Báo cáo hóa học: " Research Article Fractional Calculus and p-Valently Starlike Functions" pptx
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (479.91 KB, 9 trang )
Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 683985, 9 pages
doi:10.1155/2009/683985
Research Article
Fractional Calculus and p-Valently
Starlike Functions
Osman Altıntas¸
1
and
¨
Oznur
¨
Ozkan
2
1
Department of Matematics Education, Bas¸kent University, Ba
˘
glıca, TR-06530, Ankara, Turkey
2
Department of Statistics and Computer Sciences, Bas¸kent University, Ba
˘
glıca, TR 06530, Ankara, Turkey
Correspondence should be addressed to
¨
Oznur
¨
Ozkan,
Received 18 November 2008; Accepted 28 February 2009
Recommended by Alberto Cabada
In this investigation, the authors prove coefficient bounds, distortion inequalities for fractional
calculus of a family of multivalent functions with negative coefficients, which is defined by means
of a certain nonhomogenous Cauchy-Euler differential equation.
Copyright q 2009 O. Altıntas¸and
¨
O.
¨
Ozkan. 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.
1. Introduction and Definitions
Let T
n
p denote the class of functions fz of the f orm
fzz
p
−
∞
knp
a
k
z
k
a
k
≥ 0; n, p ∈ N {1, 2, 3, }, 1.1
which are analytic and multivalent in the unit disk U {z : z ∈ C and |z| < 1}.
The fractional calculus are defined as follows e.g., 1, 2.
Definition 1.1. The fractional integral of order δ is defined by
D
−δ
z
fz
1
Γδ
z
0
fξ
z − ξ
1−δ
dξ δ>0, 1.2
where fz is an analytic function in a simply-connected region of the z-plane containing the
origin and the multiplicity of z − ξ
δ−1
is removed by requiring log z − ξ to be real when
z − ξ>0.
2 Journal of Inequalities and Applications
Definition 1.2. The fractional derivative of order δ is defined by
D
δ
z
fz
1
Γ1 − δ
d
dz
z
0
fξ
z − ξ
δ
dξ 0 ≤ δ<1, 1.3
where fz is constrained and multiplicity of z − ξ
−δ
is removed as in Definition 1.1.
Definition 1.3. Under the hypotheses of Definition 1.1, the fractional derivative of order nδ
is defined by
D
nδ
z
fz
d
n
dz
n
D
δ
z
fz
0 ≤ δ<1,n∈ N
0
N ∪{0}
. 1.4
a
v
denotes the Pochhammer symbol (or the shifted factorial),since
1
n
n!forn ∈ N
0
: N ∪{0}, 1.5
defined for a, v ∈ C and in terms of the Gamma function by
a
v
:
Γa v
Γa
1, v 0,a∈ C \{0},
aa 1 ···a n − 1, v n ∈ N; a ∈ C.
1.6
The earlier investigations by Goodman 3, 4 and Ruscheweyh 5, we define the n, p, ε-
neighborhood of a function f ∈T
n
p by
N
ε
n,p
D
δ
z
f, D
δ
z
g
:
g ∈T
n
p : gzz
p
−
∞
knp
b
k
z
k
,
∞
knp
k 1 − δ
δ
k
a
k
− b
k
≤ ε
,
1.7
so that, obviously,
N
ε
n,p
D
δ
z
h, D
δ
z
g
:
g ∈T
n
p : gzz
p
−
∞
knp
b
k
z
k
,
∞
knp
k 1 − δ
δ
k
b
k
≤ ε
, 1.8
where
hz : z
p
. 1.9
Journal of Inequalities and Applications 3
The class S
δ
n,p
λ, α denote the subclass of T
n
p consisting of functions fz which
satisfy
Re
zF
z
Fz
>α 0 ≤ α<p, p∈ N, 1.10
where
FzλzD
1δ
z
fz1 − λD
δ
z
fz0 ≤ λ ≤ 1, 0 ≤ δ<1. 1.11
We note that the class S
0
1,1
λ, α was investigated by Altıntas¸ 6 and the class S
0
n,p
λ, α
was studied by Alt ıntas¸etal.7, 8 .
We donote by
S
0
n,p
0,αS
∗
n
p, α, S
0
n,p
1,αC
n
p, α1.12
the classes of p-valently starlike functions of order α in U 0 ≤ α<p and p-valently convex
functions of order α in U 0 ≤ α<p, respectively see, 2, 9.
