Journal of Inequalities in Pure and
Applied Mathematics
INTEGRAL MEANS FOR STARLIKE AND CONVEX FUNCTIONS
WITH NEGATIVE COEFFICIENTS
volume 6, issue 2, article 50,
2005.
SHIGEYOSHI OWA1 , MIHAI PASCU2 , DAISUKE YAGI1 AND
JUNICHI NISHIWAKI1
Department of Mathematics1
Kinki University
Higashi-Osaka, Osaka 577-8502
Japan
EMail: owa@math.kindai.ac.jp
Department of Mathematics2
Transilvania University of Brasov
R-2200 Brasov
Romania
EMail: mihai.pascu@unitbv.ro
Received 17 February, 2005;
accepted 06 April, 2005.
Communicated by: N.E. Cho
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
041-05
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Abstract
Let T be the class of functions f(z) with negative coefficients which are analytic
and univalent in the open unit disk U with f(0) = 0 and f 0 (0) = 1. The classes
T ∗ and C are defined as the subclasses of T which are starlike and convex in
U, respectively. In view of the interesting results for integral means given by
H. Silverman (Houston J. Math. 23(1977)), some generalization theorems are
discussed in this paper.
2000 Mathematics Subject Classification: Primary 30C45.
Key words: Univalent, Starlike, Convex, Integral mean.
Memorial Paper for Professor Nicolae N. Pascu
Contents
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
Title Page
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Generalization Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Integral Means for Functions in the Class C . . . . . . . . . . . . . . . 14
4
Applications for the Integrated Functions . . . . . . . . . . . . . . . . . 19
References
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
1.
Introduction
Let A denote the class of functions f (z) of the form
(1.1)
f (z) = z +
∞
X
an z n
n=2
that are analytic in the open unit disk U = {z ∈ C : |z| < 1}. Let S be
the subclass of A consisting of all univalent functions f (z) in U. A function
f (z) ∈ A is said to be starlike with respect to the origin in U if it satisfies
0 zf (z)
(1.2)
Re
>0
(z ∈ U).
f (z)
We denote by S ∗ the subclass of S consisting of all starlike functions f (z) with
respect to the origin in U. Further, a function f (z) ∈ A is said to be convex in
U if it satisfies
zf 00 (z)
(1.3)
Re 1 + 0
>0
(z ∈ U).
f (z)
We also denote by K the subclass of S consisting of f (z) which are convex in
U. By the above definitions, we know that f (z) ∈ K if and only if zf 0 (z) ∈ S ∗ ,
and that K ⊂ S ∗ ⊂ S ⊂ A.
The class T is defined as the subclass of S consisting of all functions f (z)
which are given by
(1.4)
f (z) = z −
∞
X
an z n
(an ≥ 0).
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
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n=2
J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
Further, we denote by T ∗ = S ∗ ∩ T and C = K ∩ T . It is well-known by
Silverman [6] that
Remark 1. A function f (z) ∈ T ∗ if and only if
∞
X
(1.5)
nan ≤ 1.
n=2
A function f (z) ∈ C if and only if
∞
X
(1.6)
n2 an ≤ 1.
n=2
For f (z) ∈ A and g(z) ∈ A, f (z) is said to be subordinate to g(z) in U
if there exists an analytic function ω(z) in U such that ω(0) = 0, |ω(z)| < 1
(z ∈ U), and f (z) = g(ω(z)). We denote this subordination by
f (z) ≺ g(z). (cf. Duren [1]).
(1.7)
For subordinations, Littlewood [2] has given the following integral mean.
Theorem A. If f (z) and g(z) are analytic in U with f (z) ≺ g(z), then, for
λ > 0 and |z| = r (0 < r < 1),
Z 2π
Z 2π
iθ λ
(1.8)
|f (re )| dθ ≤
|g(reiθ )|λ dθ.
