MATEMATIQKI VESNIK
originalni nauqni rad
research paper
62, 4 (2010), 271–283
December 2010
A GENERALIZED CLASS OF STARLIKE FUNCTIONS ASSOCIATED
WITH THE WRIGHT HYPERGEOMETRIC FUNCTION
J. Dziok and G. Murugusundaramoorthy
Abstract. In terms of Wright’s generalized hypergeometric function we define some classes
of analytic functions. The class generalize well known classes of starlike functions. Necessary
and sufficient coefficient bounds are given for functions in this class. Further distortion bounds,
extreme points and results on partial sums are investigated.
1. Introduction
Let A denote the class of functions of the form
∞
P
f (z) = z +
an z n
(1)
n=2
which are analytic in the open unit disc U = {z : |z| < 1}. We denote by S the
subclass of A consisting functions f which are univalent in U . Also we denote by T ,
the class of analytic functions with negative coefficients introduced by Silverman
[14] consisting of functions f of the form
∞
P
f (z) = z −
an z n (an ≥ 0, n = 2, 3, . . . ; z ∈ U ).
(2)
n=2
For functions f ∈ A given by (1) and g(z) ∈ A given by
∞
P
g(z) = z +
bn z n , z ∈ U,
n=2
we define the Hadamard product (or convolution) of f and g by
∞
P
(f ∗ g)(z) = z +
an bn z n , z ∈ U.
n=2
Definition 1. Let k ≥ 0, 0 ≤ γ < 1. A function f ∈ A is said to be in the
class S(k, γ) if it satisfies the condition
¾
½ 0
z 2 f 00 (z)
zf (z)
+k
> γ, z ∈ U.
Re
f (z)
f (z)
2010 AMS Subject Classification: 30C45.
Keywords and phrases: Analytic functions; starlike functions; coefficient estimates.
271
272
J. Dziok, G. Murugusundaramoorthy
The class S(k, γ) was studied in [8] (see also [6, 9, 11]). In particular the class
S(γ) := S(0, γ)
is the well-known class of starlike functions.
For positive real parameters α1 , A1 . . . , αp , Ap and β1 , B1 . . . , βq , Bq (p, q ∈
N = 1, 2, 3, . . . ) such that
1+
q
P
n=1
Bn −
p
P
n=1
An ≥ 0,
(3)
the Wright’s generalized hypergeometric function [20]
p Ψq [(α1 , A1 ), . . . , (αp , Ap ); (β1 , B1 ), . . . , (βq , Bq ); z]
= p Ψq [(αn , An )1,p ; (βn , Bn )1,q ; z]
is defined by
p Ψq [(αt , At )1,p ; (βt , Bt )1,q ; z]
=
∞ Q
P
Qq
zn
p
{ t=0 Γ (αt + nAt )}{ t=0 Γ (βt + nBt )}−1 ,
n!
n=0
z ∈ U . If p ≤ q + 1, At = 1(t = 1, . . . , p) and Bt = 1(t = 1, . . . , q), we have the
relationship:
Θ p Ψq [(αn , 1)1,p ; (βn , 1)1,q ; z] = p Fq (α1 , . . . , αp ; β1 , . . . , βq ; z),
z ∈ U,
(4)
where p Fq (α1 , . . . , αp ; β1 , . . . , βq ; z) is the generalized hypergeometric function and
Qp
−1 Qq
Θ = ( t=0 Γ(αt )) ( t=0 Γ(βt )) .
(5)
In [2] Dziok and Raina defined the linear operator by using Wright’s generalized
hypergeometric function. Let
p φq [(αt , At )1,p ; (βt , Bt )1,q ; z]
= Θz p Ψq [(αt , At )1,p (βt , Bt )1,q ; z],
z ∈ U,
and
W = W[(αn , An )1,p ; (βn , Bn )1,q ] : A → A
be a linear operator defined by
Wf (z) := z p φq [(αt , At )1,p ; (βt , Bt )1,q ; z] ∗ f (z),
z ∈ U.