Finally K
δ
n,p
λ, α, μ denote the subclass of the general class T
n
p consisting of
functions fz ∈T
n
p satisfying the following nonhomogeneous Cauchy-Euler differential
equation:
z
2
D
2δ
z
ω 21 μzD
1δ
z
ω μ1 μD
δ
z
ω p − δ μp − δ μ 1D
δ
z
g, 1.13
where ω fz,fz ∈T
n
p,g gz ∈S
δ
n,p
λ, α and μ>δ− p.
The main object of the present paper is to give coefficients bounds and distortion
inequalities for functions in the classes S
δ
n,p
λ, α and K
δ
n,p
λ, α, μ.
2. Coefficient Bounds and Distortion Inequalities
We begin by proving the following result.
Lemma 2.1. Let the function fz ∈T
n
p be defined by 1.1.Thenfz is in the class S
δ
n,p
λ, α
if and only i f
∞
knp
k 1 − δ
δ
k − α − δ1 λk − 1 − δa
k
≤ p 1 − δ
δ
p − α − δ1 λp − 1 − δ
0 ≤ λ ≤ 1; 0 ≤ α<p− δ;0≤ δ<1; p ∈ N.
2.1
The result is sharp for the function fz given by
fzz
p
−
p 1 − δ
δ
p − α − δ1 λp − 1 − δ
n p 1 − δ
δ
n p − α − δ1 λn p − 1 − δ
z
np
. 2.2
4 Journal of Inequalities and Applications
Proof. Let fz ∈T
n
p and Fz be defined by 1.11. Suppose that fz ∈S
δ
n,p
λ, α. Then,
in conjunction with 1.10 and 1.11 yields
Re
p1−δ
δ
p−δ1λp−1−δz
p−δ
−
∞
knp
k1−δ
δ
k − δ1λk−1−δa
k
z
k−δ
p 1 − δ
δ
1 λp − 1 − δz
p−δ
−
∞
knp
k 1 − δ
δ
1 λk − 1 − δa
k
z
k−δ
>α.
2.3
By letting z → 1
−
along the real axis, we arrive easily at the inequality in 2.1.
Lemma 2.2. Let the function fz given by 1.1 be in the class S
δ
n,p
λ, α. Then
∞
knp
k 1 − δ
δ
a
k
≤
p 1 − δ
δ
p − α − δ1 λp − 1 − δ
n p − α − δ1 λn p − 1 − δ
, 2.4
∞
knp
k 1 − δ
δ
ka
k
≤
p 1 − δ
δ
p − α − δ1 λp − 1 − δn p
n p − α − δ1 λn p − 1 − δ
. 2.5
Proof. By using Lemma 2.1,wefindfrom2.1 that
n p − α − δ1 λn p − 1 − δ
∞
knp
k 1 − δ
δ
a
k
≤
∞
knp
k 1 − δ
δ
k − α − δ1 λk − 1 − δa
k
≤ p 1 − δ
δ
p − α − δ1 λp − 1 − δ,
2.6
which immediately yields the first assertion 2.4 of Lemma 2.2.
For the proof of second assertion, by appealing to 2.1, we also have
1 λn p − δ − 1
∞
knp
k 1 − δ
δ
ka
k
− α δ
∞
knp
k 1 − δ
δ
a
k
≤ p 1 − δ
δ
p − α − δ1 λp − 1 − δ,
2.7
by using 2.4 in 2.7, we can easily get the assertion 2.5 of Lemma 2.2.
The distortion inequalities for functions in the class K
δ
n,p
λ, α, μ are given by
Theorem 2.3 below.
Journal of Inequalities and Applications 5
Theorem 2.3. Let a function fz ∈T
n
p be in the class K
δ
n,p
λ, α, μ.Then
|fz|≤|z|
p
p 1 − δ
δ
p − α − δp − δ μp − δ μ 1
n p 1 − δ
δ
n p − α − δn p − δ μ
1 λp − δ − 1
1 λn p − δ − 1
|z|
np
,
2.8
|fz|≥|z|
p
−
p 1 − δ
δ
p − α − δp − δ μp − δ μ 1
n p 1 − δ
δ
n p − α − δn p − δ μ
1 λp − δ − 1
1 λn p − δ − 1
|z|
np
.
