
(1.5)
Re 1 + 0
> α 0
(z ∈ U)
f (z)
f (z) for some α = 0 and β (0 5 β < 1). Then we note that f (z) ∈ KD(α, β) if and
only if zf 0 (z) ∈ SD(α, β). For such classes SD(α, β) and KD(α, β), Shams,
Kulkarni and Jahangiri [3] have shown some sufficient conditions for f (z) to be
in the classes SD(α, β) or KD(α, β).
Coefficient Inequalities for
Classes of Uniformly Starlike
and Convex Functions
Shigeyoshi Owa, Yaşar Polatoğlu
and Emel Yavuz
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2.
Coefficient Inequalities
Our first result is contained in
Theorem 2.1. If f (z) ∈ SD(α, β) with 0 5 α 5 β or α >
.
S ∗ β−α
1−α
1+β
2
then f (z) ∈
Proof. Since Re(w) 5 |w| for any complex number w, f (z) ∈ SD(α, β) implies that
0 0
zf (z)
zf (z)
(2.1)
Re
> α Re
− 1 + β,
f (z)
f (z)
Shigeyoshi Owa, Yaşar Polatoğlu
and Emel Yavuz
or that
(2.2)
Re
zf 0 (z)
f (z)
>
β−α
1−α
If 0 5 α 5 β, then we have that
β−α
05
< 1,
1−α
and if α >
Coefficient Inequalities for
Classes of Uniformly Starlike
and Convex Functions
1+β
, then we have
2
α−β
−1 <
5 0.
α−1
(z ∈ U).
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Corollary 2.2. If f (z) ∈ KD(α, β) with 0 5 α 5 β or α >
.
K β−α
1−α
1+β
,
2
then f (z) ∈
Next we derive
Theorem 2.3. If f (z) ∈ SD(α, β), then
|a2 | 5
(2.3)
2(1 − β)
|1 − α|
and
(2.4)
n−2
Y
2(1 − β)
2(1 − β)
|an | 5
1+
(n − 1) |1 − α| j=1
j |1 − α|
Proof. Note that, for f (z) ∈ SD(α, β),
0 zf (z)
β−α
Re
>
f (z)
1−α
Coefficient Inequalities for
Classes of Uniformly Starlike
and Convex Functions
(n = 3).
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(z ∈ U).
0
(2.5)
p(z) =
1−β
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If we define the function p(z) by
(z)
(1 − α) zff (z)
− (β − α)
Shigeyoshi Owa, Yaşar Polatoğlu
and Emel Yavuz
(z ∈ U),
then p(z) is analytic in U with p(0) = 1 and Re(p(z)) > 0 (z ∈ U). Letting
p(z) = 1 + p1 z + p2 z 2 + · · · , we have
!
∞
1−β X
0
n
(2.6)
zf (z) = f (z) 1 +
pn z .
1 − α n=1
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Therefore, (2.6) implies that
(2.7)
(n − 1)an =
1−β
(pn−1 + a2 pn−2 + · · · + an−1 p1 ).
1−α
Applying the coefficient estimates such that |pn | 5 2 (n = 1) (see [1]) for
Carathéodory functions, we obtain that
(2.8)
|an | 5
2(1 − β)
(1 + |a2 | + |a3 | + · · · + |an−1 |).
(n − 1) |1 − α|
Coefficient Inequalities for
Classes of Uniformly Starlike
and Convex Functions
Therefore, for n = 2,
|a2 | 5
2(1 − β)
,
|1 − α|
Shigeyoshi Owa, Yaşar Polatoğlu
and Emel Yavuz
which proves (2.3), and, for n = 3,
2(1 − β)
|a3 | 5
2 |1 − α|
2(1 − β)
1+
|1 − α|
.
Thus, (2.4) holds true for n = 3.
Supposing that (2.4) is true for n = 3, 4, 5, . . . , k, we see that
2(1 − β)
2(1 − β) 2(1 − β)
2(1 − β)
|ak+1 | 5
1+
+
1+
k |1 − α|
|1 − α|
2 |1 − α|
|1 − α|
)
k−2
Y
2(1 − β)
2(1 − β)
+ ··· +
1+
(k − 1) |1 − α| j=1
j |1 − α|
k−1 2(1 − β) Y
2(1 − β)
=
1+
.
k|1 − α| j=1
j|1 − α|
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Consequently, using mathematical induction, we have proved that (2.4) holds
true for any n = 3.
