Volume 2010, Article ID 178605,12pages doi:10.1155/2010/178605
Research Article
Certain Subclasses of Starlike
Functions of Complex Order Involving Generalized
Hypergeometric Functions
G. Murugusundaramoorthy
1and N. Magesh
21School of Advanced Sciences, VIT University, Vellore 632014, India
2Department of Mathematics, Government Arts College (Men), Krishnagiri 635001, India
Correspondence should be addressed to G. Murugusundaramoorthy,[email protected]
Received 24 November 2009; Accepted 18 March 2010
Academic Editor: Stanisława R. Kanas
Copyrightq2010 G. Murugusundaramoorthy and N. Magesh. 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 the generalized hypergeometric functions, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients and obtain coefficient estimates, extreme points, the radii of close-to-convexity, starlikeness and convexity, and neighborhood results for the classTSl
mα, β, γ. In particular, we obtain integral
means inequalities for the functionfthat belongs to the classTSl
mα, β, γin the unit disc.
1. Introduction
LetAdenote the class of functions of the form
fz z ∞
n2
anzn 1.1
which are analytic in the open unit discU{z:z ∈C, |z|< 1}and normalized byf0
f0−10.Also denote byTthe subclass ofAconsisting of functions of the form
fz z− ∞
n2
|an|zn, 1.2
introduced and studied by Silverman 1. For functions f ∈ A given by1.1and g ∈ A
or Convolutionoffandgby
f∗gz z ∞
n2
anbnzn, z∈U. 1.3
For positive real values ofα1, . . . , αl andβ1, . . . , βm βj/0,−1, . . .; j 1,2, . . . , mthe
generalized hypergeometric function lFmzis defined by
lFmz≡ lFm
α1, . . . , αl;β1, . . . , βm;z
:
∞
n0
α1n· · ·αln
β1
n· · ·
βm
n
zn n!
l≤m1; l, m∈N0:N∪ {0}; z∈U,
1.4
whereNdenotes the set of all positive integers andλkis the Pochhammer symbol defined by
λn ⎧ ⎨ ⎩
1, n0,
λλ1λ2· · ·λn−1, n∈N. 1.5
The notation lFm is quite useful for representing many well-known functions such as the
exponential, the Binomial, the Bessel, the Laguerre polynomial, and others; for example, see
3.
LetHα1, . . . , αl;β1, . . . , βm:A → Abe a linear operator defined by
Hα1, . . . , αl;β1, . . . , βm
fz:zlFm
α1, α2, . . . , αl;β1, β2, . . . , βm;z
∗fz
z ∞
n2
Γnanzn,
1.6
where
Γn
α1n−1· · ·αln−1
β1
n−1· · ·
βm
n−1
1
n−1!, 1.7
unless otherwise stated. For notational simplicity, we can use a shorter notationHl
mα1for
Hα1, . . . , αl;β1, . . . , βmin the sequel. The linear operatorHml α1is called Dziok-Srivastava
operatorsee3, includas its special casesvarious other linear operators introduced and studied by Carlson and Shaffer 4, Owa 5, Ruscheweyh 2, and Srivastava and Owa
6. Motivated by Goodman 7, 8, Rønning 9,10 introduced and studied the following subclasses ofA.A functionf ∈ A is said to be in the class Spα, β, uniformly β-starlike
functions if it satisfies the condition
Re zf
z fz −α
> βzf
z fz −1
and is said to be in the class UCVα, β, uniformly β-convex functions if it satisfies the condition
Re 1zf
z fz −α
> βzf
z fz
, −1< α≤1;β≥0z∈U. 1.9
Indeed it follows from1.8and1.9that
f∈UCVα, β⇐⇒zf∈Sp
α, β. 1.10
The interesting geometric properties of these function classes were extensively studied by Kanas et al., in 11–14. Motivated by Altintas et al. 15, Murugusundaramoorthy and Srivastava 16, and Murugusundaramoorthy and Magesh 17, now, we define a new subclass uniformly starlike functions of complex order.
For−1 ≤α <1, β≥0, andγ∈C\ {0}, we letTSlmα, β, γbe the class of functionsf
satisfying1.2with the analytic criterion
Re
1 1
γ
zHl
mα1fz
Hl
mα1fz
−α
> β1 1
γ
zHl
mα1fz
Hl
mα1fz
−1, z∈U 1.11
whereHl
mα1fzis given by1.6.
