Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 362312, 21 pages
doi:10.1155/2012/362312
Research Article
Properties of Certain Subclass of Multivalent
Functions with Negative Coefficients
Jingyu Yang1, 2 and Shuhai Li1
1
2
Department of Mathematics, Chifeng University, Inner Mongolia, Chifeng 024000, China
School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, China
Correspondence should be addressed to Shuhai Li, lishms66@sina.com
Received 26 December 2011; Revised 26 March 2012; Accepted 27 March 2012
Academic Editor: Marianna Shubov
Copyright q 2012 J. Yang and S. Li. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Making use of a linear operator, which is defined by the Hadamard product, we introduce and
study a subclass Ya,c
A, B; p, λ, α of the class A p. In this paper, we obtain the coefficient
inequality, distortion theorem, radius of convexity and starlikeness, neighborhood property,
modified convolution properties of this class. Furthermore, an application of fractional calculus
operator is given. The results are presented here would provide extensions of some earlier works.
Several new results are also obtained.
1. Introduction
Let A p denote the class of functions
fz zp −
∞ akp zkp
p ∈ N {1, 2, . . .}
1.1
k1
which are analytic and p-valent in the open unit disc U {z : |z| < 1} on the complex plane
C.
We denote by S∗p A, B and Kp A, B −1 ≤ B < A ≤ 1 the subclass of p-valent starlike
functions and the subclass of p-valent convex functions, respectively, that is, see for details
1, 2
S∗p A, B
zf z 1 Az
≺
fz ∈ A p :
z ∈ U; −1 ≤ B < A ≤ 1 ,
pfz
1 Bz
2
International Journal of Mathematics and Mathematical Sciences
Kp A, B 1 zf z 1 Az
≺
fz ∈ A p : ∈
U;
−1
≤
B
<
A
≤
1
.
z
p pf z
1 Bz
1.2
A function f ∈ A p is said to be analytic starlike of order ξ if it satisfies
Re
zf z
pfz
1.3
>ξ
for some ξ 0 ≤ ξ < 1 and for all z ∈ U.
Further, a function f ∈ A p is said to be analytic convex of order ξ if it satisfies
Re
1 zf z
p pf z
1.4
>ξ
for some ξ0 ≤ ξ < 1 and for all z ∈ U.
kp
and gz zp For fz zp ∞
k1 akp z
or convolution is defined by
∞
k1
bkp zkp , the Hadamard product
∞
f ∗ g z zp akp bkp zkp g ∗ f z.
1.5
k1
The linear operator Lp a, c is defined as follows see Saitoh 3:
Lp a, cfz ϕp a, c; z ∗ fz
fz ∈ A p ,
1.6
and ϕp a, c; z is defined by
ϕp a, c; z zp ∞
ak
ck
k1
zkp
a ∈ R; c \ z−0 ; z−0 : {0, −1, −2, . . .} ,
1.7
where vk is the pochharmmer symbol defined in terms of the Gamma function by
1,
n 0,
Γν n
νn Γν
νν 1 · · · ν n − 1, n ∈ N.
1.8
The operator Lp a, c was studied recently by Srivastava and Patel 4. It is easily
verified from 1.6 and 1.7 that
z Lp a, cfz aLp a 1, cfz − a − p Lp a, cfz.
1.9
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3
Moreover, for fz ∈ Ap,
zf z
,
Lp p 1, p fz p
Lp n p, 1 fz Dnp−1 fz n > −p ,
Lp a, afz fz,
1.10
where Dnp−1 fz denotes the Ruscheweyh derivative of a function fz ∈ Az of order
n p − 1 see 5.
A, B; p, λ by
Aouf, Silverman and Srivastava 6 introduced the class of Pa,c
use of linear operator Lp a, c, further investigated the properties of this class. In 7,
SokóŁ investigated several properties of the linear Aouf-Silverman-Srivastava operator and
furthermore obtained the corresponding characterizations of multivalent analytic functions
which were studied by Aouf et al. in paper 6. The properties of multivalent functions
with negative coefficients were studied in 8–10. References 11, 12 gave the results of the
univalent function with negative coefficients.
In this paper, we will use operator Lp a, c to define a new subclass of A p as follows.
