CERTAIN CLASSES OF ANALYTIC FUNCTIONS
INVOLVING SĂLĂGEAN OPERATOR
SEVTAP SÜMER EKER
SHIGEYOSHI OWA
Department of Mathematics
Faculty of Science and Letters
Dicle University
21280 - Diyarbakır, Turkey
EMail: sevtaps@dicle.edu.tr
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577 - 8502
Japan
EMail: owa@math.kindai.ac.jp
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
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27 December, 2008
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Communicated by:
G. Kohr
Page 1 of 25
2000 AMS Sub. Class.:
26D15
Key words:
Sălăgean operator, coefficient inequalities, distortion inequalities, extreme
points, integral means, fractional derivative.
Received:
21 March, 2007
Accepted:
Abstract:
Using Sălăgean differential operator, we study new subclasses of analytic functions. Coefficient inequalities and distortion theorems and extreme points of
these classes are studied. Furthermore, integral means inequalities are obtained
for the fractional derivatives of these classes.
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Contents
1
Introduction and Definitions
3
2
Coefficient Inequalities for classes Nm,n (α, β) and Msm,n (α, β)
5
3
Relation for]
Nm,n (α, β) and ]
Msm,n (α, β)
9
4
Distortion Inequalities
10
5
Extreme Points
15
6
Integral Means Inequalities
18
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
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1.
Introduction and Definitions
Let A denote the class of functions f (z) of the form
f (z) = z +
(1.1)
∞
X
aj z j
j=2
which are analytic in the open disc U = {z : |z| < 1}. Let S be the subclass of A
consisting of analytic and univalent functions f (z) in U. We denote by S ∗ (α) and
K(α) the class of starlike functions of order α and the class of convex functions of
order α, respectively, that is,
0 zf (z)
∗
S (α) = f ∈ A : Re
> α, 0 5 α < 1, z ∈ U
f (z)
and
zf 00 (z)
> α, 0 5 α < 1, z ∈ U .
K(α) = f ∈ A : Re 1 + 0
f (z)
For f (z) ∈ A, Sălăgean [1] introduced the following operator which is called the
Sălăgean operator:
Dn f (z) = D(Dn−1 f (z))
(n ∈ N = 1, 2, 3, ...).
We note that,
j=2
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D1 f (z) = Df (z) = zf 0 (z)
∞
X
vol. 10, iss. 1, art. 22, 2009
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D0 f (z) = f (z)
Dn f (z) = z +
Analytic Functions Involving
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Sevtap Sümer Eker and
Shigeyoshi Owa
j n aj z j
(n ∈ N0 = N ∪ {0}).
Close
Let Nm,n (α, β) denote the subclass of A consisting of functions f (z) which satisfy
the inequality
m
m
D f (z)
D f (z)
+α
Re
>
β
−
1
Dn f (z)
Dn f (z)
for some 0 5 α < 1, β ≥ 0, m ∈ N, n ∈ N0 and all z ∈ U. Also let Msm,n (α, β)
(s = 0, 1, 2, . . . ) be the subclass of A consisting of functions f (z) which satisfy the
condition:
f (z) ∈ Msm,n (α, β) ⇔ Ds f (z) ∈ Nm,n (α, β).
It is easy to see that if s = 0, then M0m,n (α, β) ≡ Nm,n (α, β). Furthermore,
special cases of our classes are the following:
(i) N1,0 (α, 0) = S ∗ (α) and N2,1 (α, 0) = K(α) which were studied by Silverman
[2].
(ii) N1,0 (α, β) = SD(α, β) and
Shams at all [3].
M11,0 (α, β)
= KD(α, β) which were studied by
(iii) Nm,n (α, 0) = Km,n (α) and Msm,n (α, 0) = Msm,n (α) which were studied by
Eker and Owa [4].
Therefore, our present paper is a generalization of these papers. In view of the
coefficient inequalities for f (z) to be in the classes Nm,n (α, β) and Msm,n (α, β), we
em,n (α, β) and M
fsm,n (α, β). Some distortion inequalities
introduce two subclasses N
for f (z) and some integral means inequalities for fractional calculus of f (z) in the
above classes are discussed in this paper.
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
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2.