We observe that, for f of the form (1), we have
Wf (z) = z +
∞
P
n=2
where
σn =
σn an z n ,
z ∈ U,
(6)
Θ Γ(α1 + A1 (n − 1)) . . . Γ(αp + Ap (n − 1))
,
(n − 1)!Γ(β1 + B1 (n − 1)) . . . Γ(βq + Bq (n − 1))
and Θ is given by (5). In view of the relationship (4) the linear operator (6) includes
the Dziok-Srivastava operator [3] (see [4]).
273
A generalized class of starlike functions
Corresponding to the family S(γ, k) we define the class Wqp (k, γ) of function
f of the form (1) such that
½
¾
z(Wf (z))0
z 2 (Wf (z))00
Re
+k
> γ, z ∈ U.
(7)
Wf (z)
Wf (z)
We also let
T W pq (k, γ) = T ∩ Wqp (k, γ).
Further we define some subclasses of the class Wqp (k, γ) as given below.
Suppose that: If At = 1 (t = 1, . . . , p) and Bt = 1(t = 1, . . . , q), p = 2 and
q = 1 we have
(i) α1 = δ + 1 (δ > −1), α2 = 1, β1 = 1, then
W(δ + 1, 1; 1)f (z) ≡ Dδ f (z) :=
z
∗ f (z)
(1 − z)δ+1
is called Ruscheweyh derivative of order δ (δ > −1)(see [12]). A function f ∈ A is
in RS(γ, k) if
¾
½
z 2 (Dδ f (z))00
z(Dδ f (z))0
+
k
> γ, z ∈ U,
(8)
Re
Dδ f (z)
Dδ f (z)
(ii) α1 = a (a > 0), α2 = 1, β1 = c (c > 0),
µ∞
P
W(a, 1; c)f (z) ≡ L(a, c)f (z) :=
n=0
¶
(a)n n+1
(c)n z
∗ f (z)
called Carlson and Shaffer operator [1]. A function f ∈ A is in LS(γ, k) if
½
¾
z(L(a, c)f (z))0
z 2 (L(a, c)f (z))00
Re
+k
> γ, z ∈ U,
L(a, c)f (z)
L(a, c)f (z)
(9)
(iii) α1 = 2 α2 = 1, β1 = 2 − η,
W(2, 1; 2 − η)f (z) ≡ Ωη f (z) = Γ(2 − η)z η Dzη f (z)
where (η ∈ R; η 6= 2, 3, 4, . . . ) the operator Ωη f (z) was introduced by Owa and
Srivastava [19]. A function f ∈ A is in GS(γ, k) if
½
¾
z(Ωη f (z))0
z 2 (Ωη f (z))00
Re
+k
> γ, z ∈ U.
(10)
Ωη f (z)
Ωη f (z)
In this paper we obtain a sufficient coefficient condition for functions f given
by (1) to be in the class Wqp (k, γ) and we show that it is also necessary condition
for functions to belong to the class. Distortion results and extreme points for
functions in T W pq (k, γ) are obtained. Finally, we investigate partial sums for the
class T W pq (k, γ).
274
J. Dziok, G. Murugusundaramoorthy
2. Coefficients inequalities and distortions theorem
First we obtain a sufficient condition for functions in Wqp (k, γ).
Theorem 1. Let 0 ≤ γ < 1 and k ≥ 0. Suppose also that f (z) ∈ A is given
by (1). If
∞ £
¤
P
kn2 + n − kn − γ σn |an | ≤ 1 − γ,
(11)
n=2
then f ∈
Wqp (k, γ).
Proof. If we put
z 2 (Wf (z))00
z(Wf (z))0
+k
,
Wf (z)
Wf (z)
then it is sufficient to prove that
P (z) =
z ∈ U,
|P (z) − 1| < 1 − γ, z ∈ U.
Indeed if f (z) ≡ z (z ∈ U ), then we have P (z) ≡ 1 (z ∈ U ). This implies that the
desired in equality (11). If f (z) 6= z (|z| = r < 1), then there exist a coefficient
σn an 6= 0 for some n ≥ 2. It follows that
∞
P
σn |an | rn > 0.
n=2
Further note that the sequence bn = kn2 + n − kn − γ is increasing. Therefore
∞ £
∞
¤
P
P
kn2 + n − kn − γ σn |an |rn > (1 − γ)
σn |an |rn ,
n=2
n=2
which implies that
∞
P
n=2
σn |an | < 1.