For p ∈ N, a > 0, c > 0, α ≥ 0 and for the parameters λ, A, and B such that
−1 ≤ B < A ≤ 1,
−1 ≤ B < 0,
0 ≤ λ < p,
1.11
A, B; p, λ, α if it satisfies the following
we say that a function fz ∈ A p is in the class Ya,c
subordination:
1
p−λ
Lp a, cfz
zp−1
Lp a, cfz
− α
−
1
−λ
p−1
pz
≺
1 Az
1 Bz
z ∈ U,
1.12
or equivalently, if the following inequality holds true:
Lp a, cfz /zp−1 − α Lp a, cfz /pzp−1 − 1 − p
< 1.
B L a, cfz /zp−1 − α L a, cfz /pzp−1 − 1 − pB A − B p − λ p
p
1.13
From the above definition, we can imply that the function class Pa,c
A, B; p, λ in
6 is the special case of Ya,c A, B; p, λ, α in our present paper because Ya,c A, B; p, λ, 0 A, B; p, λ.
Pa,c
A, B; p, λ, 0 Pa,c
A, B; p, λ, then, like 6, we have the following
Since Ya,c
subclasses which were studied in many earlier works:
−1, 1; p, λ, 0 Qnp−1 λ 0 ≤ λ < 1; n > −p; p ∈ N Aouf and Darwish
1 Ynp,1
8,
2 Ya,a
A, B; p, 0, 0 P ∗ p, A, B Shukla and Dashrath 9,
3 Ya,a
1, −1; p, λ, 0 Fp 1, λ 0 ≤ λ < p; p ∈ N Lee et al. 10,
4 Ya,a
−β, β; 1, λ, 0 P ∗ λ, β 0 ≤ λ < 1; 0 < β ≤ 1 Gupta and Jain 11,
4
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−1, 1; 1, λ, 0 Qn λ n ∈ N0 : N ∪ {0}; 0 ≤ λ < 1 Uralegaddi and
5 Ynp,1
Sarangi 12.
A, B; p, λ, α. We
The purpose of this paper is to give various properties of class Ya,c
extend the results of basic paper 6.
2. Necessary and Sufficient Condition for fz ∈ Ya,c
A, B; p, λ, α
Theorem 2.1. Let the function fz ∈ A p be given by 1.1, then fz ∈ Ya,c
A, B; p, λ, α if
and only if
∞
φk p, α; B, a, c akp ≤ A − B,
2.1
k1
where
p α 1 − B k p ak
φk p, α; B, a, c .
p p − λ ck
Proof. For the sufficient condition, let fz zp −
∞
k1
2.2
|akp |zkp , we have
Lp a, cfz /zp−1 − α Lp a, cfz /pzp−1 − 1 − p
B L a, cfz /zp−1 − α L a, cfz /pzp−1 − pB A − B p − λ p
p
p Lp a, cfz − αeiθ Lp a, cfz − pzp−1 − p2 zp−1
,
B pL a, cfz − αeiθ L a, cfz − pzp−1 − ppB A − Bp − λ zp−1 p
p
2.3
then
p Lp a, cfz − αeiθ Lp a, cfz − pzp−1 − p2 zp−1 − B p Lp a, cfz − αeiθ Lp a, cfz − pzp−1 − p pB A − B p − λ zp−1 ∞
∞
ak kp−1
ak kp−1 iθ − p k p akp
z
− αe −
z
k p akp
k1
k1
ck
ck
∞
∞
ak kp−1
ak kp−1 iθ
− − Bp k p akp z
− αBe −
z
k p akp k1
k1
ck
ck
p−1 −pA − B p − λ z International Journal of Mathematics and Mathematical Sciences
≤
5
∞
∞
a
a
p k p akp k |z|kp−1 αeiθ k p akp k |z|kp−1
ck
ck
k1
k1
|B|
∞
∞
a
a
p k p akp k |z|kp−1 α|B|
k p akp k |z|kp−1
ck
ck
k1
k1
− pA − B p − λ |z|p−1
∞
a
pα
k p 1 − Bakp k − pA − B p − λ
ck
k1
z ∈ ∂U.