By suitably specializing the values ofl, m, α1, α2, . . . , αl, β1, β2, . . . , βm, α,γ, andβthe
classTSl
mα, β, γ, leads to various new subclasses of starlike functions of complex order. As
for illustrations, we present some examples for the cases.
Example 1.1. Ifl2, m1 withα11, α21, β1 1, andfof the form of1.2, then
TSα, β, γ: Re 11
γ
zfz
fz −α
> β1 1
γ
zfz
fz −1
, z∈U
. 1.12
Example 1.2. Ifl2, m1 withα1 δ1δ >−1, α21, β1 1, andfof the form of1.2,
then
TRδ
α, β, γ:
Re
11
γ
zDδfz Dδfz −α
> β1 1
γ
zDδfz Dδfz −1
, z∈U
,
1.13
whereDδis called Ruscheweyh derivative of orderδ δ >−1defined by
Dδfz: z
1−zδ1 ∗fz≡H
2
Example 1.3. Ifl 2, m 1 withα1 μ1 μ >−1,α2 1, β1 μ2, andf of the form of
1.2, then
TBμ
α, β, γ
Re
1 1
γ
zJμfz
Jμfz − α
> β1 1
γ
zJμfz
Jμfz −
1, z∈U
,
1.15
whereJμis a Bernardi operator18defined by
Jμfz: μ1
zμ z
0
tμ−1ftdt≡H12μ1,1;μ2fz. 1.16
Example 1.4. Ifl 2, m 1 withα1 aa > 0, α2 1, β1 cc >0, andf of the form of
1.2, then
TLacα, β, γ:
Re
11
γ
zLa, cfz La, cfz −α
> β11
γ
zLa, cfz La, cfz −1
, z∈U
,
1.17
whereLa, cis a well-known Carlson-Shaffer linear operator4defined by
La, cfz:
∞
k0
ak
ckz k1
∗fz≡H2
1a,1;cfz. 1.18
AlsoTLa
cα, β, γwas studied by Murugusundaramoorthy and Magesh19.
The main object of this paper is to study some usual properties of the geometric function theory such as the coefficient bound, extreme points, radii of close to convexity, starlikeness, and convexity for the classTSl
mα, β, γ.Further, we obtain neighborhood results
and integral means inequalities for aforementioned class.
2. Basic Properties
First we obtain the necessary and sufficient condition for functionsfin the classTSl
mα, β, γ.
Theorem 2.1. The necessary and sufficient condition for f of the form of 1.2 to be in the class TSl
mα, β, γis ∞
n2
nγ1−β−α−βΓn|an| ≤1−α γ1−β
, 2.1
Proof. Assume thatf∈TSl
mα, β, γ, then
Re
1 1
γ
zHl
mα1fz
Hml α1fz
−α
> β1 1
γ
zHl
mα1fz
Hml α1fz
−1,
Re 1 1
γ
z1−α−∞n2n−αΓn|an|zn z−∞n2Γn|an|zn
> β1−1
γ ∞
n2n−1Γn|an|zn z−∞n2Γn|an|zn
.
2.2
Lettingz → 1−along the real axis, we have
11
γ
1−α−∞
n2n−αΓn|an|
1−∞n2Γn|an|
> β
1−1
γ ∞
n2n−1Γn|an|
1−∞n2Γn|an|
. 2.3
Hence, by maximum modulus theorem, the simple computational leads the desired inequality
∞
n2
nγ1−β−α−βΓn|an| ≤1−α γ1−β
. 2.4
Conversely, suppose that2.1is true forz∈U.Then
Re
1 1
γ
zHl
mα1fz
Hml α1fz
−α
−β11
γ
zHl
mα1fz
Hml α1fz
−1>0 2.5
if
1 1
γ
1−α−∞n2n−αΓn|an||z|n−1
1−∞n2Γn|an||z|n−1
−β
1−1
γ ∞
n2n−1Γn|an||z|n−1
1−∞n2Γn|an||z|n−1
>0.
2.6
That is if
∞
n2
nγ1−β−α−βΓn|an| ≤1−α γ1−β
, 2.7
which completes the proof.