0
0
Integral Means for Starlike and
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Shigeyoshi Owa, Mihai Pascu,
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Furthermore, Silverman [6] has shown that
J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
n
Remark 2. f1 (z) = z and fn (z) = z − zn (n ≥ 2) are extreme points of the
n
class T ∗ (or T ). f1 (z) = z and fn (z) = z − zn2 (n ≥ 2) are extreme points of
the class C.
Applying Theorem A with extreme points of T , Silverman [7] has proved
the following results.
Theorem B. Suppose that f (z) ∈ T ∗ , λ > 0 and f2 (z) = z −
z = reiθ (0 < r < 1),
Z 2π
Z 2π
λ
(1.9)
|f (z)| dθ ≤
|f2 (z)|λ dθ.
0
z2
.
2
Then, for
0
2
Theorem C. If f (z) ∈ T ∗ , λ > 0, and f2 (z) = z − z2 , then, for z = reiθ
(0 < r < 1),
Z 2π
Z 2π
0
λ
(1.10)
|f (z)| dθ ≤
|f20 (z)|λ dθ.
0
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
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0
In the present paper, we consider the generalization properties for Theorem
B and Theorem C with f (z) ∈ T ∗ and f (z) ∈ C.
Remark 3. More recently, applying Theorem A by Littlewood [2], Sekine, Tsurumi and Srivastava [4]; and Sekine, Tsurumi, Owa and Srivastava [5] have
discussed some interesting properties of integral means inequalities for fractional derivatives of some general subclasses of analytic functions f (z) in the
open unit disk U. Further, Owa and Sekine [3] have considered the integral
means with some coefficient inequalities for certain analytic functions f (z) in
U.
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
2.
Generalization Properties
Our first result for the generalization properties is contained in
Theorem 2.1. Let f (z) ∈ T ∗ , λ > 0, and fk (z) = z −
satisfies
(2.1)
k−3
X
j+1
j=0
k
zk
k
(k ≥ 2). If f (z)
(a2k+j−1 + ak+j+1 − ak−j−1 ) ≥ 0
for k ≥ 3, and if there exists an analytic function ω(z) in U given by
!
∞
X
(ω(z))k−1 = k
an z n−1 ,
n=2
then, for z = reiθ (0 < r < 1),
Z 2π
Z
λ
(2.2)
|f (z)| dθ ≤
0
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Shigeyoshi Owa, Mihai Pascu,
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2π
λ
|fk (z)| dθ.
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0
Proof. For f (z) ∈ T ∗ , we have to show that
λ
Z 2π Z 2π ∞
k−1 λ
X
z
n−1 dθ.
1 −
an z dθ ≤
1 −
k
0
0
n=2
By Theorem A, it suffices to prove that
1−
∞
X
n=2
an z n−1 ≺ 1 −
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z
k−1
k
.
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
Let us define the function ω(z) by
(2.3)
1−
∞
X
1
an z n−1 = 1 − (ω(z))k−1 .
k
n=2
It follows from (2.3) that
|ω(z)|k−1
!
∞
∞
X
X
= k
an z n−1 ≤ |z|
kan .
n=2
n=2
Thus, we only show that
∞
X
kan ≤
n=2
or
∞
X
1
an ≤
k
n=2
∞
X
nan ,
Integral Means for Starlike and
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Coefficients
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
n=2
∞
X
!
nan .
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n=2
Indeed, we see that
! ∞
1 X
k−2
k−3
2
nan = 1 −
a2 + 1 −
a3 + · · · + 1 −
ak−2
k n=2
k
k
k
1
1
2
+ 1−
ak−1 + ak + 1 +
ak+1 + 1 +
ak+2
k
k
k
k+1
k+2
a2k+1 + 1 +
a2k+2 + · · ·
+ ··· + 1 +
k
k
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
=
k−2
k−3
(a2k−2 − a2 ) +
(a2k−3 − a3 ) + · · ·
k
k
1
2
k−1
+ (ak+2 − ak−2 ) + (ak+1 − ak−1 ) + 1 +
a2k−1
k
k
k
2k−2
X
k
k+1
+ 1+
a2k + 1 +
a2k+1 + · · · +
an .
k
k
n=2
Noting that
1+
k+j
2+j
≥1+
, (j = −1, 0, 1, . . . ),
k
k
we obtain
(2.4)
1
k
∞
X
!