2.4
By the maximum modulus theorem, for any z ∈ U, we have
p Lp a, cfz − αeiθ Lp a, cfz − pzp−1 − p2 zp−1 − B p Lp a, cfz − αeiθ Lp a, cfz − pzp−1 − p pB A − B p − λ zp−1 ∞
a
≤ pα
k p 1 − Bakp k − pA − B p − λ
ck
k1
∞
φk p, α; B, a, c akp − A − B
k1
≤ 0,
2.5
this implies fz ∈ Ya,c
A, B; p, λ, α.
A, B; p, λ, α be given by 1.1, then
For necessary condition, let fz ∈ Ya,c
Lp a, cfz
zp−1
p−
∞
k1
a
k p akp k zk ,
ck
2.6
from 1.13 and 2.6, we find that
Lp a, cfz /zp−1 −α Lp a, cfz /pzp−1 −1−p
B L a, cfz /zp−1 −α L a, cfz /pzp−1 −1 − pB A−B p−λ p
p
∞ k
k
− ∞
k1 k p akp ak /ck z − α −
k1 k p /p akp ak /ck z
∞
∞
k
k
−A−B p−λ −B k1 kp akp ak /ckz −αB− k1 kp /p akp ak /ckz <1.
2.7
6
International Journal of Mathematics and Mathematical Sciences
Now, since |Rez| ≤ |z| for all z, we have
∞ k
k
− ∞
k1 k p akp ak /ck z − α −
k1 k p /p akp ak /ck z
Re
∞
∞
−A − B p − λ −B k1 kp akp ak /ck zk−αB− k1 kp /p akp ak /ck zk < 1.
2.8
We choose value of z on the real axis so that the expression Lp a, cfz /zp−1 is real.
Then, we have
∞ k k
− ∞
k1 kp akp ak /ck z −α −
k1 kp /p akp ak /ck z
Re
∞ k
k
−A−B p−λ −B ∞
k1 kp akp ak /ck z −αB − k1 kp /p akp ak /ck z
∞ k k
− ∞
k1 kp akp ak /ck z −α −
k1 kp /p akp ak /ck z
,
∞ k
k
−A−B p−λ −B ∞
k1 kp akp ak /ck z −αB − k1 kp /p akp ak /ck z
2.9
∞
k k
− ∞
k1 k p akp ak /ck z −α − k1 k p /p akp ak /ck z
∞
∞
−A−B p−λ −B k1 kp akp ak /ck zk−αB− k1 kp /p akp ak /ck zk ∞ k
k
− ∞
k1 kp akp ak /ck z − α
k1 k p /p akp ak /ck |z|
−A−B p−λ −B ∞ kp akp a /c zk−αB ∞
akp a /c |z|k kp
/p
k
k
k
k
k1
k1
< 1.
2.10
Letting z → 1− throughout real values in 2.10, we get
∞ − ∞
k1 k p akp ak /ck − α
k1 k p /p akp ak /ck akp a /c − αB ∞
akp a /c −A−B p−λ − B ∞
kp
kp
/p
k
k
k
k
k1
k1
< 1.
2.11
So we have
p k p akp ak /ck α ∞
k1 k p akp ak /ck < 1,
∞
A − B p − λ B ∞
k1 p k p akp ak /ck αB
k1 k p akp ak /ck 2.12
∞
k1
International Journal of Mathematics and Mathematical Sciences
7
it is
∞
k1
a
p α 1 − B k p akp k ≤ pA − B p − λ .
ck
2.13
This completes the proof of the theorem.
Corollary 2.2. Let the function fz ∈ A p be given by 1.1. If fz ∈ Ya,c
A, B; p, λ, α, then
pA − B p − λ ck
akp ≤ p α 1 − B k p ak
k, p ∈ N .
2.14
The result is sharp for the function given by
pA − B p − λ ck
zkp .
fz z −
p
α
k
p
−
B
1
a
k
k1
∞
p
2.15
3. Distortion Inequality of Class Ya,c
A, B; p, λ, α
Theorem 3.1. If a function fz defined by 1.1 is in the class Ya,c
A, B; p, λ, α, then
cpA − B p − λ p!
p!
|z| |z|p−m ≤ f m z
− p − m ! a p α 1 − B p 1 − m !
cpA − B p − λ p!
p!