Thus by coefficient inequality (11), we obtain
¯
¯ ∞
∞
P
¯
¯P
¯
(n − 1)(nk + 1)σn |an |
(n − 1)(nk + 1)σn an z n−1 ¯
¯ n=2
¯ n=2
¯<
|P (z) − 1| = ¯¯
∞
∞
¯
P
P
¯
¯
1+
σn an z n−1
1−
σn |an |
¯
¯
n=2
∞
P
=
n=2
(n − 1)(nk + 1)σn |an |
n=2
1−
∞
P
n=2
σn |an |
∞ £
∞
¤
P
P
kn2 + n − kn − γ σn |an | − (1 − γ)
σn |an |
<
n=2
1−
∞
P
n=2
(1 − γ) − (1 − γ)
≤
1−
∞
P
n=2
∞
P
n=2
σn |an |
n=2
σn |an |
σn |an |
≤ 1 − γ,
z ∈ U.
275
A generalized class of starlike functions
Hence we obtain
½
¾
z(Wf (z))0
z 2 (Wf (z))00
Re
= Re(P (z)) > 1 − (1 − γ) = γ
+k
Wf (z)
Wf (z)
(z ∈ U ).
That is f ∈ Wqp (k, γ). This completes the proof.
In the next theorem, we show that the condition (11) is also necessary for
functions from the class T W pq (k, γ).
Theorem 2. Let f be given by (2). Then the function f belongs to the class
T W pq (k, γ) if and only if
∞ £
¤
P
kn2 + n − kn − γ σn an ≤ 1 − γ.
(12)
n=2
Proof. In view of Theorem 1 we need only to show that f ∈ T W pq (k, γ) satisfies
the coefficient inequality (11). If f ∈ T W pq (k, γ) then the function
P (z) =
z(Wf (z))0 + kz 2 (Wf (z))00
Wf (z)
(z ∈ U )
satisfies
Re{P (z)} > γ
(z ∈ U ).
This implies that
Wf (z) = z −
∞
P
n=2
σn an z n 6= 0
(z ∈ U \ {0}).
Noting that Wfr(r) is the real continuous function in the open interval (0, 1)
with f (0) = 1, we have
1−
∞
P
n=2
Now
1−
γ < P (r) =
σn an rn−1 > 0
∞
P
n=2
(0 < r < 1).
nσn an rn−1 − k
1−
∞
P
n=2
(13)
∞ ¡
¢
P
n2 − n σn an rn−1
n=2
σn an rn−1
and consequently by (13) we obtain
∞ £
¤
P
kn2 + n − kn − γ σn an rn−1 < 1 − γ.
n=2
Letting r → 1, we get (12). This proves the converse part.
then
Corollary 1. If a function f of the form (2) belongs to the class T W pq (k, γ),
an ≤
1−γ
,
[kn2 + n − kn − γ] σn
n = 2, 3, . . . .
276
J. Dziok, G. Murugusundaramoorthy
The equality holds for the functions
1−γ
hn (z) = z −
zn,
[kn2 + n − kn − γ] σn
z ∈ U, n = 2, 3, . . . .
(14)
Next we obtain the distortion bounds for functions belonging to the class
T W pq (k, γ).
Theorem 3. Let f be in the class T W pq (k, γ), |z| = r < 1. If the sequence
©£ 2
¤ ª∞
kn + n − kn − γ σn n=2
is nondecreasing, then
1−γ
1−γ
r−
r2 ≤ |f (z)| ≤ r +
r2 .
(kγ + 2)σ2
(2k − γ + 2)σ2
o∞
n 2
σ
is nondecreasing, then
If the sequence kn +n−kn−γ
n
n
(15)
n=2
2(1 − γ)
2(1 − γ)
1−
r ≤ |f 0 (z)| ≤ 1 +
r.
