Journal of Inequalities in Pure and
Applied Mathematics
INTEGRAL MEANS OF MULTIVALENT FUNCTIONS
H. ÖZLEM GÜNEY, S. SÜMER EKER AND SHIGEYOSHI OWA
Department of Mathematics
Faculty of Science and Arts
University of Dicle
21280 - Diyarbakir, Turkey.
EMail: ozlemg@dicle.edu.tr
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577 - 8502
Japan.
EMail: owa@math.kindai.ac.jp
volume 7, issue 1, article 37,
2006.
Received 29 September, 2005;
accepted 03 January, 2006.
Communicated by: N.E. Cho
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
296-05
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Abstract
For analytic functions f(z) and g(z) which satisfy the subordination f(z) ≺
g(z), J. E. Littlewood (Proc. London Math. Soc., 23 (1925), 481–519) has shown
some interesting results for integral means of f(z) and g(z). The object of the
present paper is to derive some applications of integral means by J.E. Littlewood and show interesting examples for our theorems. We also generalize the
results of Owa and Sekine (J. Math. Anal. Appl., 304 (2005), 772–782).
2000 Mathematics Subject Classification: Primary 30C45.
Key words: Integral means, Multivalent function, Subordination, Starlike, Convex.
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
Contents
Title Page
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Integral Means Inequalities for f (z) and g(z) . . . . . . . . . . . . . 6
3
Integral Means Inequalities for f (z) and h(z) . . . . . . . . . . . . . 14
References
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1.
Introduction
Let Ap,n denote the class of functions f (z) of the form
(1.1)
f (z) = z p +
∞
X
ak z k
(p, n ∈ N = {1, 2, 3, . . . })
k=p+n
which are analytic and multivalent in the open unit disc U = {z ∈ C : |z| < 1}.
A function f (z) belonging to Ap,n is called to be multivalently starlike of order
α in U if it satisfies
0 zf (z)
(1.2)
Re
> α (z ∈ U)
f (z)
for some α(0 5 α < p). A function f (z) ∈ Ap,n is said to be multivalently
convex of order α in U if it satisfies
zf 00 (z)
> α (z ∈ U)
(1.3)
Re 1 + 0
f (z)
∗
for some α(0 5 α < p). We denote by Sp,n
(α) and Kp,n (α) the classes of
functions f (z) ∈ Ap,n which are multivalently starlike of order α in U and
multivalently convex of order α in U, respectively. We note that
0
f (z) ∈ Kp,n (α) ⇔
zf (z)
∗
∈ Sp,n
(α).
p
For functions f (z) belonging to the classes
shown the following coefficient inequalities.
∗
Sp,n
(α)
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
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and Kp,n (α), Owa [4] has
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Theorem 1.1. If a function f (z) ∈ Ap,n satisfies
∞
X
(1.4)
(k − α)|ak | 5 p − α
k=p+n
∗
for some α (0 5 α < p), then f (z) ∈ Sp,n
(α).
Theorem 1.2. If a function f (z) ∈ Ap,n satisfies
∞
X
(1.5)
k(k − α)|ak | 5 p − α
k=p+n
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
for some α(0 5 α < p), then f (z) ∈ Kp,n (α).
For analytic functions f (z) and g(z) in U, f (z) is said to be subordinate to
g(z) in U if there exists an analytic function w(z) in U such that w(0) = 0,
|w(z)| < 1 (z ∈ U), and f (z) = g(w(z)). We denote this subordination by
f (z) ≺ g(z) (cf. Duren [2]).
To discuss our problems for integral means of multivalent functions, we have to
recall here the following result due to Littlewood [3].
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Theorem 1.3. If f (z) and g(z) are analytic in U with f (z) ≺ g(z), then for
µ > 0 and z = reiθ (0 < r < 1),
Z 2π
Z 2π
µ
(1.6)
|f (z)| dθ 5
|g(z)|µ dθ.