≤ |z| |z|p−m z ∈ U; a ≥ c > 0; m ∈ N0 ; p > m .
p−m ! a pα 1−B p1−m !
3.1
The result is sharp for the function fz given by
cA − B p − λ 1p
fz z −
z
a1 − B p α
p
Proof. Since fz zp −
∞
k1
p∈N .
3.2
|akp |zkp , then
∞
kp ! p!
zp−m −
akp zkp−m .
p−m !
k1 k p − m !
f m z 3.3
From fz ∈ Ya,c
A, B; p, λ, α, using Theorem 2.1, we have
∞
∞ pα kp
a p α 1 − B 1 p 1 − Bak akp ≤ 1
k p ! akp ≤
cpA − B p − λ p 1 ! k1
pA
−
B
p
−
λ
c
k
k1
3.4
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which readily yields
cpA − B p − λ p!
.
k p ! akp ≤
a p α 1 − B
∞
k1
3.5
By 3.3 and 3.5, we can imply 3.1.
4. Radius of Starlikeness and Convexity
Theorem 4.1. Let the function fz defined by 1.1 be in the class Ya,c
A, B; p, λ, α. Then, fz is
starlike of η 0 ≤ η < 1 in |z| < rη p, k, α, A, B, a, c where
p 1 − η φk p, α; B, a, c
rη p, k, α, A, B, a, c inf
A − B k p 1 − η
1/k
4.1
and φk p, α; B, a, c is defined by 2.1.
Proof. We must show that
zf z
<1−η
−
1
pfz
for |z| < rη p, k, α, A, B, a, c .
4.2
Since
−
zf z
pfz − 1 p −
∞
k k1 k akp z ∞
k
k1 p akp z
≤
p−
∞
k
k1 k akp |z|
k,
∞
k1 p akp |z|
4.3
to prove 4.1, it is sufficient to prove
p−
∞
k
k1 k akp |z|
k
∞
k1 p akp |z|
< 1 − η.
4.4
It is equivalent to
∞ k p 1 − η a
kp
|z|k ≤ 1.
p
1
−
η
k1
4.5
By Theorem 2.1, we have
∞ φ p, α; B, a, c k
akp ≤ 1,
A−B
k1
4.6
International Journal of Mathematics and Mathematical Sciences
9
hence 4.5 will be true if
k p 1 − η akp k φk p, α; B, a, c akp .
|z| ≤
A−B
p 1−η
4.7
It is equivalent to
p 1 − η φk p, α; B, a, c
|z| ≤
A − B k p 1 − η
1/k
4.8
.
This completes the proof.
Theorem 4.2. Let the function fz defined by 1.1 be in the class Ya,c
A, B; p, λ, α. Then, fz is
convex of ξ0 ≤ ξ < 1 in |z| < rξ p, k, α, A, B, a, c where
rξ p, k, α, A, B, a, c inf
p2 1 − ξφk p, α; B, a, c
k p A − B k p1 − ξ
1/k
4.9
and φk p, α; B, a, c is defined by 2.1.
Proof. It sufficient to show that
zf z p − 1 pf z − p < 1 − ξ
for |z| < rξ p, k, α, A, B, a, c .
4.10
Since
zf z p − 1 −
pf z − p p2 −
∞
k k1 k k p akp z
∞
akp zk p
k
p
k1
≤
∞
k
k1 k k p akp |z|
k,
p2 − ∞
k1 p k p akp |z|
4.11
to prove 4.9, it is sufficient to prove
∞
k
k1 k k p akp |z|
k
p2 − ∞
k1 p k p akp |z|
< 1 − ξ.
4.12
|z|k ≤ 1.
4.13
It is equivalent to
∞ k p k p1 − ξ a
kp
k1
p2 1 − ξ
By Theorem 2.1, we have
∞ φ p, α; B, a, c k
akp ≤ 1,
A−B
k1
4.14
10
International Journal of Mathematics and Mathematical Sciences
hence 4.13 will be true if
k p k p1 − ξ akp p2 1 − ξ
φk p, α; B, a, c akp .
|z| ≤
A−B
k
4.15
It is equivalent to
|z| ≤
p2 1 − ξφk p, α; B, a, c
k p A − B k p1 − ξ
1/k
.
