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 Altıntaş1 and Öznur Özkan2
1
2
Department of Matematics Education, Başkent University, Bağlıca, TR-06530, Ankara, Turkey
Department of Statistics and Computer Sciences, Başkent University, Bağlıca, TR 06530, Ankara, Turkey
Correspondence should be addressed to Öznur Özkan, oznur@baskent.edu.tr
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. Altıntaş and Ö. Özkan. 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 Tn p denote the class of functions fz of the form
fz zp −
∞
ak zk
ak ≥ 0; n, p ∈ N {1, 2, 3, . . .},
1.1
knp
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
Dz−δ fz 1
Γδ
z
0 z
fξ
− ξ1−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 − ξδ−1 is removed by requiring log z − ξ to be real when
z − ξ > 0.
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Journal of Inequalities and Applications
Definition 1.2. The fractional derivative of order δ is defined by
Dzδ
d
1
fz Γ1 − δ dz
z
fξ
0 z
− ξδ
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
Dznδ fz dn δ
D fz
dzn z
0 ≤ δ < 1, n ∈ N0 N ∪ {0} .
1.4
av denotes the Pochhammer symbol (or the shifted factorial), since
1n n!
for n ∈ N0 : N ∪ {0},
1.5
defined for a, v ∈ C and in terms of the Gamma function by
Γa v
av :
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, εneighborhood of a function f ∈ Tn p by
Nεn,p
Dzδ f, Dzδ
g :
g ∈ Tn p : gz z −
p
∞
k
bk z ,
knp
∞
k 1 − δδ k ak − bk ≤ ε ,
knp
1.7
so that, obviously,
Nεn,p
δ
Dz h, Dzδ g :
g ∈ Tn p : gz z −
p
∞
knp
k
bk z ,
∞
k 1 − δδ k bk ≤ ε ,
1.8
knp
where
hz : zp .
1.9
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3
δ
λ, α denote the subclass of Tn p consisting of functions fz which
The class Sn,p
satisfy
Re
zF z
> α 0 ≤ α < p, p ∈ N,
Fz
1.10
where
Fz λzDz1δ fz 1 − λDzδ fz
0 ≤ λ ≤ 1, 0 ≤ δ < 1.
1.11
0
0
We note that the class S1,1
λ, α was investigated by Altıntaş 6 and the class Sn,p
λ, α
was studied by Altıntaş et al. 7, 8 .
We donote by
0
0, α Sn∗ p, α,
Sn,p
0
Sn,p
1, α Cn p, α
1.12
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 Kδn,p λ, α, μ denote the subclass of the general class Tn p consisting of
functions fz ∈ Tn p satisfying the following nonhomogeneous Cauchy-Euler differential
equation:
z2 Dz2δ ω 21 μzDz1δ ω μ1 μDzδ ω p − δ μp − δ μ 1Dzδ g,
1.13
δ
where ω fz, fz ∈ Tn p, g gz ∈ Sn,p
λ, α and μ > δ − p.
The main object of the present paper is to give coefficients bounds and distortion
δ
λ, α and Kδn,p λ, α, μ.
inequalities for functions in the classes Sn,p
2. Coefficient Bounds and Distortion Inequalities
We begin by proving the following result.
δ
Lemma 2.1. Let the function fz ∈ Tn p be defined by 1.1. Then fz is in the class Sn,p
λ, α
if and only if
∞
k 1 − δδ k − α − δ1 λk − 1 − δak ≤ p 1 − δδ p − α − δ1 λp − 1 − δ
knp
0 ≤ λ ≤ 1; 0 ≤ α < p − δ; 0 ≤ δ < 1; p ∈ N.