Integral Means for Starlike and
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Coefficients
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
nan
n=2
k−2
k−3
≥
(a2k−2 − a2 ) +
(a2k−3 − a3 ) + · · ·
k
k
2
1
+ (ak+2 − ak−2 ) + (ak+1 − ak−1 )
k
k
2
1
+ 1+
a2k−1 + 1 +
a2k + · · ·
k
k
2k−2
X
k−3
k−2
+ 1+
a3k−5 + 1 +
a3k−4 + · · · +
an
k
k
n=2
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
≥
=
≥
1
2
(a2k−1 + ak+1 − ak−1 ) + (a2k + ak+2 − ak−2 ) + · · ·
k
k
∞
X
k−2
+
(a3k−4 + a2k−2 − a2 ) +
an
k
n=2
k−3
X
j+1
j=0
∞
X
k
(a2k+j−1 + ak+j+1 − ak−j−1 ) +
∞
X
an
n=2
an
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
n=2
with the following condition
k−3
X
j=0
j+1
(a2k+j−1 + ak+j+1 − ak−j−1 ) ≥ 0.
k
Thus, we observe that the function ω(z) defined by (2.3) is analytic in U with
ω(0) = 0, |ω(z)| < 1 (z ∈ U). This completes the proof of the theorem.
Remark 4. Taking k = 2 in Theorem 2.1, we have Theorem B by Silverman
[7].
Example 2.1. Let us define
(2.5)
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37 2
1
1
1 5
f (z) = z −
z − z3 − z4 −
z
1200
18
48
100
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and
(2.6)
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
1
f3 (z) = z − z 3
3
Page 9 of 29
J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
with k = 3 in Theorem 2.1. Since f (z) satisfies
∞
X
nan =
n=2
217
< 1,
600
we have f (z) ∈ T ∗ . Furthermore, f (z) satisfies,
1
1
1
37
1
(a5 + a4 − a2 ) =
+
−
= 0.
3
3 100 48 1200
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Thus, f (z) satisfies the conditions in Theorem 2.1 with k = 3.
If we take λ = 2, then we have
Z 2π
20
1 4
2
2
|f (z)| dθ ≤ 2πr 1 + r < π = 6.9813 . . . .
9
9
0
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
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zk
k
∗
Corollary 2.2. Let f (z) ∈ T , 0 < λ ≤ 2, and fk (z) = z − (k ≥ 2). If f (z)
satisfies the conditions in Theorem 2.1, then, for z = reiθ (0 < r < 1),
Z
(2.7)
2π
|f (z)|λ dθ ≤ 2πrλ 1 +
0
1 2(k−1)
r
k2
λ2
< 2π 1 +
Proof. It follows that
Z
.
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2π
λ
Z
|fk (z)| dθ =
0
1
k2
λ2
Contents
0
2π
λ
z k−1 dθ.
|z| 1 −
k Quit
λ
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
Applying Hölder’s inequality for 0 < λ < 2, we obtain that
Z
0
2π
 λ2
λ ! λ2
k−1
1 − z dθ
k 
λ
Z 2π
2−λ
Z 2π
k−1
2
2
z λ 2−λ
λ

|z| 1 −
dθ ≤
(|z| ) dθ
k 0
0
! λ2
k−1 2
z
dθ
1 −
=
|z| dθ
k
0
0
λ2
2−λ
2λ
1
2
2(k−1)
2π 1 + 2 r
= 2πr 2−λ
k
λ2
1
= 2πrλ 1 + 2 r2(k−1)
k
λ
1 2
.