4.16
This completes the proof.
5. δ-Neighborhood of Ya,c
A, B; p, λ, α
Based on the earlier works by Aouf et al. 6, Altintas et al. 13–15, and Aouf 16, we
introduce the δ-neighborhood of a function fz ∈ A p of the form 1.1 and present the
relationship between δ-neighborhood and corresponding function class.
Definition 5.1. For δ > 0, a > 0, c > 0, the δ-neighborhood of a function fz ∈ A p is
defined as follows:
Nδ
f g : gz zp −
∞ ∞
bkp zkp ∈ A p ,
tk bkp − akp < δ ,
k1
5.1
k1
where
p α 1 − B k p ak
tk .
pA − B p − λ ck
5.2
Theorem 5.2. Let the function fz defined by 1.1 be in the class Ya1,c
A, B; p, λ, α. Then,
Nδ f ⊂ Ya,c
A, B; p, λ, α
1
.
a1
5.3
This result is the best possible in the sense that δ cannot be increased.
A, B; p, λ, α, then by Theorem 2.1, we have
Proof. Let fz ∈ Ya1,c
∞ pα
1 − B k p a 1k akp ≤ 1
pA − B p − λ ck
k1
5.4
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11
which is equivalent to
∞ pα
1 − B k p ak akp ≤ a .
a1
pA − B p − λ ck
k1
5.5
For any
gz zp −
∞ bkp zkp ∈ N f ,
δ
5.6
k1
we find from 5.1 that
∞ pα
1 − B k p ak bkp − akp ≤ δ.
pA − B p − λ ck
k1
5.7
By 5.5 and 5.7, we get
∞ pα
1 − B k p ak bkp pA − B p − λ ck
k1
∞ pα
∞ pα
1 − B k p ak 1 − B k p ak akp bkp − akp ≤
pA − B p − λ ck
pA − B p − λ ck
k1
k1
≤
a
δ 1.
a1
5.8
A, B; p, λ, α.
By Theorem 2.1, it implies that gz ∈ Ya,c
To show the sharpness of the assertion of Theorem 5.2, we consider the functions fz
and gz given by
cpA − B p − λ
fz z −
A, B; p, λ, α ,
zp1 ∈ Ya1,c
a 1 p α 1 − B p 1
cpA − B p − λ δ cpA − B p − λ
p
gz z −
a p α 1 − B 1 p
a 1 p α 1 − B p 1
p
where δ > δ 1/a 1.
5.9
p1
z
,
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Since
p α p 1 1 − Ba1 b1p − a1p pA − B p − λ c1
cpA − B p − λ δ
p α p 1 1 − Ba1
· pA − B p − λ c1
a p α 1 − B 1 p
5.10
δ ,
so gz ∈ Nδ f.
But
cpA − B p − λ δ
cpA − B p − λ
p α p 1 1 − Ba1
pA − B p − λ c1
a 1 p α p 1 1 − B a p α 1 − B 1 p
5.11
a
δ
a1
> 1,
A, B; p, λ, α.
by Theorem 2.1 gz∈Ya,c
6. Properties Associated with Modified Hadamard Product
Following early works by Aouf et al. in 6, we provide the properties of modified Hadamard
A, B; p, λ, α.
product of Ya,c
For the function
fj z zp −
∞ akp,j zkp
j 1, 2; p ∈ N ,
6.1
k1
the modified Hadamard product of the functions f1 z and f2 z was denoted by f1 · f2 z
and defined as follows:
∞ f1 · f2 z zp − akp,1 akp,2 zkp f2 · f1 z.