0
0
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Applying Theorem 1.3 by Littlewood [3], Owa and Sekine [5] have considered some integral means inequalities for certain analytic functions. In the
present paper, we discuss the integral means inequalities for multivalent functions which are the generalization of the paper by Owa and Sekine [5].
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
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2.
Integral Means Inequalities for f (z) and g(z)
In this section, we discuss the integral means inequalities for f (z) ∈ Ap,n and
g(z) defined by
(2.1)
g(z) = z p + bj z j + b2j−p z 2j−p
(j = n + p).
We first derive
Theorem 2.1. Let f (z) ∈ Ap,n and g(z) be given by (2.1). If f (z) satisfies
(2.2)
∞
X
|ak | 5 |b2j−p | − |bj |
(|bj | < |b2j−p |)
k=p+n
and there exists an analytic function w(z) such that
∞
X
2(j−p)
j−p
b2j−p (w(z))
+ bj (w(z)) −
ak z k−p = 0,
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
Title Page
k=p+n
then for µ > 0 and z = reiθ (0 < r < 1),
Z 2π
Z
µ
|f (z)| dθ 5
0
Contents
2π
µ
|g(z)| dθ.
0
Proof. By putting z = reiθ (0 < r < 1), we see that
µ
Z 2π
Z 2π ∞
X
p
k
|f (z)|µ dθ =
z
+
a
z
k dθ
0
0
k=p+n
µ
Z 2π ∞
X
= rpµ
ak z k−p dθ
1 +
0
k=p+n
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and
2π
Z
Z
2π
p
z + bj z j + b2j−p z 2j−p µ dθ
0
Z 2π
pµ
1 + bj z j−p + b2j−p z 2j−2p µ dθ.
=r
µ
|g(z)| dθ =
0
0
Applying Theorem 1.3, we have to show that
∞
X
1+
ak z k−p ≺ 1 + bj z j−p + b2j−p z 2(j−p) .
k=p+n
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
Let us define the function w(z) by
1+
∞
X
ak z k−p = 1 + bj (w(z))j−p + b2j−p (w(z))2(j−p)
k=p+n
(2.3)
b2j−p (w(z))
j−p
+ bj (w(z))
−
∞
X
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or by
2(j−p)
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ak z k−p = 0.
k=p+n
Since, for z = 0,
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n
o
(w(0))j−p b2j−p (w(0))j−p + bj = 0,
there exists an analytic function w(z) in U such that w(0) = 0.
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Next we prove the analytic function w(z) satisfies |w(z)| < 1 (z ∈ U) for
∞
X
|ak | 5 |b2j−p | − |bj |
(|bj | < |b2j−p |) .
k=p+n
By the inequality (2.3), we know that
∞
∞
X
X
2(j−p)
j−p k−p b
(w(z))
+
b
(w(z))
=
a
z
<
|ak |
2j−p
j
k
k=p+n
k=p+n
for z ∈ U, hence
(2.4)
2(j−p)
|b2j−p | |w(z)|
j−p
− |bj | |w(z)|
−
∞
X
|ak | < 0.
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
k=p+n
Letting t = |w(z)|j−p (t = 0) in (2.4), we define the function G(t) by
∞
X
2
G(t) = |b2j−p | t − |bj | t −
Contents
|ak | .
k=p+n
If G(1) = 0, then we have t < 1 for G(t) < 0. Indeed we have
G(1) = |b2j−p | − |bj | −
∞
X
|ak | = 0.
k=p+n
that is,
∞
X
k=p+n
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|ak | 5 |b2j−p | − |bj | .
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Consequently, if the inequality (2.2) holds true, there exists an analytic function
w(z) with w(0) = 0, |w(z)| < 1 (z ∈ U) such that f (z) = g (w(z)). This
completes the proof of Theorem 2.1.
Theorem 2.1 gives us the following corollary.
Corollary 2.2. Let f (z) ∈ Ap,n and g(z) be given by (2.1). If f (z) satisfies the
conditions of Theorem 2.1, then for 0 < µ 5 2 and z = reiθ (0 < r < 1)
Z 2π
µ
|f (z)|µ dθ 5 2πrpµ 1 + |bj |2 r2(j−p) + |b2j−p |2 r4(j−p) 2
0
µ
< 2π 1 + |bj |2 + |b2j−p |2 2 .