2.9
Proof. Suppose that a function fz ∈T
n
p is given by 1.1 and also let the function gz ∈
S
δ
n,p
λ, α occurring in the nonhomogenous differential equation 1.13 be given as in the
Definitions 1.2 or 1.3 with of course
b
k
≥ 0 k n p, n p 1, . 2.10
Then we easily see from 1.13 that
a
k
p − δ μp − δ μ 1
k − δ μk − δ μ 1
b
k
k n p, n p 1, . 2.11
So that
fzz
p
−
∞
knp
a
k
z
k
z
p
−
∞
knp
p − δ μp − δ μ 1
k − δ μk − δ μ 1
b
k
z
k
, 2.12
|fz|≤|z|
p
|z|
np
∞
knp
p − δ μp − δ μ 1
k − δ μk − δ μ 1
b
k
. 2.13
Since gz ∈S
δ
n,p
λ, α, the first assertion 2.4 of Lemma 2.2 yields the following inequality:
b
k
≤
p 1 − δ
δ
p − α − δ1 λp − δ − 1
n p 1 − δ
δ
n p − α − δ1 λn p − δ − 1
. 2.14
From 2.13 and 2.14 we have
|fz|≤|z|
p
|z|
np
p 1 − δ
δ
p − α − δ1 λp − δ − 1
n p 1 − δ
δ
n p − α − δ1 λn p − δ − 1
·
∞
knp
1
k − δ μk − δ μ 1
,
2.15
6 Journal of Inequalities and Applications
andalsonotethat
∞
knp
1
k − δ μk − δ μ 1
∞
knp
1
k − δ μ
−
1
k − δ μ 1
1
n p − δ μ
,
2.16
where μ ∈ R \{−n− p, − n− p − 1, }. The assertion 2.8 of Theorem 2.3 follows at once from
2.15. The assertion 2.9 of Theorem 2.3 can be proven by similarly applying 2.12, 2.14,
and 2.15,andalso2.16.
By setting δ : 0inTheorem 2.3, we obtain the following Corollary 2.4.
Corollary 2.4 See Altıntas¸etal.8, Theorem 1. If the functions f and g satisfy the nonhomoge-
neous Cauchy-Euler differential equation 1.13,then
|fz|≤|z|
p
p − αp μp μ 11 λp − 1
n p − αn p μ1 λn p − 1
|z|
np
,
|fz|≥|z|
p
−
p − αp μp μ 11 λp − 1
n p − αn p μ1 λn p − 1
|z|
np
.
2.17
By letting δ : 0, λ : 0andδ : 0, λ : 1inTheorem 2.3. We arrive at Corollaries 2.5
and 2.6 see, 8.
Corollary 2.5. If the functions f and g satisfy the nonhomogeneous Cauchy-Euler differential equa-
tion 1.13 with g ∈S
∗
n
p, α, then
|fz|≤|z|
p
p − αp μp μ 1
n p − αn p μ
|z|
np
,
|fz|≥|z|
p
−
p − αp μp μ 1
n p − αn p μ
|z|
np
.
2.18
Corollary 2.6. If the functions f and g satisfy the nonhomogeneous Cauchy-Euler differential equa-
tion 1.13 with g ∈C
n
p, α, then
|fz|≤|z|
p
pp − αp μp μ 1
n p − αn p μn p
|z|
np
,
|fz|≥|z|
p
−
pp − αp μp μ 1
n p − αn p μn p
|z|
np
.
2.19
3. Neighborhoods for the Classes S
δ
n,p
λ, α and K
δ
n,p
λ, α, μ
In this section, we determine inclusion relations for the classes S
δ
n,p
λ, α and K
δ
n,p
λ, α, μ
concerning the n, p, ε-neighborhoods is defined by 1.7 and 1.8.
Journal of Inequalities and Applications 7
Theorem 3.1. Let a function fz ∈T
n
p be in the class S
δ
n,p
λ, α.Then
S
δ
n,p
λ, α ⊂N
ε
n,p
D
δ
z
h, D
δ
z
f
, 3.1
where hz is given by 1.9 and the parameter ε is the given by
ε :
n pp − δ
δ
p − α − δ1 λp − 1 − δ
n p − α − δ1 λn p − 1 − δ
. 3.2
Proof. Assertion 3.1 would follow easily from the definition of N
ε
n,p
D
δ
z
h, D
δ
z
f, which is
given by 1.8 with gz replaced by fz and the second assertion 2.5 of Lemma 2.2.