Coefficient Inequalities for classes Nm,n (α, β) and Msm,n (α, β)
Theorem 2.1. If f (z) ∈ A satisfies
∞
X
(2.1)
Ψ(m, n, j, α, β) |aj | 5 2(1 − α)
j=2
where
(2.2)
Ψ(m, n, j, α, β) = |j m − j n − αj n | + (j m + j n − αj n ) + 2β |j m − j n |
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
for some α(0 5 α < 1), β ≥ 0, m ∈ N and n ∈ N0 ,then f (z) ∈ Nm,n (α, β).
Proof. Suppose that (2.1) is true for α(0 5 α < 1), β ≥ 0, m ∈ N , n ∈ N0 . For
f (z) ∈ A, let us define the function F (z) by
m
D f (z)
Dm f (z)
− β n
− 1 − α.
F (z) = n
D f (z)
D f (z)
It suffices to show that
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F (z) − 1 F (z) + 1 < 1
(z ∈ U).
We note that
F (z) − 1 F (z) + 1 m
D f (z) − βeiθ |Dm f (z) − Dn f (z)| − αDn f (z) − Dn f (z) = m
D f (z) − βeiθ |Dm f (z) − Dn f (z)| − αDn f (z) + Dn f (z) Go Back
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−α + P∞ (j m − j n − αj n )aj z j−1 − βeiθ | P∞ (j m − j n )aj z j−1 | j=2
j=2
P
P
=
∞
m + j n − αj n )a z j−1 − βeiθ |
m − j n )a z j−1 | (2 − α) + ∞
(j
(j
j
j
j=2
j=2
P
P
m
n
n
j−1
m
n
j−1
α+ ∞
+ β|e|iθ ∞
j=2 |j − j − αj | |aj ||z|
j=2 |j − j ||aj ||z|
P∞ m
P
5
m
n
n
j−1 − β|e|iθ
n
j−1
(2 − α) − ∞
j=2 (j + j − αj ) |aj ||z|
j=2 |j − j ||aj ||z|
P∞ m
P
m
n
α + j=2 |j − j n − αj n | |aj | + β ∞
j=2 |j − j ||aj |
P∞
P
5
.
m
n
(2 − α) − j=2 (j m + j n − αj n ) |aj | − β ∞
j=2 |j − j ||aj |
The last expression is bounded above by 1, if
α+
∞
X
|j m − j n − αj n | |aj | + β
j=2
5 (2 − α) −
∞
X
j=2
∞
X
vol. 10, iss. 1, art. 22, 2009
|j m − j n ||aj |
m
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
n
n
(j + j − αj ) |aj | − β
j=2
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∞
X
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|j m − j n ||aj |
j=2
which is equivalent to our condition (2.1). This completes the proof of our theorem.
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By using Theorem 2.1, we have:
Theorem 2.2. If f (z) ∈ A satisfies
∞
X
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s
j Ψ(m, n, j, α, β) |aj | 5 2(1 − α),
j=2
where Ψ(m, n, j, α, β) is defined by (2.2) for some α(0 5 α < 1), β ≥ 0, m ∈ N
and n ∈ N0 , then f (z) ∈ Msm,n (α, β).
Proof. From
f (z) ∈ Msm,n (α, β) ⇔ Ds f (z) ∈ Nm,n (α, β),
replacing aj by j s aj in Theorem 2.1, we have the theorem.
Example 2.1. The function f (z) given by
f (z) = z +
∞
X
j=2
∞
X
2(2 + δ)(1 − α)j
zj = z +
Aj z j
(j + δ)(j + 1 + δ)Ψ(m, n, j, α, β)
j=2
with
2(2 + δ)(1 − α)j
(j + δ)(j + 1 + δ)Ψ(m, n, j, α, β)
belongs to the class Nm,n (α, β) for δ > −2, 0 5 α < 1, β ≥ 0, j ∈ C and |j | = 1.
Because, we know that
∞
∞
X
X
2(2 + δ)(1 − α)
Ψ(m, n, j, α, β) |Aj | 5
(j + δ)(j + 1 + δ)
j=2
j=2
∞
∞ X
X
1
1
=
2(2 + δ)(1 − α)
−
j+δ j+1+δ
j=2
j=2
Aj =
j=2
∞
X
2(2 + δ)(1 − α)j
j
z
=
z
+
Bj z j
j s (j + δ)(j + 1 + δ)Ψ(m, n, j, α, β)
j=2
with
Bj =
j s (j
2(2 + δ)(1 − α)j
+ δ)(j + 1 + δ)Ψ(m, n, j, α, β)
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Example 2.2. The function f (z) given by
f (z) = z +
vol. 10, iss. 1, art. 22, 2009
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= 2(1 − α).