2.1
The result is sharp for the function fz given by
fz zp −
p 1 − δδ p − α − δ1 λp − 1 − δ
znp .
n p 1 − δδ n p − α − δ1 λn p − 1 − δ
2.2
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Journal of Inequalities and Applications
δ
λ, α. Then,
Proof. Let fz ∈ Tn p and Fz be defined by 1.11. Suppose that fz ∈ Sn,p
in conjunction with 1.10 and 1.11 yields
Re
p1−δδ p−δ1λp−1−δzp−δ −
∞
p 1 − δδ 1 λp − 1 − δzp−δ −
knp k1−δδ k
∞
knp k
− δ1λk−1−δak zk−δ
1 − δδ 1 λk − 1 − δak zk−δ
> α.
2.3
By letting z → 1− along the real axis, we arrive easily at the inequality in 2.1.
δ
Lemma 2.2. Let the function fz given by 1.1 be in the class Sn,p
λ, α. Then
∞
p 1 − δδ p − α − δ1 λp − 1 − δ
,
n p − α − δ1 λn p − 1 − δ
2.4
p 1 − δδ p − α − δ1 λp − 1 − δn p
.
n p − α − δ1 λn p − 1 − δ
2.5
k 1 − δδ ak ≤
knp
∞
k 1 − δδ kak ≤
knp
Proof. By using Lemma 2.1, we find from 2.1 that
n p − α − δ1 λn p − 1 − δ
∞
k 1 − δδ ak
knp
≤
∞
2.6
k 1 − δδ k − α − δ1 λk − 1 − δak
knp
≤ p 1 − δδ p − α − δ1 λp − 1 − δ,
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
∞
k 1 − δδ kak − α δ
knp
∞
knp
k 1 − δδ ak
2.7
≤ p 1 − δδ p − α − δ1 λp − 1 − δ,
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 Kδn,p λ, α, μ are given by
Theorem 2.3 below.
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5
Theorem 2.3. Let a function fz ∈ Tn p be in the class Kδn,p λ, α, μ. Then
p 1 − δδ p − α − δp − δ μp − δ μ 1 1 λp − δ − 1
|z|np ,
n p 1 − δδ n p − α − δn p − δ μ 1 λn p − δ − 1
2.8
|fz| ≤ |z|p p 1 − δδ p − α − δp − δ μp − δ μ 1 1 λp − δ − 1
|z|np .
n p 1 − δδ n p − α − δn p − δ μ 1 λn p − δ − 1
2.9
|fz| ≥ |z|p −
Proof. Suppose that a function fz ∈ Tn p is given by 1.1 and also let the function gz ∈
δ
λ, α occurring in the nonhomogenous differential equation 1.13 be given as in the
Sn,p
Definitions 1.2 or 1.3 with of course
bk ≥ 0 k n p, n p 1, . . ..
2.10
Then we easily see from 1.13 that
ak p − δ μp − δ μ 1
bk
k − δ μk − δ μ 1
k n p, n p 1, . . ..
2.11
So that
fz zp −
∞
ak zk zp −
knp
|fz| ≤ |z|p |z|np
∞
p − δ μp − δ μ 1
bk zk ,
k
− δ μk − δ μ 1
knp
∞
p − δ μp − δ μ 1
bk .
k − δ μk − δ μ 1
knp
2.12
2.13
δ
Since gz ∈ Sn,p
λ, α, the first assertion 2.4 of Lemma 2.2 yields the following inequality:
bk ≤
p 1 − δδ p − α − δ1 λp − δ − 1
.
n p 1 − δδ 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 − δδ p − α − δ1 λp − δ − 1
n p 1 − δδ n p − α − δ1 λn p − δ − 1
1
,
k
−
δ
μk
− δ μ 1
knp
2.15
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Journal of Inequalities and Applications
and also note that
∞ 1
1
1
−
k − δ μk − δ μ 1 knp k − δ μ k − δ μ 1
knp
∞
1
,
np−δμ
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, and also 2.16.
By setting δ : 0 in Theorem 2.3, we obtain the following Corollary 2.4.