< 2π 1 + 2
k
Z
2π
2λ
2−λ
2−λ
2
Z
2π
Integral Means for Starlike and
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Shigeyoshi Owa, Mihai Pascu,
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Further, it is clear for λ = 2.
For the generalization of Theorem C by Silverman [7], we have
Theorem 2.3. Let f (z) ∈ T ∗ , λ > 0, and fk (z) = z −
exists an analytic function ω(z) in U given by
zk
k
(k ≥ 2). If there
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∞
k−1 X
ω(z)
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nan z n−1 ,
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
then, for z = reiθ (0 < r < 1),
Z 2π
Z
0
λ
(2.8)
|f (z)| dθ ≤
0
2π
|fk0 (z)|λ dθ.
0
Proof. For f (z) ∈ T ∗ , it is sufficient to show that
(2.9)
1−
∞
X
nan z n−1 ≺ 1 − z k−1 .
n=2
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Let us define the function ω(z) by
(2.10)
1−
∞
X
k−1
nan z n−1 = 1 − ω(z)
,
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
n=2
Title Page
or, by
ω(z)k−1 =
∞
X
nan z n−1 .
n=2
Since f (z) satisfies
∞
X
nan ≤ 1,
n=2
the function ω(z) is analytic in U, ω(0) = 0, and |ω(z)| < 1 (z ∈ U).
Remark 5. If we take k = 2 in Theorem 2.3, then we have Theorem C by
Silverman [7].
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
Using the Hölder inequality for Theorem 2.3, we have
k
Corollary 2.4. Let f (z) ∈ T ∗ , 0 < λ ≤ 2, and fk (z) = z − zk (k ≥ 2). If f (z)
satisfies the conditions in Theorem 2.3, then, for z = reiθ (0 < r < 1),
Z 2π
λ
2+λ
|f 0 (z)|λ dθ ≤ 2π 1 + r2(k−1) 2 < 2 2 π.
0
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
3.
Integral Means for Functions in the Class C
In this section, we discuss the integral means for functions f (z) in the class C.
Theorem 3.1. Let f (z) ∈ C, λ > 0, and fk (z) = z −
satisfies
(3.1)
k−1
X
(k + j)(k − j)
k2
j=2
zk
k2
(k ≥ 2). If f (z)
(a2k−j − aj ) ≥ 0
for k ≥ 3, and if there exists an analytic function ω(z) in U given by
(ω(z))k−1 = k 2
∞
X
an z n−1 ,
n=2
then, for z = reiθ (0 < r < 1),
Z 2π
Z
λ
(3.2)
|f (z)| dθ ≤
0
Shigeyoshi Owa, Mihai Pascu,
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2π
|fk (z)|λ dθ.
Contents
0
Proof. For the proof, we need to show that
(3.3)
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
1−
∞
X
an z n−1 ≺ 1 −
n=2
z k−1
k2
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by Theorem A. Define the function ω(z) by
(3.4)
1−
∞
X
n=2
an z n−1 = 1 −
Quit
1
ω(z)k−1 ,
2
k
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
or by
(3.5)
(ω(z))k−1 = k 2
∞
X
!
an z n−1
.
n=2
Therefore, we have to show that
∞
X
1
an ≤ 2
k
n=2
∞
X
!
n 2 an .
n=2
Using the same technique as in the proof of Theorem 2.1, we see that
! k−1
∞
∞
X (k + j)(k − j)
X
1 X 2
n
a
≥
(a
−
a
)
+
an
n
2k−j
j
2
k 2 n=2
k
n=2
j=2
≥
∞
X
an .
Example 3.1. Consider the functions
f (z) = z −
1 2
1
1
z − z3 − z4
40
18
40
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(3.7)
Shigeyoshi Owa, Mihai Pascu,
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n=2
(3.6)
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
1
f3 (z) = z − z 3
9
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
with k = 3 in Theorem 3.1. Then we have that
∞
X
n 2 an =
n=2
4
9
16
+
+
= 1,
40 18 40
which implies f (z) ∈ C, and that
5
(a4 − a2 ) = 0.