∞
X
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
Close
belongs to the class Msm,n (α, β) for δ > −2, 0 5 α < 1, β ≥ 0, j ∈ C and |j | = 1.
Because, the function f (z) gives us that
∞
X
∞
X
2(2 + δ)(1 − α)
j Ψ(m, n, j, α, β) |Bj | 5
= 2(1 − α).
(j + δ)(j + 1 + δ)
j=2
j=2
s
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
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vol. 10, iss. 1, art. 22, 2009
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em,n (α, β) and M
fsm,n (α, β)
Relation for N
3.
In view of Theorem 2.1 and Theorem 2.2, we now introduce the subclasses
em,n (α, β) ⊂ Nm,n (α, β) and M
fs (α, β) ⊂ Ms (α, β)
N
m,n
m,n
which consist of functions
f (z) = z +
(3.1)
∞
X
aj z j
(aj ≥ 0)
j=2
whose Taylor-Maclaurin coefficients satisfy the inequalities (2.1) and (2.2), respecem,n (α, β) and M
fs (α, β),
tively. By the coefficient inequalities for the classes N
m,n
we see:
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
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Theorem 3.1.
em,n (α, β2 ) ⊂ N
em,n (α, β1 )
N
for some β1 and β2 , 0 5 β1 5 β2 .
Proof. For 0 5 β1 5 β2 we obtain
∞
∞
X
X
Ψ(m, n, j, α, β1 )aj 5
Ψ(m, n, j, α, β2 )aj .
j=2
j=2
em,n (α, β2 ), then f (z) ∈ N
em,n (α, β1 ). Hence we get the
Therefore, if f (z) ∈ N
required result.
By using Theorem 3.1, we also have
Corollary 3.2.
fsm,n (α, β2 ) ⊂ M
fsm,n (α, β1 )
M
for some β1 and β2 , 0 5 β1 5 β2 .
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4.
Distortion Inequalities
em,n (α, β), then we have
Lemma 4.1. If f (z) ∈ N
P
∞
X
2(1 − α) − pj=2 Ψ(m, n, j, α, β)aj
aj 5
.
Ψ(m, n, p + 1, α, β)
j=p+1
Analytic Functions Involving
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Sevtap Sümer Eker and
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Proof. In view of Theorem 2.1, we can write
(4.1)
∞
X
Ψ(m, n, j, α, β)aj 5 2(1 − α) −
j=p+1
p
X
Ψ(m, n, j, α, β)aj .
j=2
Clearly Ψ(m, n, j, α, β) is an increasing function for j. Then from (2.2) and (4.1),
we have
Ψ(m, n, p + 1, α, β)
∞
X
aj 5 2(1 − α) −
X
Ψ(m, n, j, α, β)aj .
j=2
Thus, we obtain
j=p+1
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p
j=p+1
∞
X
vol. 10, iss. 1, art. 22, 2009
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Pp
aj 5
2(1 − α) − j=2 Ψ(m, n, j, α, β)aj
= Aj .
Ψ(m, n, p + 1, α, β)
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em,n (α, β), then
Lemma 4.2. If f (z) ∈ N
P
∞
X
2(1 − α) − pj=2 Ψ(m, n, j, α, β)aj
jaj 5
= Bj .
Ψ(m
−
1,
n
−
1,
p
+
1,
α,
β)
j=p+1
s
fm,n
Corollary 4.3. If f (z) ∈ M
(α), then
P
∞
X
2(1 − α) − pj=2 j s Ψ(m, n, j, α, β)aj
= Cj
aj 5
(p + 1)s Ψ(m, n, p + 1, α, β)
j=p+1
and
P
2(1 − α) − pj=2 j s Ψ(m, n, j, α, β)aj
= Dj .
jaj 5
(p + 1)s Ψ(m − 1, n − 1, p + 1, α, β)
j=p+1
∞
X
em,n (α, β). Then for |z| = r < 1
Theorem 4.4. Let f (z) ∈ N
r−
p
X
aj |z|j − Aj rp+1 5 |f (z)| 5 r +
j=2
and
1−
p
X
p
X
vol. 10, iss. 1, art. 22, 2009
aj |z|j + Aj rp+1
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j=2
jaj |z|j−1 − Bj rp 5 |f 0 (z)| 5 1 +
j=2
p
X
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jaj |z|j + Bj rp
j=2
where Aj and Bj are given by Lemma 4.1 and Lemma 4.2.