Theorem 3.2. Let a function fz ∈T
n
p be in the class K
δ
n,p
λ, α, μ.Then
K
δ
n,p
λ, α, μ ⊂N
ε
n,p
D
δ
z
g,D
δ
z
f
, 3.3
where gz is given by 1.13 and the parameter ε is the given by
ε :
n pp − δ
δ
p − α − δ1 λp − 1 − δn p − δ μp − δ μ 2
n p − α − δn p − δ μ1 λn p − 1 − δ
. 3.4
Proof. Suppose that fz ∈K
δ
n,p
λ, α, μ. Then, upon substituting from 2.11 into the follo-
wing coefficient inequality:
∞
knp
k − δ
δ
k
b
k
− a
k
≤
∞
knp
k − δ
δ
kb
k
∞
knp
k − δ
δ
ka
k
, 3.5
where a
k
≥ 0andb
k
≥ 0, we obtain that
∞
knp
k − δ
δ
k
b
k
− a
k
≤
∞
knp
k − δ
δ
kb
k
∞
knp
p − δ μp − δ μ 1
k − δ μk − δ μ 1
k − δ
δ
kb
k
.
3.6
Since gz ∈S
δ
n,p
λ, α, the second assertion 2.5 of Lemma 2.2 yields that
k − δ
δ
kb
k
≤
n pp − δ
δ
p − α − δ1 λp − 1 − δ
n p − α − δ1 λn p − 1 − δ
k n p, n p 1, . 3.7
8 Journal of Inequalities and Applications
Finally, by making use of 2.5 as well as 3.7 on the right-hand side of 3.6,wefindthat
∞
knp
k − δ
δ
k
b
k
− a
k
≤
n pp − δ
δ
p − α − δ1 λp − 1 − δ
n p − α − δ1 λn p − 1 − δ
1
p − δ μp − δ μ 1
k − δ μk − δ μ 1
,
3.8
which, by virtue of the identity 2.16, immediately yields that
∞
knp
k − δ
δ
k
b
k
− a
k
≤
n pp − δ
δ
p − α − δ1 λp − 1 − δ
n p − α − δ1 λn p − 1 − δ
·
n p − δ μp − δ μ 2
n p − δ μ
: ε.
3.9
Thus, by definition 1.7 with gz interchanged by fz, fz ∈N
ε
n,p
D
δ
z
g,D
δ
z
f.This
evidently completes the proof of Theorem 3.2.
By setting δ 0inTheorem 3.2, we receive the following result.
Corollary 3.3. If the function fz ∈T
n
p is in the class K
0
n,p
λ, α, μ.Then
K
0
n,p
λ, α, μ ⊂N
ε
n,p
g,f, 3.10
where gz is given by 1.13 and the parameter ε is the given by
ε :
n pp − α1 λp − 1n p μp μ 2
n p − αn p μ1 λn p − 1
. 3.11
Acknowledgment
This present investigation was supported by Bas¸kent University Ankara, TURKEY.
References
1 S. Owa, “On the distortion theorems—I,” Kyungpook Mathematical Journal, vol. 18, no. 1, pp. 53–59, 1978.
2 H. M. Srivastava and S. Owa, Eds., Univalent Functions, Fractional Calculus, and Their Applications, Ellis
Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK, 1989.
3 A. W. Goodman, Univalent Functions. Vol. I, Mariner, Tampa, Fla, USA, 1983.
4 A. W. Goodman, UnivalentFunctions. Vol. II, Mariner, Tampa, Fla, USA, 1983.
5 S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathematical
Society, vol. 81, no. 4, pp. 521–527, 1981.
6 O. Altıntas¸, “On a subclass of certain starlike functions with negative coefficients,” Mathematica
Japonica, vol. 36, no. 3, pp. 489–495, 1991.
7 O. Altıntas¸, H. Irmak, and H. M. Srivastava, “Fractional calculus and certain starlike functions with
negative coefficients,” Computers & Mathematics with Applications, vol. 30, no. 2, pp. 9–15, 1995.
Journal of Inequalities and Applications 9
8 O. Altıntas¸,
¨
O.
¨
Ozkan, and H. M. Srivastava, “Neighborhoods of a certain family of multivalent
functions with negative coefficients,” Computers & Mathematics with Applications, vol. 47, no. 10-11,
pp. 1667–1672, 2004.
9 P. L. Duren, Univalent Functions, vol. 259 of Grundlehren der Mathematischen Wissenschaften,Springer,
New York, NY, USA, 1983.