6.2
k1
Theorem 6.1. Let fj z j 1, 2 given by 6.1 be in the class Ya,c
A, B; p, λ, α, then
f1 · f2 z ∈ Ya,c
A, B; p, γ, α ,
6.3
2
cpA − B p − λ
γ : p − .
a p α 1 − B 1 p
6.4
where
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13
The result is sharp for the functions fj z j 1, 2 given by
cpA − B p − λ
fj z zp − zp1 .
a p α 1 − B 1 p
6.5
Proof. By Theorem 2.1, we need to find the largest γ such that
∞ pα
1 − B k p ak akp,1 · akp,2 ≤ 1.
pA − B p − γ ck
k1
6.6
Since fj z ∈ Ya,c
A, B; p, λ, α, j 1, 2, then we see that
∞ pα
1 − B k p ak akp,j ≤ 1 j 1, 2 .
pA − B p − λ ck
k1
6.7
By Cauchy-Schwartz inequality, we obtain
∞ pα
1 − B k p ak akp,1 · akp,2 ≤ 1.
pA − B p − λ ck
k1
6.8
This implies that we only need to show that
akp,1 · akp,2 p−γ
≤
akp,1 · akp,2 p−λ
k ∈ N
6.9
or equivalently that
akp,1 · akp,2 ≤ p − γ .
p−λ
6.10
By making use of the inequality 6.8, it is sufficient to prove that
pA − B p − λ ck
p−γ
≤
p
−λ
p α 1 − B k p ak
k ∈ N.
6.11
2
pA − B p − λ ck
γ ≤p− p α 1 − B k p ak
k ∈ N.
6.12
From 6.11, we have
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Define the function Φk by
2
pA − B p − λ ck
Φk : p − p α 1 − B k p ak
k ∈ N.
6.13
We see that Φk is an increasing function of k. Therefore, we conclude that
2
pA − B p − λ c1
.
p α 1 − B k p a1
6.14
γ ≤ Φ1 p − By using arguments similar to those in the proof of Theorem 6.1, we can derive the
following result.
Theorem 6.2. Let f1 z defined by 6.1 be in the class Ya,c
A, B; p, λ, α, f2 z defined by 6.1 be
in the class Ya,c A, B; p, γ, α, then
A, B; p, ξ, α ,
f1 · f2 z ∈ Ya,c
6.15
cpA − B p − λ p − γ
ξ : p − .
a p α 1 − B 1 p
6.16
where
The result is sharp for the functions fj z j 1, 2 given by
cpA − B p − λ
f1 z z − zp1
a p α 1 − B 1 p
cpA − B p − γ
p
f2 z z − zp1
a p α 1 − B 1 p
p
p∈N ,
6.17
p∈N .
A, B; p, λ, α, then the function
Theorem 6.3. Let fj z j 1, 2 defined by 6.1 be in the class Ya,c
hz defined by
hz zp −
∞ akp,1 2 akp,2 2
6.18
k1
belongs to the class Ya,c
A, B; p, χ, α where
2pcA − B p − λ
χ : p − .
a p α 1 − B 1 p
This result is sharp for the functions given by 6.5.
6.19
International Journal of Mathematics and Mathematical Sciences
15
Proof. By Theorem 2.1, we want to find the largest χ such that
∞ pα
1 − B k p ak akp,1 2 akp,2 2 ≤ 1.
pA − B p − χ ck
k1
6.20
Since fj z ∈ Ya,c
A, B; p, λ, α j 1, 2, we readily see that
∞ pα
1 − B k p ak akp,j ≤ 1 j 1, 2 .
pA − B p − χ ck
k1
6.21
From 6.21, we have
2 ∞ pα 2
2
1 − B2 k p
ak 2 ≤1
a
kp,j
2
ck
p2 A − B2 p − χ
k1
j 1, 2 ,
6.22
then we have
2 ∞ pα 2
2 2 1 − B2 k p
ak 2 akp,1 akp,2 ≤ 2.
2
ck
p2 A − B2 p − χ
k1
6.23
From 6.23, if we want to prove 6.20, it is sufficient to prove there exists the largest
χ such that
p α k p 1 − Bak
1
≤
2
p−χ
2pA − B p − λ ck
k ∈ N,
6.24
2
2pA − B p − λ ck
p α k p 1 − Bak
k ∈ N.
6.25
that is
χ≤p− Now we define Ψk by
2
2pA − B p − λ ck
p α k p 1 − Bak
Ψk p − k ∈ N.
6.26
We see that Ψk is an increasing function of k. Therefore, we conclude that
2pcA − Bp − λ2
χ ≤ Ψ1 p − a p α k p 1 − B
which completes the proof of Theorem 6.1.