Proof. Let f (z) given by (1.1). For |z| = r < 1,using Lemma 4.1, we have
|f (z)| 5 |z| +
p
X
j
aj |z| +
j=2
p
5 |z| +
X
∞
X
5r+
j=2
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j
aj |z|
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j=p+1
j
p+1
aj |z| + |z|
j=2
p
X
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Sevtap Sümer Eker and
Shigeyoshi Owa
aj |z|j + Aj rp+1
∞
X
j=p+1
Close
aj
and
|f (z)| ≥ |z| −
p
X
j=2
p
≥ |z| −
X
∞
X
aj |z|j −
aj |z|j
j=p+1
j
∞
X
p+1
aj |z| − |z|
j=2
p
≥r−
X
aj
j=p+1
aj |z|j − Aj rp+1 .
j=2
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
Furthermore, for |z| = r < 1 using Lemma 4.2, we obtain
|f 0 (z)| 5 1 +
p
X
jaj |z|j−1 +
j=2
p
51+
X
∞
X
X
Contents
j=p+1
j−1
jaj |z|
∞
X
p
+ |z|
j=2
p
51+
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jaj |z|j−1
jaj
j=p+1
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jaj |z|j−1 + Bj rp
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and
|f 0 (z)| ≥ 1 −
p
X
jaj |z|j−1 −
j=2
p
≥1−
X
j=2
∞
X
jaj |z|j−1
j=p+1
j−1
jaj |z|
p
− |z|
∞
X
j=p+1
jaj
Close
≥1−
p
X
jaj |z|j−1 − Bj rp .
j=2
This completes the assertion of Theorem 4.4.
fs (α, β). Then
Theorem 4.5. Let f (z) ∈ M
m,n
r−
p
X
aj |z|j − Cj rp+1 5 |f (z)| 5 r +
j=2
and
1−
p
X
p
X
aj |z|j + Cj rp+1
j=2
j−1
jaj |z|
p
0
− Dj r 5 |f (z)| 5 1 +
j=2
p
X
Analytic Functions Involving
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Sevtap Sümer Eker and
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vol. 10, iss. 1, art. 22, 2009
jaj |z|j + Dj rp
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j=2
where Cj and Dj are given by Corollary 4.3.
Proof. Using a similar method to that in the proof of Theorem 4.4 and making use
Corollary 4.3, we get our result.
Taking p = 1 in Theorem 4.4 and Theorem 4.5, we have:
em,n (α, β). Then for |z| = r < 1
Corollary 4.6. Let f (z) ∈ N
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2(1 − α)
2(1 − α)
r−
r2 5 |f (z)| 5 r +
r2
Ψ(m, n, 2, α, β)
Ψ(m, n, 2, α, β)
and
1−
2(1 − α)
2(1 − α)
r 5 |f 0 (z)| 5 1 +
r.
Ψ(m − 1, n − 1, 2, α, β)
Ψ(m − 1, n − 1, 2, α, β)
Close
fsm,n (α, β). Then for |z| = r < 1
Corollary 4.7. Let f (z) ∈ M
r−
2(1 − α)
2s Ψ(m, n, 2, α, β)
r2 5 |f (z)| 5 r +
2(1 − α)
2s Ψ(m, n, 2, α, β)
r2
and
1−
2(1 − α)
2(1 − α)
r 5 |f 0 (z)| 5 1 + s
r.
s
2 Ψ(m − 1, n − 1, 2, α, β)
2 Ψ(m − 1, n − 1, 2, α, β)
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5.
Extreme Points
The determination of the extreme points of a family F of univalent functions enables
us to solve many extremal problems for F . Now, let us determine extreme points of
em,n (α, β) and M
fsm,n (α, β).
the classes N
Theorem 5.1. Let f1 (z) = z and
fj (z) = z +
2(1 − α)
zj
Ψ(m, n, j, α, β)
(j = 2, 3, ...).
Analytic Functions Involving
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Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
em,n (α, β) if and only if it can
where Ψ(m, n, j, α, β) is defined by (2.2). Then f ∈ N
be expressed in the form
∞
X
λj fj (z),
f (z) =
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j=1
where λj > 0 and
P∞
j=1
λj = 1.
Proof. Suppose that
f (z) =
∞
X
λj fj (z) = z +
j=1
∞
X
j=2
λj
2(1 − α)
zj .