Corollary 2.2. Let the functionfdefined by1.2belong toTSl
mα, β, γ.Then
|an| ≤
1−α γ1−β
nγ1−β−α−βΓn
n≥2, −1≤α <1, β≥0 and,γ∈C\ {0}, with equality for
fz z−
1−α γ1−β
nγ1−β−α−βΓn
zn. 2.9
Next we state the following theorem on extreme points for the class TSl
mα, β, γ
without proof.
Theorem 2.3Extreme Points. Let
f1z z, fnz z−
1−α γ1−β
nγ1−β−α−βΓn
zn forn2,3,4, . . . . 2.10
Thenf ∈ TSl
mα, β, γif and only iff can be expressed in the formfz
∞
n1λnfnz, where λn≥0 and
∞
n1λn1.
3. Close-to-Convexity, Starlikeness, and Convexity
We determine the radii of close-to-convexity, starlikeness, and convexity results for functions in the classTSl
mα, β, γin the following theorems.
Theorem 3.1. Letf ∈ TSl
mα, β, γ.Thenf is close-to-convex of orderδ 0 ≤ δ < 1in the disc |z|< r1, where
r1inf
n≥2
1−δ n
nγ1−β−α−β
1−α γ1−β Γn 1/n−1
. 3.1
Proof. Letfbelong toT.It is known20thatfis close-to-convex of orderδ, if it satisfies the condition
fz−1<1−δ. 3.2
For the left-hand side of3.2we have
fz−1≤∞ n2
n|an||z|n−1. 3.3
The last expression is less than 1−δif
∞
n2
n
1−δ|an||z|
n−1<1.
Using the fact thatf∈TSl
mα, β, γif and only if
∞
n2
nγ1−β−α−β
1−α γ1−β Γn|an|<1. 3.5
we can say that3.2is true if
n
1−δ|z| n−1≤
nγ1−β−α−β
1−α γ1−β Γn. 3.6
Or, equivalently,
|z| ≤
1−δnγ1−β−α−β n1−α γ1−β Γn
1/n−1
, 3.7
which completes the proof.
Theorem 3.2. Letf ∈TSl
mα, β, γ.Then the following are given.
1fis starlike of orderδ 0≤δ <1in the disc|z|< r2,where
r2inf
n≥2
1−δ
n−δ
nγ1−β−α−β
1−α γ1−β Γn 1/n−1
. 3.8
2fis convex of orderδ 0≤δ <1in the unit disc|z|< r3, where
r3 inf
n≥2
1−δ nn−δ
nγ1−β−α−β
1−α γ1−β Γn 1/n−1
. 3.9
Each of these results is sharp for the extremal functionfgiven by2.10.
Proof. Letf ∈T.It is known1thatfis starlike of orderδ, if it satisfies the condition
zffzz−1<1−δ. 3.10
For the left-hand side of3.10we have
zffzz −1≤
∞
n2n−1|an||z|n−1
1−∞n2|an||z|n−1
The last expression is less than 1−δif
∞
n2
n−δ
1−δ|an||z|
n−1<1.
3.12
Using the fact thatf∈TSl
mα, β, γif and only if ∞
n2
nγ1−β−α−β
1−α γ1−β Γn|an|<1. 3.13
we can say that3.10is true if
n−δ
1−δ|z| n−1 <
nγ1−β−α−β
1−α γ1−β Γn. 3.14
Or, equivalently,
|z|n−1< 1−δ
nγ1−β−α−β
n−δ1−α γ1−β Γn, 3.15
which yields the starlikeness of the family.
Using the fact thatf is convex if and only ifzfis starlike, we can prove2, on lines similar to those the proof of1.
4. Integral Means
In order to find the integral means inequality and to verify the Silverman Conjuncture21 forf∈TSl
mα, β, γwe need the following subordination result due to Littlewood22.
Lemma 4.1see22. If the functionsfandgare analytic inUwithg≺f, then
2π
0
greiθηdθ≤ 2π
0
freiθηdθ, η >0, zreiθ,0< r <1. 4.1
Applying Theorem 2.1 with the extremal function and Lemma 4.1, we prove the following theorem.