Further, we have that f (z) ∈ Hq (U ) for 0 < q 5 2, where Hq denotes the
Hardy space (cf. Duren [1]).
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
Title Page
Proof. Since,
Z 2π
Z
µ
2π
|g(z)| dθ =
0
Contents
µ
|z | 1 + bj z j−p + b2j−p z 2(j−p) dθ,
p µ
0
applying Hölder’s inequality for 0 < µ < 2, we obtain that
Z 2π
|g(z)|µ dθ
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0
Z
5
2π
2
(|z p |µ ) 2−µ dθ
2−µ
Z
2
2π
µ2 µ2
µ
1 + bj z j−p + b2j−p z 2(j−p) dθ
0
0
Z
2pµ
2−µ
= r
0
2−µ
Z
2
2π
dθ
0
2π
1 + bj z j−p + b2j−p z 2(j−p) 2 dθ
µ2
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n
o 2−µ
µ
2pµ
2 = 2πr 2−µ
2π 1 + |bj |2 r2(j−p) + |b2j−p |2 r4(j−p) 2
µ
= 2πrpµ 1 + |bj |2 r2(j−p) + |b2j−p |2 r4(j−p) 2
µ
< 2π 1 + |bj |2 + |b2j−p |2 2 .
Further, it is easy to see that for µ = 2,
Z 2π
|f (z)|2 dθ 5 2πrpµ 1 + |bj |2 r2(j−p) + |b2j−p |2 r4(j−p)
0
< 2π 1 + |bj |2 + |b2j−p |2 .
From the above, we also have that, for 0 < µ 5 2,
Z 2π
µ
1
sup
|f (z)|µ dθ 5 2πrpµ 1 + |bj |2 + |b2j−p |2 2 < ∞
z∈U 2π 0
which observe that f (z) ∈ H2 (U). Noting that Hq ⊂ Hr (0 < r < q < ∞), we
complete the proof of the corollary.
Example 2.1. Let f (z) ∈ Ap,n satisfy the coefficient inequality (1.4) and
n
g(z) = z p +
εz j + δz 2j−p (|ε| = |δ| = 1)
n+p−α
nε
with 0 5 α < p. Note that bj =
and b2j−p = δ.
n+p−α
By virtue of (1.4), we observe that
∞
X
p−α
n
|ak | ≤
=1−
= |b2j−p | − |bj | .
p
+
n
−
α
p
+
n
−
α
k=p+n
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
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Therefore, if there exists the function w(z) satisfying the condition in Theorem
2.1, then f (z) and g(z) satisfy the conditions in Theorem 2.1. Thus we have for
0 < µ 5 2 and z = reiθ (0 < r < 1),
Z
0
2π
) µ2
2
n
r2(j−p) + r4(j−p)
|f (z)|µ dθ 5 2πrpµ 1 +
p+n−α
(
2 ) µ2
n
.
< 2π 2 +
p+n−α
(
Using the same technique as in the proof of Theorem 2.1, we also derive
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
Theorem 2.3. Let f (z) ∈ Ap,n and g(z) be given by (2.1). If f (z) satisfies
(2.5)
∞
X
Title Page
k|ak | 5 (2j − p)|b2j−p | − j|bj |
(|bj | < |b2j−p |)
Contents
k=p+n
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and there exists an analytic function w(z) such that
2(j−p)
(2j − p)b2j−p (w(z))
j−p
+ jbj (w(z))
∞
X
−
k=p+n
then for µ > 0 and z = reiθ (0 < r < 1)
Z 2π
Z
0
µ
|f (z)| dθ 5
0
0
kak z k−p = 0,
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2π
0
µ
|g (z)| dθ.