(16)
(2k − γ + 2)σ2
(2k − γ + 2)σ2
The result is sharp. The extremal function is the function h2 of the form (14).
Proof. Since f ∈ T W pq (k, γ), we apply Theorem 2 to obtain
∞
∞ £
¤
P
P
(2k − γ + 2)σ2
|an | ≤
kn2 + n − kn − γ σn |an | ≤ 1 − γ.
n=2
n=2
Thus
|f (z)| ≤ |z| + |z|2
∞
P
n=2
Also we have
∞
P
|an | ≤ r +
1−γ
r2 .
(2k − γ + 2)σ2
1−γ
r2
(2k − γ + 2)σ2
n=2
and (15) follows. In similar manner for f 0 , the inequalities
∞
∞
P
P
|f 0 (z)| ≤ 1 +
n|an ||z|n−1 ≤ 1 + |z|
nan
|f (z)| ≥ |z| − |z|2
|an | ≥ r −
n=2
and
∞
P
n=2
n|an | ≤
n=2
2(1 − γ)
(2k − γ + 2)σ2
lead to (16). This completes the proof.
Corollary 2. Let f be in the class T W pq (k, γ), |z| = r < 1. If
p > q, αq+1 ≥ 1, αj ≥ βj and Aj ≥ Bj ( j = 2, . . . , q),
then the assertions (15), (16) holds true.
Proof. From (17) we have that the sequences
¾∞
½ 2
©£ 2
¤ ª∞
kn + n − kn − γ
σn
kn + n − kn − γ σn n=2 and
n
n=2
are nondecreasing. Thus, by Theorem 3, we have Corollary 2.
(17)
A generalized class of starlike functions
277
Theorem 4. Let h1 (z) = z and hn be defined by (14). Then f ∈ T W pq (k, γ)
if and only if f can be expressed in the form
∞
∞
P
P
f (z) =
(18)
µn hn (z), µn ≥ 0 and
µn = 1.
n=1
n=1
Proof. If a function f is of the form (18), then
∞ £
¤
P
1−γ
kn2 + n − kn − γ σn
µn
f (z) =
2
[kn + n − kn − γ] σn
n=2
∞
P
=
µn (1 − γ) = (1 − µ1 )(1 − γ) ≤ 1 − γ.
n=2
Hence f ∈ T
W pq (k, γ).
[kn2 +n−kn−γ ]σn
Conversely, if f is in the class T W pq (k, γ), then we may set µn =
,
1−γ
∞
P
n ≥ 2 and µ1 = 1 −
µn . Then the function f is of the form (18) and this completes the proof.
n=2
3. Partial sums
For a function f ∈ A given by (1) Silverman [15] and Silvia [18] investigated
the partial sums f1 and fm defined by
m
P
f1 (z) = z; and fm (z) = z +
an z n , m = 2, 3 . . . .
(19)
n=2
We consider in this section partial sums of functions in the class T W pq (k, γ) and
0
obtain sharp lower bounds for the ratios of real part of f to fm and f 0 to fm
.
Theorem 5. Let a function f of the form (2) belong to the class T W pq (k, γ)
and assume (17). Then
½
¾
f (z)
1
Re
≥1−
, z ∈ U, m ∈ N
(20)
fm (z)
dm+1
and
½
Re
fm (z)
f (z)
¾
≥
where
dn :=
dm+1
,
1 + dm+1
z ∈ U, m ∈ N,
kn2 + n − kn − γ
σn .
1−γ
(21)
(22)
Proof. By (17) it is not difficult to verify that
dn+1 > dn > 1,
Thus by Theorem 2 we have
m
P
an + dm+1
n=2
∞
P
n=m+1
n = 2, 3 . . . .
an ≤
∞
P
n=2
dn an ≤ 1.