9
Thus f (z) satisfies the conditions of Theorem 3.1. If we make λ = 2, then we
see that
Z 2π
1 4
164
2
2
|f (z)| dθ ≤ 2πr 1 + r <
π = 6.3607 · · · .
81
81
0
k
Corollary 3.2. Let f (z) ∈ C, 0 < λ ≤ 2, and fk (z) = z − zk2 (k ≥ 2). If
f (z) satisfies the condition in Theorem 3.1, then, for k ≥ 3, then, for z = reiθ
(0 < r < 1),
Z
(3.8)
0
2π
1
|f (z)| dθ ≤ 2πr 1 + 4 r2(k−1)
k
λ
1 2
< 2π 1 + 4
.
k
Further, we may have
λ
λ
λ2
Integral Means for Starlike and
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
Theorem 3.3. Let f (z) ∈ C, λ > 0, and fk (z) = z −
satisfies
2k−2
X
(3.9)
zk
k2
(k ≥ 2). If f (z)
j(k − j)aj ≤ 0,
j=2
and if there exists an analytic function ω(z) in U given by
k−1
(ω(z))
=k
∞
X
nan z n−1 ,
n=2
then, for z = reiθ (0 < r < 1),
Z 2π
Z
0
λ
(3.10)
|f (z)| dθ ≤
0
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
2π
|fk0 (z)|λ dθ.
0
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Example 3.2. Take the functions
(3.11)
1
1
1
f (z) = z − z 2 − z 3 − z 4
24
18
48
and
1
f3 (z) = z − z 3
9
(3.12)
with k = 3 in Theorem 3.3. Since
∞
X
n=2
Integral Means for Starlike and
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Coefficients
n 2 an =
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4
9
16
5
+
+
= <1
24 18 48
6
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
and
1
1
−
= 0,
12 12
f (z) satisfies the conditions in Theorem 3.3. If we take λ = 2, then we have
Z 2π
1 4
20
0
2
|f (z)| dθ ≤ 2π 1 + r < π.
9
9
0
2(3 − 2)a2 + 3(3 − 3)a3 + 4(3 − 4)a4 =
k
Corollary 3.4. Let f (z) ∈ C, 0 < λ ≤ 2, and fk (z) = z − zk2 (k ≥ 2). If
f (z) satisfies the condition in Theorem 3.3, then, for k ≥ 2, then, for z = reiθ
(0 < r < 1),
Z
0
2π
1
|f 0 (z)|λ dθ ≤ 2π 1 + r2(k−1)
k
λ2
< 2π 1 +
1
k
λ2
.
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
4.
Applications for the Integrated Functions
For f (z) ∈ T , we define
I0 f (z) = f (z) = z −
∞
X
Z
an z n
n=2
z
∞
1 2 X an n+1
z
f (t)dt = z −
2
n+1
n=2
If (z) = I1 f (z) =
0
∞
X
1
n!
Ik f (z) = I(Ik−1 f (z)) =
z k+1 −
an z n+k (k = 1, 2, 3, . . . ).
(k + 1)!
(n
+
k)!
n=2
∗
Theorem 4.1. Let f (z) ∈ T , λ > 0, and fj (z) = z −
If f (z) satisfies
zj
j
(j = 2, 3, 4, . . . ).
j 2 +j−1
(4.1)
X j2 + j − k
(a2j 2 +2j−k − ak ) ≥ 0
j(j + 1)
k=2
for j = 2, 3, 4, . . . , and if there exists an analytic function ω(z) in U given by
!
∞
X
1
j−1
an z n−1 ,
(ω(z))
= j(j + 1)
n
+
1
n=2
then
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
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Z
(4.2)
0
2π
|If (z)|λ dθ ≤
Z
2π
|Ifj (z)|λ dθ.
