Volume 2008, Article ID 134932,12pages doi:10.1155/2008/134932
Research Article
A New Subclass of Analytic Functions Defined by
Generalized Ruscheweyh Differential Operator
Serap Bulut
Civil Aviation College, Kocaeli University, Arslanbey Campus, 41285 ˙Izmit-Kocaeli, Turkey
Correspondence should be addressed to Serap Bulut,[email protected]
Received 1 July 2008; Accepted 3 September 2008
Recommended by Narendra Kumar Govil
We investigate a new subclass of analytic functions in the open unit diskUwhich is defined by generalized Ruscheweyh differential operator. Coefficient inequalities, extreme points, and the integral means inequalities for the fractional derivatives of orderpη0≤ p ≤n, 0≤ η < 1 of functions belonging to this subclass are obtained.
Copyrightq2008 Serap Bulut. 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
Throughout this paper, we use the following notations:
N:{1,2,3, . . .},
N0:N∪ {0},
R−1:{u∈R:u >−1},
R0
−1:R−1\ {0}.
1.1
LetAdenote the class of all functions of the form
fz z
∞
n2
anzn, 1.2
which are analytic in the open unit diskU:{z∈C:|z|<1}. Forfj∈ Agiven by
fjz z
∞
n2
the Hadamard productor convolutionf1∗f2off1andf2is defined by
f1∗f2z z ∞
n2
an,1an,2zn. 1.4
Using the convolution1.4, Shaqsi and Darus1 introduced the generalization of the Ruscheweyh derivative as follows.
Forf∈ A, λ≥0, andu∈R−1, we consider
Ruλfz z
1−zu1∗Rλfz z∈U, 1.5
whereRλfz 1−λfz λzfz, z∈U.
Iff∈ Ais of the form1.2, then we obtain the power series expansion of the form
Ruλfz z
∞
n2
1 n−1λ Cu, nanzn, 1.6
where
Cu, n 1un−1
n−1! n∈N, 1.7
and where an is the Pochhammer symbol or shifted factorial defined in terms of the Gamma functionby
an: Γan
Γa
1, ifn0, a∈C\ {0},
aa1· · ·an−1, ifn∈N, a∈C. 1.8
In the casem∈N0, we have
Rmλfz zz
m−1fzm
m! , 1.9
and forλ0, we obtainuth Ruscheweyh derivative introduced in2 ,Rm 0 Rm.
Using the generalized Ruscheweyh derivative operatorRuλ, we define the following classes.
Definition 1.1. LetSλu, v;αbe the class of functionsf∈ Asatisfying
Re Ru
λfz
Rv λfz
> α 1.10
for some 0≤α <1, u∈R0
In this paper, basic properties of the classSλu, v;αare studied, such as coefficient
bounds, extreme points, and integral means inequalities for the fractional derivative.
2. Coefficient inequalities
Theorem 2.1. Let0≤α <1, u∈R0−1, v∈R−1, andλ≥0. Iff ∈ Asatisfies
∞
n2
Bnu, v, α|an| ≤21−α, 2.1
where
Bnu, v, α: 1 n−1λ {|Cu, n−1αCv, n|Cu, n 1−αCv, n}, 2.2
thenf ∈ Sλu, v;α.
Proof. Let2.1be true for 0 ≤ α < 1, u ∈ R0−1, v ∈ R−1, andλ ≥ 0. Forf ∈ A, define the functionFby
Fz: R
u λfz
Rvλfz−α. 2.3
It is sufficient to show that
FFzz−11<1 2.4
forz∈U.
So, we have
FFzz−11R u
λfz−1αRvλfz
Ru
λfz 1−αRvλfz
α− ∞
n21 n−1λ Cu, n−1αCv, n anzn−1 2−α ∞n21 n−1λ Cu, n 1−αCv, n anzn−1
≤ α
∞
n21 n−1λ |Cu, n−1αCv, n||an||z|n−1 2−α−∞n21 n−1λ Cu, n 1−αCv, n |an||z|n−1
< α
∞
n21 n−1λ |Cu, n−1αCv, n||an| 2−α−∞n21 n−1λ Cu, n 1−αCv, n |an|
<1 by2.1.
2.5
Theorem 2.2. Iff ∈ Sλu, v;α, then
|an| ≤ 21−
α
1 n−1λ |Cu, n−Cv, n|
n−1
u1
1 n−u−1λ Cv, n−u|an−u| 2.6
forn≥2, witha11.
Proof. Define the function
Gz: 1
1−α
Ru λfz
Rvλfz −α :1
∞
n1
anzn. 2.7
Since Re{Gz}>0, we get
|an| ≤2 2.8
forn1,2, . . ..