6.27
16
International Journal of Mathematics and Mathematical Sciences
7. Application of Fractional Calculus Operator
References 17–19 have studied the fractional calculus operators extensively. In this part, we
only investigate the application of fractional calculus operator which was defined by 6 in
A, B; p, λ, α.
the class of Ya,c
Definition 7.1 see 6. The fractional integral of order μ is defined, for a function fz, by
1
−μ
Dz fz Γ μ
z
0
fz
z − ξ
1−μ
μ>0 ,
dξ
7.1
where the function fz is analytic in a simply connected domain of the complex z-plane
containing the origin and the multiplicity of z − ξμ−1 is removed by requiring logz − ξ to
be real when z − ξ > 0.
Definition 7.2 see 6. The fractional integral of order μ is defined, for a function fz, by
μ
Dz fz
1
Γ 1−μ
z
0
fz
dξ
z − ξμ
0≤μ<1 ,
7.2
where the function fz is constrained, the multiplicity of z − ξ−μ is removed as in
Definition 7.1.
In our investigation, we will use the operators Jδ,p defined by cf, 20–22
δp
Jδ,p f z :
zp
z
tδ−1 ftdt
f ∈ A p ; δ > −p; p ∈ N
7.3
0
μ
as well as Dz for which it is well known that see 23
μ
Dz zρ
Γ ρ1
zρ−μ
Γ ρ−μ1
ρ > −1; μ ∈ R
7.4
in terms of Gamma.
Lemma 7.3 see Chen et al. 18. For a function fz ∈ Ap,
∞
Γ p1
δp Γ kp1
p−μ
z
akp zkp−μ ,
Γ p−μ1
k1 δ k p Γ k p − μ 1
δp Γ p1
μ
Dz fz zp−μ
δp−μ Γ p−μ1
∞
δp Γ kp1
akp zkp−μ
k1 δ k p − μ Γ k p − μ 1
μ Dz Jδ,p f z
Jδ,p
μ ∈ R; δ > −p; p ∈ N provided that no zeros appear in the denominators in 7.5.
7.5
International Journal of Mathematics and Mathematical Sciences
17
Remark 7.4. Throughout this section, we assume further that a ≥ c > 0.
A, B; p, λ, α, then
Theorem 7.5. Let function defined by 1.1 be in the class Ya,c
Γ p1
cp δ p A − B p − λ
−μ pμ
1− Jδ,p f z ≥ |z|
|z| ,
Dz
Γ pμ1
a p α p δ 1 p μ 1 1 − B
7.6
Γ p1
cp δ p A − B p − λ
−μ pμ
≤
1
f
J
z
|z|
|z|
Dz
δ,p
Γ pμ1
a p α p δ 1 p μ 1 1 − B
7.7
μ > 0; δ > −p; p ∈ N; z ∈ U, each of the assertions is sharp.
Proof. Since
−μ Dz
Jδ,p f z
∞
Γ p1
δp Γ kp1
pμ
−
z
akp zkpμ ,
Γ p−μ1
k1 δ k p Γ k p μ 1
7.8
then
∞
kp Γ p1
δp Γ kp1 Γ pμ1 −μ μ p
Jδ,p f z |z |z −
akp z .
Dz
Γ p−μ1
k1 δ k p Γ k p μ 1 Γ p 1
7.9
Let Gz is defined by
∞
δp Γ kp1 Γ pμ1 Gz z −
akp zkp zp − Qkakp zkp ,
k1 δ k p Γ k p μ 1 Γ p 1
k1
7.10
p
∞
where
δp Γ kp1 Γ pμ1
Qk δkp Γ kpμ1 Γ p1
k, p ∈ N; μ > 0 .
7.11
Since Qk is a decreasing function of k when μ > 0, we get
δp p1
0 < Qk ≤ Q1 ,
δkp 1pμ
7.12
18
International Journal of Mathematics and Mathematical Sciences
A, B; p, λ, α, by Theorem 2.1, we get
since fz ∈ Ya,c
∞
∞ pα kp
a p α 1 − B p 1 1 − Bak akp ≤
akp ≤ 1.
cpA − B p − λ
pA − B p − λ ck
k1
k1
7.13
It is that
∞ cpA − B p − λ
akp ≤ ,
a α p 1 − B p 1
k1
7.14
then
∞ ∞
p |Gz| z − Qkakp zkp ≥ |z|p − Q1|z|p1 akp k1
k1
cp p δ A − B p − λ
≥ |z| − |z|p1 ,
a p α δ k p p 1 μ 1 − B
∞ ∞
p |Gz| z − Qkakp zkp ≤ |z|p Q1|z|p1 akp k1
k1
p
7.15
cp p δ A − B p − λ
≤ |z| |z|p1 .
a p α δ k p p 1 μ 1 − B
p
From 7.15, we obtain 7.6 and 7.7, respectively.