Page 19 of 29
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
Proof. We have to prove
λ
λ
Z 2π Z 2π ∞
X
2
2
n−1 j−1
dθ.
1 −
an z dθ ≤
z
1 −
n
+
1
j(j
+
1)
0
0
n=2
If
1−
∞
X
n=2
2
2
an z n−1 ≺ 1 −
z j−1 ,
n+1
j(j + 1)
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
then the proof is completed by Theorem A.
Let us define the function ω(z) by
1−
∞
X
n=2
2
2
an z n−1 = 1 −
(ω(z))j−1 .
n+1
j(j + 1)
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
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Then
j−1
|ω(z)|
∞
X
1
= j(j + 1)
an z n−1 n
+
1
n=2
!
∞
X
1
≤ |z| j(j + 1)
an .
n
+
1
n=2
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Thus, we only show that
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j(j + 1)
∞
X
n=2
1
an ≤
n+1
∞
X
nan
Page 20 of 29
n=2
J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
or
∞
X
n=2
∞
X
an ≤
n
n=2
1
1
+
j(j + 1) n + 1
an .
Indeed,
∞
X
n
n=2
1
1
+
an
j(j + 1) n + 1
1
1
1
1
=2
+
a2 + 3
+
a3 + · · ·
j(j + 1) 3
j(j + 1) 4
1
1
1
1
+ (j − 1)
+
aj−1 + j
+
aj
j(j + 1) j
j(j + 1) j + 1
1
1
+ (j + 1)
+
aj+1
j(j + 1) j + 2
1
1
2
+
a2j 2 +2j−3
+ · · · + (2j + 2j − 3)
j(j + 1) 2j 2 + 2j − 2
1
1
2
+
a2j 2 +2j−1 + · · ·
+ (2j + 2j − 2)
j(j + 1) 2j 2 + 2j − 1
j(j + 1) − 2
j(j + 1) − 3
a2 + 1 −
a3 + · · ·
≥ 1−
j(j + 1)
j(j + 1)
j(j + 1) − j
j(j + 1) − (j − 1)
aj−1 + 1 −
aj
+ 1−
j(j + 1)
j(j + 1)
j(j + 1) − (j + 1)
+ 1−
aj+1
j(j + 1)
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
j(j + 1) − (2j 2 + 2j − 3)
+ ··· + 1 −
a2j 2 +2j−3
j(j + 1)
j(j + 1) − (2j 2 + 2j − 2)
+ 1−
a2j 2 +2j−2 + · · ·
j(j + 1)
j2 + j − 2
j2 + j − 3
=
(a2j 2 +2j−2 − a2 ) +
(a2j 2 +2j−3 − a3 )
j(j + 1)
j(j + 1)
j2 + 1
j2
+ ··· +
(a2j 2 +j+1 − aj−1 ) +
(a2j 2 +j − aj )
j(j + 1)
j(j + 1)
j2 − 1
+
(a2j 2 +j−1 − aj+1 ) + · · · + a2 + a3 + · · · + a2j 2 +2j−2 + · · ·
j(j + 1)
j 2 +j−1
∞
X
X j2 + j − k
=
(a2j 2 +2j−k − ak ) +
an
j(j
+
1)
n=2
k=2
≥
∞
X
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
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an
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n=2
for
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
j 2 +j−1
X j2 + j − k
(a2j 2 +2j−k − ak ) ≥ 0.
j(j
+
1)
k=2
This completes the proof of Theorem 4.1.
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Finally, we derive
Theorem 4.2. Let f (z) ∈ T ∗ ,λ > 0,and fj (z) = z −
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zj
j
(j = 2, 3, 4, . . . ). If
Page 22 of 29
J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
f (z) satisfies
(4.3)
∞
X
6
an ≥
5
n=2
(j+k)!
−1
2(j−1)!
X
n=2
2n(j − 1)!