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Further, with the help of Hölder’s inequality, we have
Corollary 2.4. Let f (z) ∈ Ap,n and g(z) be given by (2.1). If. f (z) satisfies
conditions of Theorem 2.3, then for 0 < µ 5 2 and z = reiθ (0 < r < 1)
Z
2π
µ
|f 0 (z)|µ dθ 5 2πr(p−1)µ p2 + j 2 |bj |2 r2(j−p) + (2j − p)2 |b2j−p |2 r4(j−p) 2
0
µ
< 2π p2 + j 2 |bj |2 + (2j − p)2 |b2j−p |2 2 .
Example 2.2. Let f (z) ∈ Ap,n satisfy the coefficient inequality (1.5) and
δ
n
g(z) = z +
εz j +
z 2j−p
j(n + p − α)
2j − p
p
(|ε| = |δ| = 1)
with 0 5 α < p. Then
bj =
k=p+n
k|ak | 5
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
Title Page
nε
j(n + p − α)
and
b2j−p =
δ
.
2j − p
Since
∞
X
Integral Means of Multivalent
Functions
p−α
n
=1−
= (2j − p)|b2j−p | − j|bj |,
p+n−α
p+n−α
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if there exists the function w(z) satisfying the condition in Theorem 2.3, then
f (z) and g(z) satisfy the conditions in Theorem 2.3. Thus by Corollary 2.4, we
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have for 0 < µ 5 2 and z = reiθ (0 < r < 1),
Z
0
2π
) µ2
2
n
|f 0 (z)|µ dθ 5 2πr(p−1)µ p2 +
r2(j−p) + r4(j−p)
p+n−α
(
2 ) µ2
n
< 2π p2 + 1 +
.
p+n−α
(
Integral Means of Multivalent
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H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
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3.
Integral Means Inequalities for f (z) and h(z)
In this section, we introduce an analytic and multivalent function h(z) defined
by
(3.1)
h(z) = z p + bj z j + b2j−p z 2j−p + b3j−2p z 3j−2p
(j = n + p).
For the above function h(z), we show
Theorem 3.1. Let f (z) ∈ Ap,n and h(z) be given by (3.1). If f (z) satisfies
(3.2)
∞
X
|ak | 5 |b3j−2p | − |b2j−p | − |bj |
(|bj | + |b2j−p | < |b3j−2p |)
k=p+n
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
and there exists an analytic function w(z) such that
Title Page
(3.3)
3(j−p)
b3j−2p (w(z))
2(j−p)
+ b2j−p (w(z))
+ bj (w(z))j−p −
Contents
∞
X
ak z k−p = 0,
k=p+n
then for µ > 0 and z = reiθ (0 < r < 1)
Z 2π
Z
µ
|f (z)| dθ 5
0
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2π
|h(z)|µ dθ.
0
Proof. In the same way as in the proof of Theorem 2.1, we have to show that
there exists an analytic function w(z) with w(0) = 0 and |w(z)| < 1 (z ∈ U)
such that f (z) = h(w(z)). Note that this function w(z) is defined by (3.3).
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Since, for z = 0,
o
n
(w(0))j−p b3j−2p (w(0))2(j−p) + b2j−p (w(0))j−p + bj = 0,
we consider w(z) satisfies w(0) = 0.
On the other hand, we have that
∞
X
|b3j−2p ||w(z)|3(j−p) − |b2j−p ||w(z)|2(j−p) − |bj ||w(z)|j−p −
|ak | < 0.
k=n+p
Putting t = |w(z)|j−p (t = 0), we define the function H(t) by
3
∞
X
2
H(t) = |b3j−2p |t − |b2j−p |t − |bj |t −
|ak |.
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
k=n+p
Title Page
It follows that H(0) 5 0 and
H 0 (t) = 3|b3j−2p |t2 − 2|b2j−p |t − |bj |.
Since the discriminant of H 0 (t) = 0 is greater than 0, if H 0 (1) = 0, then t < 1
for H(t) < 0. Therefore, we need the following inequality:
H(1) = |b3j−2p | − |b2j−p | − |bj | −
∞
X
|ak | = 0
k=p+n
or
∞
X
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|ak | 5 |b3j−2p | − |b2j−p | − |bj |.