Ψ(m, n, j, α, β)
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Then
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∞
X
2(1 − α)
Ψ(m, n, j, α, β)
λj
Ψ(m,
n,
j,
α,
β)
j=2
=
∞
X
j=2
2(1 − α)λj = 2(1 − α)
∞
X
j=2
λj = 2(1 − α)(1 − λ1 ) < 2(1 − α)
em,n (α, β) from the definition of the class of N
em,n (α, β).
Thus, f (z) ∈ N
em,n (α, β). Since
Conversely, suppose that f ∈ N
aj 5
2(1 − α)
Ψ(m, n, j, α, β)
(j = 2, 3, ...),
we may set
Ψ(m, n, j, α, β)
λj =
aj
2(1 − α)
and
λ1 = 1 −
∞
X
vol. 10, iss. 1, art. 22, 2009
λj .
j=2
Then,
f (z) =
∞
X
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
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λj fj (z).
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This completes the proof of the theorem.
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Corollary 5.2. Let g1 (z) = z and
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j=1
gj (z) = z +
2(1 − α)
zj
s
j Ψ(m, n, j, α, β)
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(j = 2, 3, ...).
fsm,n (α, β) if and only if it can be expressed in the form
Then g ∈ M
g(z) =
∞
X
j=1
where λj > 0 and
P∞
j=1
λj = 1.
λj gj (z),
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em,n (α, β) are the functions f1 (z) = z and
Corollary 5.3. The extreme points of N
fj (z) = z +
2(1 − α)
zj
Ψ(m, n, j, α, β)
(j = 2, 3, ...).
fsm,n (α, β) are given by g1 (z) = z and
Corollary 5.4. The extreme points of M
2(1 − α)
zj
gj (z) = z + s
j Ψ(m, n, j, α, β)
(j = 2, 3, ...).
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Sevtap Sümer Eker and
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vol. 10, iss. 1, art. 22, 2009
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6.
Integral Means Inequalities
We shall use the following definitions for fractional derivatives by Owa [6] (also
Srivastava and Owa [7]).
Definition 6.1. The fractional derivative of order λ is defined, for a function f (z),
by
Z z
1
d
f (ξ)
λ
(6.1)
Dz f (z) =
dξ (0 5 λ < 1),
Γ(1 − λ) dz 0 (z − ξ)λ
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
where the function f (z) is analytic in a simply-connected region of the complex zplane containing the origin, and the multiplicity of (z −ξ)−λ is removed by requiring
log(z − ξ) to be real when (z − ξ) > 0.
Definition 6.2. Under the hypotheses of Definition 6.1, the fractional derivative of
order (p + λ) is defined, for a function f (z), by
Dzp+λ f (z) =
dp λ
D f (z)
dz p z
where 0 5 λ < 1 and p ∈ N0 = N ∪ {0}.
(6.2)
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It readily follows from (6.1) in Definition 6.1 that
Γ(k + 1) k−λ
Dzλ z k =
z
Γ(k − λ + 1)
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(0 5 λ < 1).
Further, we need the concept of subordination between analytic functions and a subordination theorem by Littlewood [5] in our investigation.
Let us consider two functions f (z) and g(z), which are analytic in U. The function f (z) is said to be subordinate to g(z) in U if there exists a function w(z) analytic
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in U with
w(0) = 0 and |w(z)| < 1
(z ∈ U),
such that
(z ∈ U).
f (z) = g(w(z))
We denote this subordination by
f (z) ≺ g(z).
Theorem 6.3 (Littlewood [5]). 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
|f (z)| dθ 5
|g (z)|µ dθ.
0
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
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0
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em,n (α, β) and suppose
Theorem 6.4. Let f (z) ∈ A given by (3.1) be in the class N
that
∞
X
2(1 − α)Γ(k + 1)Γ(3 − λ − p)
(j − p)p+1 aj 5
Ψ(m, n, k, α, β)Γ(k + 1 − λ − p)Γ(2 − p)
j=2
for some 0 5 p 5 2, 0 5 λ < 1 where (j − p)p+1 denotes the Pochhammer symbol
defined by (j − p)p+1 = (j − p)(j − p + 1) · · · j. Also given is the function fk (z) by
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(6.3)
fk (z) = z +
2(1 − α)
zk
Ψ(m, n, k, α, β)
(k ≥ 2).