Corollary 2.4 See Altıntaş et al. 8, Theorem 1. If the functions f and g satisfy the nonhomogeneous Cauchy-Euler differential equation 1.13, then
|fz| ≤ |z|p p − αp μp μ 11 λp − 1 np
|z| ,
n p − αn p μ1 λn p − 1
p − αp μp μ 11 λp − 1 np
|z| .
|fz| ≥ |z|p −
n p − αn p μ1 λn p − 1
2.17
By letting δ : 0, λ : 0 and δ : 0, λ : 1 in Theorem 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 equation 1.13 with g ∈ Sn∗ p, α, then
|fz| ≤ |z|p p − αp μp μ 1 np
|z| ,
n p − αn p μ
p − αp μp μ 1 np
|fz| ≥ |z| −
|z| .
n p − αn p μ
2.18
p
Corollary 2.6. If the functions f and g satisfy the nonhomogeneous Cauchy-Euler differential equation 1.13 with g ∈ Cn p, α, then
|fz| ≤ |z|p pp − αp μp μ 1
|z|np ,
n p − αn p μn p
pp − αp μp μ 1
|fz| ≥ |z| −
|z|np .
n p − αn p μn p
2.19
p
δ
3. Neighborhoods for the Classes Sn,p
λ, α and Kδn,p λ, α, μ
δ
In this section, we determine inclusion relations for the classes Sn,p
λ, α and Kδn,p λ, α, μ
concerning the n, p, ε-neighborhoods is defined by 1.7 and 1.8.
Journal of Inequalities and Applications
7
δ
λ, α. Then
Theorem 3.1. Let a function fz ∈ Tn p be in the class Sn,p
δ
Sn,p
λ, α ⊂ Nεn,p Dzδ h, Dzδ f ,
3.1
where hz is given by 1.9 and the parameter ε is the given by
ε :
n pp − δδ 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 Nεn,p Dzδ h, Dzδ 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 ∈ Tn p be in the class Kδn,p λ, α, μ. Then
Kδn,p λ, α, μ ⊂ Nεn,p Dzδ g, Dzδ f ,
3.3
where gz is given by 1.13 and the parameter ε is the given by
ε :
n pp − δδ 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 ∈ Kδn,p λ, α, μ. Then, upon substituting from 2.11 into the following coefficient inequality:
∞
∞
∞
k − δδ kbk − ak ≤
k − δδ kbk k − δδ kak ,
knp
knp
3.5
knp
where ak ≥ 0 and bk ≥ 0, we obtain that
∞
knp
∞
k − δδ kbk − ak ≤
k − δδ kbk
knp
∞
p − δ μp − δ μ 1
k − δδ kbk .
k − δ μk − δ μ 1
knp
3.6
δ
λ, α, the second assertion 2.5 of Lemma 2.2 yields that
Since gz ∈ Sn,p
k − δδ kbk ≤
n pp − δδ p − α − δ1 λp − 1 − δ
n p − α − δ1 λn p − 1 − δ
k n p, n p 1, . . ..
3.7
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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, we find that
∞
k − δδ kbk − ak knp
n pp − δδ p − α − δ1 λp − 1 − δ
≤
n p − α − δ1 λn p − 1 − δ
p − δ μp − δ μ 1
1
k − δ μk − δ μ 1
3.8
,
which, by virtue of the identity 2.16, immediately yields that
∞
k − δδ kbk − ak knp
≤
n pp − δδ p − α − δ1 λp − 1 − δ n p − δ μp − δ μ 2
·
: ε.
n p − α − δ1 λn p − 1 − δ
n p − δ μ
3.9
Thus, by definition 1.7 with gz interchanged by fz, fz ∈ Nεn,p Dzδ g, Dzδ f. This
evidently completes the proof of Theorem 3.2.
By setting δ 0 in Theorem 3.2, we receive the following result.
Corollary 3.3. If the function fz ∈ Tn p is in the class K0n,p λ, α, μ. Then
K0n,p λ, α, μ ⊂ Nεn,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 Başkent University Ankara, TURKEY.
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