From the definition ofGz, we obtain
Ru
λfz−αRvλfz
1−α R
vfz
1 ∞
n1
anzn
. 2.9
So, by1.6, we have
z 1λ
1−αCu,2−αCv,2 a2z
2 12λ
1−αCu,3−αCv,3 a3z
3· · ·
za1z2a2z3a3z4· · ·
1λCv,2a2z2 1λCv,2a2a1z3 1λCv,2a2a2z4· · · 12λCv,3a3z3 12λCv,3a3a1z4· · ·
2.10
or
z 1λ
1−αCu,2−Cv,2 a2z
212λ
1−αCu,3−Cv,3 a3z
3· · ·
za1z2 1λCv,2a2a1a2 z3
12λCv,3a3a1 1λCv,2a2a2a3 z4· · ·
or, equivalently,
z
∞
n2
1 n−1λ
1−α Cu, j−Cv, j anz
n
z
∞
n2 n−1
u1
1 n−u−1λ Cv, n−uan−uau
zn.
2.12
When we consider the coefficients ofznof both series in the above equality, we have
an
1−α
1 n−1λ Cu, n−Cv, n
n−1
u1
1 n−u−1λ Cv, n−uan−uau. 2.13
Therefore,
|an| ≤
1−α
1 n−1λ |Cu, n−Cv, n|
n−1
u1
1 n−u−1λ Cv, n−u|an−u||au|
≤ 1 21−α
n−1λ |Cu, n−Cv, n|
n−1
u1
1 n−u−1λ Cv, n−u|an−u|,
2.14
since|au| ≤2, u1,2, . . ..
3. Extreme points
Definition 3.1. LetSλu, v;αbe the subclass ofSλu, v;αwhich consists of function
fz z
∞
n2
anzn an≥0 3.1
whose coefficients satisfy inequality2.1.
Theorem 3.2. Letf1z zand
fkz z 21−α
Bku, v, αz
k k2,3, . . ., 3.2
Thenf ∈Sλu, v;αif and only if it can be expressed in the form
fz
∞
k1
δkfkz, 3.3
whereδk≥0and∞k1δk1.
Proof. Assume that
fz
∞
k1
δkfkz. 3.4
Then
fz δ1f1z ∞
k2
δkfkz
δ1z ∞
k2
δk
z 21−α
Bku, v, αz
k
∞
k1
δk
z
∞
k2
δk 21−
α
Bku, v, αz
k
z
∞
k2
δk 21−
α
Bku, v, αz
k.
3.5
Thus
∞
k2
δk 21−
α
Bku, v, αBku, v, α 21−α
∞
k2
δk21−α1−δ1≤21−α. 3.6
Therefore, we havef∈Sλu, v;α.
Conversely, suppose thatf∈Sλu, v;α. Since
ak≤ B21k −α
u, v, α k2,3, . . ., 3.7
we can set
δk: B21ku, v, α−
α ak k2,3, . . .,
δ1 :1− ∞
k2
δk.
Then
fz z
∞
k2
akzk
∞
k1
δk
z
∞
k2
δk 21−
α
Bku, v, αz
k
δ1z ∞
k2
δk
z 21−α
Bku, v, αz
k
δ1f1z ∞
k2
δkfkz
∞ k1
δkfkz.
3.9
This completes the proof ofTheorem 3.2.
Corollary 3.3. The extreme points ofSλu, v;αare given by
f1z z, fkz zB21k −α
u, v, αz
k k2,3, . . ., 3.10
whereBku, v, αis given by2.2.
4. The main integral means inequalities for the fractional derivative
We discuss the integral means inequalities for functionsf∈Sλu, v;α.
The following definitions of fractional derivatives by Owa3 also by Srivastava and Owa4 will be required in our investigation.
Definition 4.1. The fractional derivative of orderηis defined, for a functionf, by
Dηzfz
1 Γ1−η
d dz
z
0
fξ
z−ξηdξ 0≤η <1, 4.1
where the function f is analytic in a simply connected region of the complex z-plane containing the origin, and the multiplicity ofz−ξ−η is removed by requiring logz−ξ
to be real whenz−ξ >0.
Definition 4.2. Under the hypothesis ofDefinition 4.1, the fractional derivative of orderpη
is defined, for a functionf, by
Dpzηfz
dp dzpD
η
zfz, 4.2
It readily follows from4.1inDefinition 4.1that
Dzηzk Γ
k1
Γk1−ηz
k−η 0≤η <1, k∈N. 4.3
We will also need the concept of subordination between analytic functions and a subordination theorem of Littlewood5 in our investigation.
Definition 4.3. Given two functionsfandg, which are analytic inU, the functionfis said to be subordinate toginUif there exists a functionwanalytic inUwith
w0 0, |wz|<1 z∈U, 4.4
such that
fz gwz z∈U. 4.5
We denote this subordination by
fz≺gz. 4.6
Lemma 4.4. If the functionsfandgare analytic inUwith
fz≺gz, 4.7
then, forμ >0andzreiθ 0< r <1, 2π
0
|fz|μdθ≤
2π
0
|gz|μdθ. 4.8
Our main theorem is contained in the following.