Equalities in 7.6 and 7.7 are attained for the function fz given by
−μ Dz
Jδ,p f z
cp δ p A − B p − λ
Γ p1
pμ
1− z .
z
Γ p−μ1
a p α δ k p p μ 1 1 − B
7.16
Theorem 7.6. Let function defined by 1.1 be in the class Ya,c
A, B; p, λ, α, then
Γ p1
cp δ p A − B p − λ
μ
p−μ
1− |z|
|z| ,
Dz Jδ,p f z ≥ Γ p−μ1
a p α p δ 1 p − μ 1 1 − B
7.17
Γ p1
cp δ p A − B p − λ
μ
p−μ
1 |z|
|z| ,
Dz Jδ,p f z ≤ Γ p−μ1
a p α p δ 1 p − μ 1 1 − B
7.18
μ > 0; δ > −p; p ∈ N; z ∈ U, each of the assertions is sharp.
International Journal of Mathematics and Mathematical Sciences
19
Proof. Since
μ Dz Jδ,p f z
∞
Γ p1
δp Γ kp1
p−μ
−
z
akp zkp−μ ,
Γ p−μ1
k1 δ k p Γ k p − μ 1
7.19
then
∞
kp Γ p 1 −μ p δp Γ kp1 Γ p−μ1 μ
z z −
akp z .
Dz Jδ,p f z Γ p−μ1
k1 δ k p Γ k p − μ 1 Γ p 1
7.20
Let Hz be defined by
Hz zp −
∞
δp Γ kp1 Γ p−μ1 akp zkp
δkp Γ kp−μ1 Γ p1
k1
∞
z − Ek k p akp zkp ,
7.21
p
k1
where
δp Γ kp Γ p−μ1
Ek δkp Γ kp−μ1 Γ p1
k, p ∈ N; μ > 0 .
7.22
Since Ek is a decreasing function of k when 0 ≤ μ < 1, then
δp
0 < Ek ≤ E1 ,
δ1p 1p−μ
7.23
since fz ∈ Ya,c
A, B; p, λ, α, by Theorem 2.1, we get
∞
∞ pα kp
a p α 1 − B 1 − Bak akp ≤ 1.
k p akp ≤
cpA − B p − λ k1
pA − B p − λ ck
k1
7.24
It is that
cpA − B p − λ
,
k p akp ≤ a α p 1 − B
∞
k1
7.25
20
International Journal of Mathematics and Mathematical Sciences
then
∞
∞
kp p k p akp |Hz| z − Ek k p akp z ≥ |z|p − E1|z|p1
k1
k1
cp p δ A − B p − λ
≥ |z| − |z|p1 ,
a p α δ k p p 1 − μ 1 − B
∞
∞
kp p k p akp |Hz| z − Ek k p akp z ≤ |z|p E1|z|p1
k1
k1
p
7.26
cp p δ A − B p − λ
≤ |z| |z|p1 .
a p α δ k p p 1 − μ 1 − B
p
From 7.26, we obtain 7.17 and 7.18, respectively.
Equalities in 7.17 and 7.18 are attained for the function fz given by
μ Dz Jδ,p f z
cp δ p A − B p − λ
Γ p1
p−μ
1− z
z
Γ p−μ1
a p α δ k p p − μ 1 1 − B
7.27
or equivalently by
cp δ p A − B p − λ
Jδ,p f z z − zp1 .
a p α δ k p p − μ 1 1 − B
p
7.28
Acknowledgments
The author thanks the referee for numerous suggestions that helped make this paper more
readable. The present investigation was partly supported by the Natural Science Foundation
of Inner Mongolia under Grant 2009MS0113.
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