(23)
(24)
278
J. Dziok, G. Murugusundaramoorthy
Setting
½
g(z) = dm+1
µ
f (z)
1
− 1−
fm (z)
dm+1
dm+1
¶¾
=1+
∞
P
n=m+1
m
P
1+
n=2
an z n−1
,
(25)
an z n−1
it suffices to show that Re g(z) ≥ 0, z ∈ U . Applying (24), we find that
¯
¯
¯ g(z) − 1 ¯
¯
¯
¯ g(z) + 1 ¯ ≤
dm+1
n
P
2−2
n=2
∞
P
n=m+1
|an |
|an | − dm+1
≤ 1, z ∈ U,
∞
P
n=m+1
|an |
which readily yields the assertion (20) of Theorem 5. In order to see that
f (z) = z +
z m+1
,
dm+1
z ∈ U,
(26)
gives the sharp result, we observe that for z = reiπ/m we have
f (z)
z m z→1−
1
=1+
−→ 1 −
.
fm (z)
dm+1
dm+1
Similarly, if we take
½
¾
fm (z)
dm+1
−
f (z)
1 + dm+1
∞
P
(1 + dn+1 )
an z n−1
n=m+1
, z ∈ U,
=1−
∞
P
1+
an z n−1
h(z) = (1 + dm+1 )
n=2
and making use of (24), we can deduce that
¯
¯
¯ h(z) − 1 ¯
¯
¯
¯ h(z) + 1 ¯ ≤
(1 + dm+1 )
2−2
m
P
n=2
∞
P
n=m+1
|an |
|an | − (1 − dm+1 )
∞
P
n=m+1
≤ 1, z ∈ U,
|an |
which leads us immediately to the assertion (21) of Theorem 5. The bound in (21)
is sharp for each m ∈ N with the extremal function f given by (26), and the proof
is complete.
Theorem 6. Let a function f of the form (1) belong to the class T W pq (k, γ)
and assume (17). Then
¾
½ 0
m+1
f (z)
≥1−
(27)
Re
0
fm (z)
dm+1
279
A generalized class of starlike functions
and
½
Re
0
fm
(z)
f 0 (z)
¾
≥
dm+1
,
m + 1 + dm+1
(28)
where dm is defined by (22).
Proof. By setting
½
g(z) = dm+1
¶¾
µ
f 0 (z)
m+1
− 1−
,
0 (z)
fm
dm+1
and
½
h(z) = [(m + 1) + dm+1 ]
z ∈ U,
0
dm+1
fm
(z)
−
0
f (z)
m + 1 + dm+1
¾
,
z ∈ U,
the proof is analogous to that of Theorem 5, and we omit the details.
4. Integral transform and integral means
First we prove that the class T W pq (k, γ) is closed under some integral transforms.
For f ∈ A we define the Komatu operator [10]
µ
¶δ−1
Z
1
(c + 1)δ 1 c−1
t
log
f (tz)dt, c > −1, δ ≥ 0.
Vδ,c (f )(z) =
Γ (δ)
t
0
In particular, for δ = 1 we obtain well-known the Bernardi operator.
A simple calculation gives
µ
¶δ
∞
P
c+1
Vδ,c (f )(z) = z −
an z n .
n=2 c + n
First we show that the class T W pq (k, γ) is closed under Vδ,c (f )(z).
Theorem 7. Let f (z) ∈ T W pq (k, γ). Then Vδ,c (f )(z) ∈ T W pq (k, γ).
Proof. We need to prove that
where
´δ
∞ Φ(λ, γ, k, n) ³
P
c+1
an ≤ 1,
c+n
(1 − γ)
n=2
(29)
Φ(γ, k, n) = [kn2 + n − kn − γ]σn .
(30)
On the other hand by Theorem 2, f ∈ T
W pq (k, γ)
if and only if
∞ Φ(λ, γ, k, n)
P
an ≤ 1,
(1 − γ)
n=2
where Φ(γ, k, n) is defined in (30). Since
complete.
c+1
c+n
< 1, then (29) holds and the proof is
280
J. Dziok, G. Murugusundaramoorthy
Theorem 8. Let f ∈ T W pq (k, γ). Then Vδ,c (f )(z) is starlike of order ξ,
0 ≤ ξ < 1 in |z| < R1 , where
1
"µ
# n−1
¶δ
c+n
(1 − ξ)Φ(γ, k, n)
R1 = inf
(n ≥ 2),
n
c+1
(n − ξ)(1 − γ)
where Φ(γ, k, n) is defined in (30).