1−
(j + k)!
an − a (j+k)! −n
(j−1)!
for k = 2, 3, 4, . . . , and if there exists an analytic function ω(z) in U given by
∞
n!
(j + k)! X
=
an z n−1 ,
(j − 1)! n=2 (n + k)!
j−1
(ω(z))
then
Z
2π
λ
Z
2π
λ
|Ik f (z)| dθ ≤
(4.4)
0
|Ik fj (z)| dθ.
0
Proof. We have to show that
1−
∞
X
n!(k + 1)!
n=2
(n + k)!
1−
n=2
(n + k)!
an z n−1 ≺ 1 −
an z n−1 = 1 −
or by
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
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(j − 1)!(k + 1)! j−1
z .
(j + k)!
Define ω(z) by
∞
X
n!(k + 1)!
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
(j − 1)!(k + 1)!
(ω(z))j−1
(j + k)!
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(ω(z))j−1 =
∞
(j + k)! X
(j − 1)!
n=2
n!
an z n−1 .
(n + k)!
Page 23 of 29
J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
Then we have to show that
∞
∞
X
(j + k)! X
n!
an ≤
nan ,
(j − 1)! n=2 (n + k)!
n=2
that is, that
∞
X
n=2
Since
∞
X
n=2
∞
n!
(j − 1)! X
an ≤
nan .
(n + k)!
(j + k)! n=2
∞
X
n!
1
an =
an
(n + k)!
(n + 1)(n + 2) · · · (n + k)
n=2
∞ X
1
1
1
1
=
−
−
· · · an
n
+
1
n
+
2
n
+
3
n
+
4
n=2
[ k2 ]
∞ X
1
1
−
an
≤
n+1 n+2
n=2
∞ X
1
1
≤
−
an ,
n
+
1
n
+
2
n=2
we obtain
∞
∞ X
1
(j − 1)! X
1
−
an ≤
nan .
n+1 n+2
(j + k)! n=2
n=2
Furthermore, we have
∞
∞ X
X
2n(j − 1)!
2n
n
an ≤
+
−
an .
(j + k)!
n+1 n+2
n=2
n=2
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
Let the function h(n) be given by
h(n) =
n
2
2n
−
=1− 2
.
n+1 n+2
n + 3n + 2
Since h(n) is increasing for n ≥ 2,
5
h(n) ≥ .
6
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Thus, we only show that
∞
X
n=2
In fact,
∞ X
n=2
an ≤
∞ X
11
n=2
(j + k)! − 2n(j − 1)!
−
6
(j + k)!
an .
11 (j + k)! − 2n(j − 1)!
−
an
6
(j + k)!
11 (j + k)! − 4(j − 1)!
=
−
a2
6
(j + k)!
11 (j + k)! − 6(j − 1)!
−
a3 + · · ·
+
6
(j + k)!
11 4(j − 1)!
11 2(j − 1)!
+
−
a (j+k)! −2 +
−
a (j+k)! −1
2(j−1)!
2(j−1)!
6
(j + k)!
6
(j + k)!
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
11 2(j − 1)!
11
+
− 0 a (j+k)! +
−
a (j+k)! +1
2(j−1)!
2(j−1)!
6
6
(j + k)!
11 4(j − 1)!
+
+
a (j+k)! +2 + · · ·
2(j−1)!
6
(j + k)!
11 (j + k)! − 6(j − 1)!
+
a (j+k)! −3
+
(j−1)!
6
(j + k)!
11 (j + k)! − 4(j − 1)!
+
+
a (j+k)! −2 + · · ·
(j−1)!
6
(j + k)!
∞
11 X
(j + k)! − 4(j − 1)!
≥
an +
a (j+k)! −2 − a2
(j−1)!
6 n=2
(j + k)!
(j + k)! − 6(j − 1)!
+
a (j+k)! −3 − a3
(j−1)!
(j + k)!
4(j − 1)!
+
a (j+k)! +2 − a (j+k)! −2
2(j−1)!