Theorem 4.2. Letη >0.Iff∈TSlmα, β, γand{Φα, β, γ, n}∞n2is nondecreasing sequence, then, forzreiθand 0< r <1,one has
2π
0
freiθηdθ≤ 2π
0
f2reiθ
η
dθ, 4.2
Proof. Letfof the form of1.2andf2z z−1−α |γ|1−βz2/Φα, β, γ,2, then we
must show that
2π
0
1−
∞
n2
anzn−1 η
dθ≤ 2π
0
1−
1−α γ1−β
Φα, β, γ,2 z
η
dθ. 4.3
ByLemma 4.1, it suffices to show that
1−
∞
n2
|an|zn−1≺1−
1−α γ1−β
Φα, β, γ,2 z. 4.4
Setting
1−
∞
n2
|an|zn−11−
1−α γ1−β
Φα, β, γ,2 wz, 4.5
from4.5and2.1we obtain
|wz|
∞
n2
Φα, β, γ, n
1−α γ1−βanz n−1
≤ |z|∞ n2
Φα, β, γ, n
1−α γ1−β|an|
≤ |z|<1.
4.6
This completes the proof ofTheorem 4.2.
5. Inclusion Relations Involving
N
δe
To study about the inclusion relations involvingNδewe need the following definitions due
to Goodman23and Ruscheweyh24. Then, δneighborhood of functionf ∈Tis given by
Nδ
f
g∈T :gz z− ∞
n2
|bn|zn, ∞
n2
n|an−bn| ≤δ
. 5.1
Particularly for the identity functionez z, we have
Nδe
g ∈T:gz z− ∞
n2
|bn|zn, ∞
n2
n|bn| ≤δ
Theorem 5.1. Let
δ 2
1−α γ1−β
2γ1−β−α−βΓ2
, 5.3
whereΓ2 α1· · ·α2/β1· · ·β2.ThenTSlmα, β, γ⊂Nδe.
Proof. Forf∈TSlmα, β, γ,Theorem 2.1yields
2γ1−β−α−βΓ2
∞
n2
|an| ≤1−α γ1−β
5.4
so that
∞
n2
|an| ≤
1−α γ1−β
2γ1−β−α−βΓ2
. 5.5
On the other hand, from2.1and5.5we have
1−βΓ2
∞
n2
n|an| ≤1−α γ1−β
α−β−γ1−βΓ2
∞
n2
|an|
≤1−α γ1−βα−β−γ1−β
×Γ2
1−α γ1−β
2γ1−β−α−βΓ2
≤
1−α γ1−β21−β
2γ1−β−α−β , ∞
n2
n|an| ≤
21−α γ1−β
2γ1−β−α−βΓ2
.
5.6
Now we determine the neighborhood for each of the class TSl
mα, β, γ which we
define as follows. A function f ∈ T is said to be in the class TSl
mα, β, γ if there exists a
functiong∈TSl
mα, β, γsuch that
fgzz−1<1−η, z∈U,0≤η <1. 5.7
Theorem 5.2. Ifg∈TSl
mα, β, γand
η1− δ
2γ1−β−α−βΓ2
22γ1−β−α−βΓ2−
1−α γ1−β, 5.8
Proof. Suppose thatf ∈Nδg, then we find from5.6that
∞
n2
|an−bn| ≤δ, 5.9
which implies the coefficient inequality
∞
n2
|an−bn| ≤ δ
2. 5.10
Next, sinceg∈TSl
mα, β, γ, we have ∞
n2
|bn| ≤
21−α γ1−β
2γ1−β−α−βΓ2
5.11
so that
fgzz−1< ∞
n2|an−bn|
1−∞n2|bn|
≤ δ
2 ×
2γ1−β−α−βΓ2
2γ1−β−α−βΓ2−
1−α γ1−β
≤1−η,
5.12
provided thatηis given precisely by5.8. Thus by definition,f ∈TSl
mα, β, γ, ηforηgiven
by5.8, which completes the proof.
Concluding Remarks
By suitably specializing the various parameters involved in Theorems 2.1 to 5.2, we can state the corresponding results for the new subclasses defined in Examples1.1to1.4and also for many relatively more familiar function classes.
Acknowledgment
The authors would like to record their sincerest thanks to the refereesfor their valuable suggestions and comments.
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