Page 15 of 22
k=p+n
This completes the proof of Theorem 3.1.
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Corollary 3.2. Let f (z) ∈ Ap,n and h(z) be given by (3.1). If f (z) satisfies
conditions of Theorem 3.1, then for 0 < µ 5 2 and z = reiθ (0 < r < 1)
Z 2π
µ
|f (z)|µ dθ 5 2πrpµ 1 + |bj |2 r2(j−p) + |b2j−p |2 r4(j−p) + |b3j−2p |2 r6(j−p) 2
0
µ
< 2π 1 + |bj |2 + |b2j−p |2 + |b3j−2p |2 2 .
Proof. Since
Z 2π
Z
µ
|h(z)| dθ =
0
2π
|z p |µ |1 + bj z j−p + b2j−p z 2(j−p) + b3j−2p z 3(j−p) |µ dθ,
Integral Means of Multivalent
Functions
0
applying Hölder’s inequality for 0 < µ < 2, we obtain that
Z 2π
|h(z)|µ dθ
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
0
Z
2π
p µ
(|z | )
5
2
2−µ
Title Page
2−µ
2
dθ
Contents
0
2π
Z
×
|1 + bj z
j−p
+ b2j−p z
2(j−p)
+ b3j−2p z
3(j−p) µ
|
µ2
JJ
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µ2
dθ
0
=
r
2pµ
2−µ
Z
2−µ
Z
2
2π
dθ
0
2π
|1 < +bj z j−p + b2j−p z 2(j−p) + b2j−2p z 3(j−p) |2 dθ
0
o 2−µ
n
µ
2pµ
2 = 2πr 2−µ
2π(1 + |bj |2 r2(j−p) + |b2j−p |2 r4(j−p) + |b3j−2p |2 r6(j−p) ) 2
µ
= 2πrpµ 1 < +|bj |2 r2(j−p) + |b2j−p |2 r4(j−p) + |b3j−2p |2 r6(j−p) 2
µ
< 2π 1 + |bj |2 + |b2j−p |2 + |b3j−2p |2 2 .
µ2
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Further, we have that f (z) ∈ Hq (U) for 0 < q < 2.
We consider the example for Theorem 3.1.
Example 3.1. Let f (z) ∈ Ap,n satisfy the coefficient inequality (1.4) and
h(z) = z p +
nt
n(1 − t) 2j−p
+ σz 3j−2p
εz j +
δz
p+n−α
p+n−α
(|ε| = |δ| = σ| = 1; 0 5 t 5 1)
Integral Means of Multivalent
Functions
with 0 5 α < p. Then
nt
n(1 − t)
bj =
ε, b2j−p =
δ
p+n−α
p+n−α
and
b3j−2p = σ.
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
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In view of (1.4), we see that
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∞
X
p−α
|ak | 5
p+n−α
k=p+n
n(1 − t)
nt
−
p+n−α p+n−α
= |b3j−2p | − |b2j−p | − |bj |.
=1−
Therefore, if there exists the function w(z) satisfying the condition in Theorem
3.1, then f (z) and g(z) satisfy the conditions in Theorem 3.1. Thus applying
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Corollary 3.2, we have for 0 < µ 5 2 and z = reiθ (0 < r < 1),
Z 2π
|f (z)|µ dθ
0
(
2
nt
p+n−α
(
2
< 2π 2 + (2t − 2t + 1)
5 2πrpµ
1+
n(1 − t)
p+n−α
)
2 µ2
.
r2(j−p) +
n
p+n−α
2
) µ2
r4(j−p) + r6(j−p)
Integral Means of Multivalent
Functions
Next, we derive
Theorem 3.3. Let f (z) ∈ Ap,n and h(z) be given by (3.1). If f (z) satisfies
(3.4)
∞
X
k|ak | 5 (3j − 2p)|b3j−2p | − (2j − p)|b2j−p | − j|bj |
k=p+n
(j|bj | + (2j − p)|b2j−p | < (3j − 2p)|b3j−2p |)
and there exists an analytic function w(z) such that
(3j − 2p)b3j−2p (w(z))3(j−p) + (2j − p)b2j−p (w(z))2(j−p)
∞
X
j−p
+ jbj (w(z)) −
kak z k−p = 0,
k=p+n
then for µ > 0 and z = reiθ (0 < r < 1)
Z 2π
Z
0
µ
|f (z)| dθ 5
0
0
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
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2π
0
µ
|h (z)| dθ.