Theorem 4.5. Letf ∈Sλu, v;αand suppose that
∞
n2
n−pp1an≤ 21−
αΓk1Γ3−η−p
Bku, v, αΓk1−η−pΓ2−p 4.9
for0≤p≤n, k≥p, 0≤η <1, wheren−pp1denotes the Pochhammer symbol defined by
n−pp1 n−pn−p1· · ·n. 4.10
Also let the functionfkbe defined by
fkz z 21−α
Bku, v, αz
If there exists an analytic functionwdefined by
wzk−1: Bku, v, αΓk1−η−p
21−αΓk1
∞
n2
n−pp1Ψnanzn−1 4.12
with
Ψn Γn−p
Γn1−η−p, 0≤η <1, n2,3, . . ., 4.13
then, forμ >0andzreiθ 0< r <1,
2π
0 Dpη
z fz μ
dθ≤
2π
0 Dpη
z fkz μ
dθ, 0≤η <1. 4.14
Proof. By means of4.3andDefinition 4.2, we find from3.1that
Dpzηfz
z1−η−p
Γ2−η−p
1
∞
n2
Γ2−η−pΓn1
Γn1−η−p anz
n−1
Γ2z−1−η−p
η−p
1
∞
n2
Γ2−η−pn−pp1Ψnanzn−1
,
4.15
where
Ψn Γn−p
Γn1−η−p, 0≤η <1, n2,3, . . .. 4.16
SinceΨis a decreasing function ofn, we get
0<Ψn≤Ψ2 Γ2−p
Γ3−η−p. 4.17
Similarly, from4.11,4.3, andDefinition 4.2, we have
Dpzηfkz
z1−η−p Γ2−η−p
1 21−α Bku, v, α
Γ2−η−pΓk1
Γk1−η−p z
Forμ >0 andzreiθ 0< r <1, we want to show that 2π 0 1 ∞
n2
Γ2−η−pn−pp1Ψnanzn−1 μ dθ ≤ 2π 0
1 21−α Bku, v, α
Γ2−η−pΓk1
Γk1−η−p z
k−1μdθ.
4.19
So, by applyingLemma 4.4, it is enough to show that
1 ∞
n2
Γ2−η−pn−pp1Ψnanzn−1≺1 21−
α
Bku, v, α
Γ2−η−pΓk1
Γk1−η−p z
k−1.
4.20
If the above subordination holds true, then we have an analytic functionwwithw0 0 and |wz|<1 such that
1 ∞
n2
Γ2−η−pn−pp1Ψnanzn−1 1B21k −α
u, v, α
Γ2−η−pΓk1
Γk1−η−p wz
k−1.
4.21
By the condition of the theorem, we define the functionwby
wzk−1 Bku, v, αΓk1−η−p
21−αΓk1
∞
n2
n−pp1Ψnanzn−1, 4.22
which readily yieldsw0 0. For such a functionw, we have
|wz|k−1≤ Bku, v, αΓk1−η−p
21−αΓk1
∞
n2
n−pp1Ψnan|z|n−1
≤ |z|Bku, v, αΓk1−η−p
21−αΓk1 Ψ2
∞
n2
n−pp1an
|z|Bku, v, αΓk1−η−p
21−αΓk1
Γ2−p
Γ3−η−p
∞
n2
n−pp1an
≤ |z|<1
4.23
by means of the hypothesis of the theorem. Thus the theorem is proved.
Corollary 4.6. Letf ∈Sλu, v;αand suppose that
∞
n2
nan≤ 21−
αΓk1Γ3−η
Bku, v, αΓk1−η k2,3, . . .. 4.24
If there exists an analytic functionwdefined by
wzk−1 Bku, v, αΓk1−η
21−αΓk1
∞
n2
nΨnanzn−1 4.25
with
Ψn Γn
Γn1−η, 0≤η <1, n2,3, . . ., 4.26
then, forμ >0andzreiθ 0< r <1,
2π
0 Dη
zfz μ
dθ≤
2π
0 Dη
zfkz μ
dθ, 0≤η <1. 4.27
Lettingp1 inTheorem 4.5, we have the following.
Corollary 4.7. Letf ∈Sλu, v;αand suppose that
∞
n2
nn−1an≤ 21−
αΓk1Γ2−η
Bku, v, αΓk−η k2,3, . . .. 4.28
If there exists an analytic functionwdefined by
wzk−1 Bku, v, αΓk−η
21−αΓk1
∞
n2
nn−1Ψnanzn−1 4.29
with
Ψn Γn−1
Γn−η, 0≤η <1, n2,3, . . ., 4.30
then, forμ >0andzreiθ 0< r <1,
2π
0
D1zηfz μ
dθ≤
2π
0
D1zηfkz μ
References
1 K. A. Shaqsi and M. Darus, “On univalent functions with respect toK-symmetric points given by a generalised Ruscheweyh derivatives operator,” submitted.
2 S. Ruscheweyh, “New criteria for univalent functions,”Proceedings of the American Mathematical Society, vol. 49, no. 1, pp. 109–115, 1975.
3 S. Owa, “On the distortion theorems. I,”Kyungpook Mathematical Journal, vol. 18, no. 1, pp. 53–59, 1978. 4 H. M. Srivastava and S. Owa, Eds.,Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series in Mathematics and Its Applications, Ellis Horwood, Chichester, UK; John Wiley & Sons, New York, NY, USA, 1989.