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If there exists an analytic function w(z) given by
{w(z)}k−1 =
Ψ(m, n, k, α, β)Γ(k + 1 − λ − p)
2(1 − α)Γ(k + 1)
∞
X
Γ(j − p)
×
(j − p)p+1
aj z j−1 ,
Γ(j
+
1
−
λ
−
p)
j=2
then for z = reiθ (0 < r < 1) and µ > 0,
Z 2π
Z
p+λ
Dz f (z)µ dθ 5
0
2π
p+λ
Dz fk (z)µ dθ.
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
0
Proof. By virtue of the fractional derivative formula (6.2) and Definition 6.2, we find
from (1.1) that
(
)
∞
X
z 1−λ−p
Γ(2 − λ − p)Γ(j + 1) j−1
p+λ
Dz f (z) =
1+
aj z
Γ(2 − λ − p)
Γ(j + 1 − λ − p)
j=2
(
)
∞
X
z 1−λ−p
=
1+
Γ(2 − λ − p)(j − p)p+1 Φ(j)aj z j−1
Γ(2 − λ − p)
j=2
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where
Φ(j) =
Γ(j − p)
.
Γ(j + 1 − λ − p)
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Since Φ(j) is a decreasing function of j, we have
0 < Φ(j) 5 Φ(2) =
Γ(2 − p)
Γ(3 − λ − p)
(0 5 λ < 1
; 0 5 p 5 2 5 j).
Similarly, from (6.2), (6.3) and Definition 6.2, we obtain
z 1−λ−p
2(1 − α)Γ(2 − λ − p)Γ(k + 1) k−1
p+λ
Dz fk (z) =
z
1+
.
Γ(2 − λ − p)
Ψ(m, n, k, α, β)Γ(k + 1 − λ − p)
For z = reiθ , 0 < r < 1, we must show that
µ
Z 2π ∞
X
j−1 1
+
Γ(2
−
λ
−
p)(j
−
p)
Φ(j)a
z
dθ
p+1
j
0
j=2
µ
Z 2π 2(1 − α)Γ(2 − λ − p)Γ(k + 1) k−1 5
1 + Ψ(m, n, k, α, β)Γ(k + 1 − λ − p) z dθ (µ > 0).
0
Thus by applying Littlewood’s subordination theorem, it would suffice to show that
(6.4)
1+
∞
X
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2(1 − α)Γ(2 − λ − p)Γ(k + 1) k−1
≺1+
z .
Ψ(m, n, k, α, β)Γ(k + 1 − λ − p)
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By setting
1+
vol. 10, iss. 1, art. 22, 2009
Γ(2 − λ − p)(j − p)p+1 Φ(j)aj z j−1
j=2
∞
X
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
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Γ(2 − λ − p)(j − p)p+1 Φ(j)aj z j−1
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j=2
=1+
2(1 − α)Γ(2 − λ − p)Γ(k + 1)
{w(z)}k−1
Ψ(m, n, k, α, β)Γ(k + 1 − λ − p)
we find that
∞
{w(z)}k−1 =
Ψ(m, n, k, α, β)Γ(k + 1 − λ − p) X
(j − p)p+1 Φ(j)aj z j−1
2(1 − α)Γ(k + 1)
j=2
Close
which readily yields w(0) = 0.
Therefore, we have
∞
Ψ(m, n, k, α, β)Γ(k + 1 − λ − p) X
k−1
j−1 |w(z)|
=
(j − p)p+1 Φ(j)aj z 2(1 − α)Γ(k + 1)
j=2
∞
Ψ(m, n, k, α, β)Γ(k + 1 − λ − p) X
5
(j − p)p+1 Φ(j)aj |z|j−1
2(1 − α)Γ(k + 1)
j=2
∞
X
Ψ(m, n, k, α, β)Γ(k + 1 − λ − p)
Φ(2)
(j − p)p+1 aj
5 |z|
2(1 − α)Γ(k + 1)
j=2
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
∞
Ψ(m, n, k, α, β)Γ(k + 1 − λ − p) Γ(2 − p) X
(j − p)p+1 aj
= |z|
2(1 − α)Γ(k + 1)
Γ(3 − λ − p) j=2
5 |z| < 1
by means of the hypothesis of Theorem 6.4.