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Corollary 3.4. Let f (z) ∈ Ap,n and h(z) be given by (3.1). If f (z) satisfies
conditions in Theorem 3.3, then for 0 < µ 5 2 and z = reiθ (0 < r < 1)
Z
2π
|f 0 (z)|µ dθ
0
5 2πr(p−1)µ p2 + j 2 |bj |2 r2(j−p) + (2j − p)2 |b2j−p |2 r4(j−p)
µ
+ (3j − 2p)2 |b3j−2p |2 r6(j−p) 2
µ
< 2π p2 + j 2 |bj |2 + (2j − p)2 |b2j−p |2 + (3j − 2p)2 |b3j−2p |2 2 .
Finally, we show
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
Example 3.2. Let f (z) ∈ Ap,n satisfy the coefficient inequality (1.5) and
h(z) = z p +
n(1 − t)
σ
nt
εz j +
δz 2j−p +
z 3j−2p
j(p + n − α)
(2j − p)(p + n − α)
3j − 2p
(|ε| = |δ| = |σ| = 1; 0 5 t 5 1)
with 0 5 α < p. Then
nt
n(1 − t)
ε, b2j−p =
δ
p+n−α
(2j − p)(p + n − α)
σ
=
.
3j − 2p
bj =
b3j−2p
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Since
∞
X
p−α
p+n−α
n
=1−
p+n−α
= (3j − 2p)|b3j−2p | − (2j − p)|b2j−p | − j|bj |,
k|ak | 5
k=p+n
if there exists the function w(z) satisfying the condition in Theorem 3.3, then
f (z) and g(z) satisfy the conditions in Theorem 3.3. Thus by Corollary 3.4, we
have for 0 < µ 5 2 and z = reiθ (0 < r < 1),
Z 2π
|f 0 (z)|µ dθ
0
(
2
nt
n(1 − t)
r2(j−p) +
p+n−α
p+n−α
(
2 ) µ2
n
< 2π p2 + 1 + (2t2 − 2t + 1)
.
p+n−α
5 2πr(p−1)µ
p2 +
2
) µ2
r4(j−p) + r6(j−p)
Remark 1. We have not been able to prove that the analytic function w(z)
satisfying each condition of the theorems in this paper exists. However, if we
consider some special function f (z) in our theorems, then we know that there
is the analytic function w(z) satisfying each condition of our theorems. Thus, if
we prove that such a function w(z) exists for any function f (z) ∈ Ap,n , then we
do not need to give the condition for w(z) in our theorems.
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
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Remark 2. In the above theorems and examples, if we take p = 1 , we obtain
the results by Owa and Sekine [5]. Therefore, the results of our paper are a
generalization of the results in [5].
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
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References
[1] P.L. DUREN, Theory of Hp Space, Academic Press, New York, 1970.
[2] P.L. DUREN, Univalent Functions, Springer-Verlag, New York, 1983.
[3] J.E. LITTLEWOOD, On inequalities in the theory of functions, Proc. London Math. Soc., (2) 23 (1925), 481–519.
[4] S. OWA, On certain classes of p-valent functions with negative coefficients,
Simon Stevin, 59 (1985), 385–402.
[5] S. OWA AND T. SEKINE, Integral means for analytic functions, J. Math.
Anal. Appl., 304 (2005), 772–782.
Integral Means of Multivalent
Functions
H. Özlem Güney, S. Sümer Eker
and Shigeyoshi Owa
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