2(j−1)!
(j + k)!
2(j − 1)!
+
a (j+k)! +1 − a (j+k)! −1
2(j−1)!
2(j−1)!
(j + k)!
∞
∞
X
5X
(j + k)! − 4(j − 1)!
an +
an +
=
a (j+k)! −2 − a2
2(j−1)!
6
(j
+
k)!
n=2
n=2
(j + k)! − 6(j − 1)!
+
a (j+k)! −3 − a3 + · · ·
2(j−1)!
(j + k)!
(j + k)! − {(j + k)! − 4(j − 1)!}
a (j+k)! +2 − a (j+k)! −2
+
2(j−1)!
2(j−1)!
(j + k)!
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Shigeyoshi Owa, Mihai Pascu,
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
(j + k)! − {(j + k)! − 2(j − 1)!}
+
(j + k)!
∞
X
∞
5X
=
an +
an +
6 n=2
n=2
≥
∞
X
(j+k)!
−1
2(j−1)!
X
n=2
a (j+k)! +1 − a (j+k)! −1
2(j−1)!
2(j−1)!
(j + k)! − 2n(j − 1)!
(j + k)!
a (j+k)! −n − an
(j−1)!
an
n=2
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
for
∞
X
6
an ≥
5
n=2
(j+k)!
−1
2(j−1)!
X
n=2
2n(j − 1)!
1−
(j + k)!
an − a (j+k)! −n .
(j−1)!
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
This completes the proof of Theorem 4.2.
Remark 6. Letting k = 2, if f (z) satisfies,
(4.5)
∞
X
n=2
an ≥
6
5
j(j+1)(j+2)
−1
2
X
n=2
Contents
1−
2n
j(j + 1)(j + 2)
for j = 2, 3, 4, . . . , then
Z 2π
Z
λ
(4.6)
|I2 f (z)| dθ ≤
0
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0
(an − aj(j+1)(j+2)−n )
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2π
|I2 fj (z)|λ dθ.
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J. Ineq. Pure and Appl. Math. 6(2) Art. 50, 2005
Remark 7. Letting k = 3, if f (z) satisfies,
(4.7)
∞
X
6
an ≥
5
n=2
j(j+1)(j+2)(j+3)
−1
2
X
n=2
2n
1−
j(j + 1)(j + 2)(j + 3)
× (an − aj(j+1)(j+2)(j+3)−n )
for j = 2, 3, 4, . . . , then
Z 2π
Z
λ
(4.8)
|I3 f (z)| dθ ≤
0
2π
λ
|I3 fj (z)| dθ.
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
0
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Daisuke Yagi and
Junichi Nishiwaki
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References
[1] P.L. DUREN, Univalent Functions, Springer-Verlag, New York, 1983.
[2] J.E. LITTLEWOOD, On inequalities in the theory of functions, Proc. London Math. Soc., 23 (1925), 481–519.
[3] S. OWA AND T. SEKINE, Integral means of analytic functions, J. Math.
Anal. Appl. (in press).
[4] T. SEKINE, K. TSURUMI AND H.M. SRIVASTAVA, Integral means for
generalized subclasses of analytic functions, Sci. Math. Jpon., 54 (2001),
489–501.
[5] T. SEKINE, K. TSURUMI, S. OWA AND H.M. SRIVASTAVA, Integral
means inequalities for fractional derivatives of some general subclasses of
analytic functions, J. Inequal. Pure Appl. Math., 3 (2002), Art. 66.
[6] H. SILVERMAN, Univalent functions with negative coefficients, Proc.
Amer. Math. Soc., 51 (1975), 109–116.
[7] H. SILVERMAN, Integral means for univalent functions with negative coefficients, Houston J. Math., 23 (1997), 169–174.
Integral Means for Starlike and
Convex Functions with Negative
Coefficients
Shigeyoshi Owa, Mihai Pascu,
Daisuke Yagi and
Junichi Nishiwaki
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