Remark 1. If we take α = 0 in Theorem 2.3, then we have
Qn
j=2 (j − 2β)
|an | 5
(n = 2)
(n − 1)!
which was given by Robertson [2].
Since f (z) ∈ KD(α, β) if and only if zf 0 (z) ∈ SD(α, β), we have
Coefficient Inequalities for
Classes of Uniformly Starlike
and Convex Functions
Corollary 2.4. If f (z) ∈ KD(α, β), then
|a2 | 5
(2.9)
Shigeyoshi Owa, Yaşar Polatoğlu
and Emel Yavuz
1−β
|1 − α|
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and
(2.10)
Contents
|an | 5
2(1 − β)
n(n − 1) |1 − α|
n−2
Y
1+
j=1
2(1 − β)
j |1 − α|
Remark 2. Letting α = 0 in Corollary 2.4, we see that
Qn
j=2 (j − 2β)
|an | 5
(n = 2),
n!
given by Robertson [2].
Further applying Theorem 2.3 we derive:
(n = 3).
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Theorem 2.5. If f (z) ∈ SD(α, β), then
)
!
2(1 − β)
1+
|z|n
j|1
−
α|
j=1
!
∞
n−2
Y
X
2(1 − β) 2
2(1 − β)
2(1 − β)
5 |f (z)| 5 |z|+
|z| +
1+
|z|n
|1 − α|
(n − 1)|1 − α| j=1
j|1 − α|
n=3
(
∞
2(1 − β) 2 X 2(1 − β)
|z| −
max 0, |z| −
|1 − α|
(n − 1)|1 − α|
n=3
n−2
Y
and
∞
X
4(1 − β)
2n(1 − β)
max 0, 1 −
|z| −
|1 − α|
(n − 1)|1 − α|
n=3
(
)
!
2(1 − β)
1+
|z|n−1
j|1
−
α|
j=1
!
n−2
∞
X 2n(1 − β)
Y
4(1
−
β)
2(1
−
β)
5 |f 0 (z)| 5 1+
|z|+
1+
|z|n−1 .
|1 − α|
(n
−
1)|1
−
α|
j|1
−
α|
n=3
j=1
n−2
Y
Corollary 2.6. If f (z) ∈ KD(α, β), then
(
)
!
2(1 − β)
1+
|z|n
j|1 − α|
j=1
!
∞
n−2
Y
1−β 2 X
2(1 − β)
2(1 − β)
5 |f (z)| 5 |z|+
|z| +
1+
|z|n
|1 − α|
n(n
−
1)|1
−
α|
j|1
−
α|
n=3
j=1
∞
1−β 2 X
2(1 − β)
max 0, |z| −
|z| −
|1 − α|
n(n − 1)|1 − α|
n=3
n−2
Y
Coefficient Inequalities for
Classes of Uniformly Starlike
and Convex Functions
Shigeyoshi Owa, Yaşar Polatoğlu
and Emel Yavuz
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and
∞
X
2(1 − β)
2(1 − β)
max 0, 1 −
|z| −
|1 − α|
(n − 1)|1 − α|
n=3
(
)
!
2(1 − β)
1+
|z|n−1
j|1
−
α|
j=1
!
n−1
∞
Y
X 2(1 − β)
2(1
−
β)
2(1
−
β)
5 |f 0 (z)| 5 1+
|z|+
1+
|z|n−1 .
|1 − α|
(n
−
1)|1
−
α|
j|1
−
α|
n=3
j=1
n−1
Y
Coefficient Inequalities for
Classes of Uniformly Starlike
and Convex Functions
Shigeyoshi Owa, Yaşar Polatoğlu
and Emel Yavuz
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References
[1] C. CARATHÉODORY, Über den variabilitätsbereich der Fourier’schen
konstanten von possitiven harmonischen funktionen, Rend. Circ. Palermo,
32 (1911), 193–217.
[2] M.S. ROBERTSON, On the theory of univalent functions, Ann. Math., 37
(1936), 374–408.
[3] S. SHAMS, S.R. KULKARNI AND J.M. JAHANGIRI, Classes of uniformly starlike and convex functions, Internat. J. Math. Math. Sci., 55
(2004), 2959–2961.
Coefficient Inequalities for
Classes of Uniformly Starlike
and Convex Functions
Shigeyoshi Owa, Yaşar Polatoğlu
and Emel Yavuz
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