For the special case p = 0, Theorem 6.4 readily yields the following result.
em,n (α, β) and suppose
Corollary 6.5. Let f (z) ∈ A given by (3.1) be in the class N
that
∞
X
2(1 − α)Γ(k + 1)Γ(3 − λ)
jaj 5
Ψ(m, n, k, α, β)Γ(k + 1 − λ)
j=2
for 0 5 λ < 1. Also let the function fk (z) be given by (6.3). If there exists an
analytic function w(z) given by
∞
{w(z)}k−1 =
Ψ(m, n, k, α, β)Γ(k + 1 − λ) X Γ(j + 1)
aj z j−1 ,
2(1 − α)Γ(k + 1)
Γ(j
+
1
−
λ)
j=2
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then for z = reiθ and 0 < r < 1,
Z 2π
Z 2π
λ
µ
λ
Dz f (z) dθ 5
Dz fk (z)µ dθ
0
(0 5 λ < 1, µ > 0).
0
fs (α, β) and suppose
Corollary 6.6. Let f (z) ∈ A given by (3.1) be in the class M
m,n
that
∞
X
2(1 − α)Γ(k + 1)Γ(3 − λ − p)
(j − p)p+1 aj 5 s
k Ψ(m, n, k, α, β)Γ(k + 1 − λ − p)Γ(2 − p)
j=2
vol. 10, iss. 1, art. 22, 2009
for some 0 5 p 5 2, 0 5 λ < 1. Also let the function
gk (z) = z +
(6.5)
2(1 − α)
k s Ψ(m, n, k, α, β)
zk ,
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
(k ≥ 2).
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If there exists an analytic function w(z) given by
{w(z)}k−1 =
k s Ψ(m, n, k, α, β)Γ(k + 1 − λ − p)
2(1 − α)Γ(k + 1)
∞
X
Γ(j − p)
×
(j − p)p+1
aj z j−1 ,
Γ(j + 1 − λ − p)
j=2
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iθ
then for z = re (0 < r < 1) and µ > 0,
Z 2π
Z
p+λ
Dz f (z)µ dθ 5
0
Close
2π
p+λ
Dz gk (z)µ dθ.
0
For the special case p = 0, Corollary 6.6 readily yields,
fsm,n (α, β) and suppose
Corollary 6.7. Let f (z) ∈ A given by (3.1) be in the class M
that
∞
X
2(1 − α)Γ(k + 1)Γ(3 − λ)
jaj 5 s
k Ψ(m, n, k, α, β)Γ(k + 1 − λ)
j=2
for 0 5 λ < 1. Also let the function gk (z) be given by (6.5). If there exists an
analytic function w(z) given by
∞
{w(z)}
k−1
k s Ψ(m, n, k, α, β)Γ(k + 1 − λ) X Γ(j + 1)
=
aj z j−1 ,
2(1 − α)Γ(k + 1)
Γ(j
+
1
−
λ)
j=2
then for z = reiθ (0 < r < 1) and µ > 0,
Z 2π
Z
λ
Dz f (z)µ dθ 5
0
0
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
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2π
λ
Dz gk (z)µ dθ.
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References
[1] G.S. SĂLĂGEAN , Subclasses of univalent functions, Complex analysis - Proc.
5th Rom.-Finn. Semin., Bucharest 1981, Part 1, Lect. Notes Math., 1013 (1983),
362–372.
[2] H. SILVERMAN, Univalent functions with negative coefficients, Proc. Amer.
Math. Soc., 51(1) (1975), 109–116.
[3] S. SHAMS, S.R. KULKARNI AND J.M. JAHANGIRI, Classes of uniformly
starlike and convex functions Internat. J. Math. Math. Sci., 2004 (2004), Issue
55, 2959–2961.
[4] S. SÜMER EKER AND S. OWA, New applications of classes of analytic functions involving the Sălăgean Operator, Proceedings of the International Symposium on Complex Function Theory and Applications, Transilvania University of
Braşov Printing House, Braşov, Romania, 2006, 21–34.
[5] J.E. LITTLEWOOD, On inequalities in the theory of functions, Proc. London
Math. Soc., 23 (1925), 481–519.
[6] S. OWA, On the distortion theorems I, Kyungpook Math. J., 18 (1978), 53–59.
[7] H.M. SRIVASTAVA AND S. OWA (Eds.), Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press (Ellis Horwood Limited, Chichester) John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1989
Analytic Functions Involving
Sălăgean Operator
Sevtap Sümer Eker and
Shigeyoshi Owa
vol. 10, iss. 1, art. 22, 2009
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