Proof. It is sufficient to prove
¯
¯
¯
¯ z(Vδ,c (f )(z))0
¯
¯
¯ Vδ,c (f )(z) − 1¯ < 1 − ξ,
|z| < R1 .
(31)
For the left hand side of (30) we have,
¯
¯
µ
¶δ
¯
¯P
∞
c+1
¯
¯
n−1
¯
¯ ¯
(1 − n)
an z
¯
¯ z(Vδ,c (f )(z))0
¯ ¯ n=2
c
+
n
¯
¯
¯
¯
¶
µ
¯ Vδ,c (f )(z) − 1¯ = ¯¯
δ
∞
¯
P
c+1
n−1 ¯
¯ 1−
a
z
n
¯
¯
n=2 c + n
³
´
∞
δ
P
c+1
(1 − n) c+n
an |z|n−1
n=2
≤
.
¶δ
µ
∞
P
c+1
n−1
1−
an |z|
n=2 c + n
The last expression is less than 1 − ξ if
µ
¶δ
c+n
(1 − ξ)Φ(γ, k, n)
|z|n−1 <
.
c+1
(n − ξ)(1 − γ)
Therefore, the proof is complete.
Using the fact that f (z) is convex if and only if zf 0 (z) is starlike, we obtain
the following.
Theorem 9. Let f ∈ T W pq (k, γ). Then Vδ,c (f )(z) is convex of order 0 ≤ ξ < 1
in |z| < R2 where
1
# n−1
"µ
¶δ
(1 − ξ)Φ(γ, k, n)
c+n
R2 = inf
(n ≥ 2),
n
c+1
n(n − ξ)(1 − γ)
where Φ(λ, γ, k, 2) is defined in (30).
2
In [14], Silverman found that the function f2 (z) = z − z2 is often extremal
over the family T . He applied this function to resolve his integral means inequality,
conjectured in [16] and settled in [17], that
Z 2π
Z 2π
¯
¯
¯
¯
¯f (reiθ )¯η dθ ≤
¯f2 (reiθ )¯η dθ,
0
0
A generalized class of starlike functions
281
for all f ∈ T , η > 0 and 0 < r < 1. In [17], he also proved his conjecture for the
subclasses T ∗ (γ) and C(γ) of T .
Now, we prove the Silverman’s conjecture for the functions in the family
T W pq (k, γ).
Lemma 1. [7] If the functions f and g are analytic in U with g ≺ f , then for
η > 0, and 0 < r < 1,
Z 2π
Z 2π
¯
¯
¯
¯
¯g(reiθ )¯η dθ ≤
¯f (reiθ )¯η dθ.
(32)
0
0
Applying Lemma 1, Theorem 2 and Theorem 4 we prove the following result.
by
Theorem 10. Suppose f ∈ T W pq (k, γ), η > 0, 0 ≤ λ ≤ 1 and f2 (z) is defined
f2 (z) = z −
1−γ
z2,
Φ(γ, k, 2)
where Φ(γ, k, 2) is defined in (30). Then for z = reiθ , 0 < r < 1, we have
Z 2π
Z 2π
η
η
|f (z)| dθ ≤
|f2 (z)| dθ.
0
Proof. By (2), it is equivalent to prove that
¯η
Z 2π ¯
Z 2π ¯
∞
¯
¯
¯
P
n−1 ¯
¯1 −
¯1 −
an z
¯
¯ dθ ≤
¯
0
(33)
0
n=2
0
¯η
¯
(1 − γ)
z ¯ dθ.
Φ(λ, γ, k, 2) ¯
By Lemma 1, it suffices to show that
1−
∞
P
n=2
Setting
∞
P
1−
n=2
an z n−1 ≺ 1 −
an z n−1 = 1 −
1−γ
z.
Φ(λ, γ, k, 2)
1−γ
w(z),
Φ(λ, γ, k, 2)
(34)
and using (11), we obtain
¯
¯
∞
∞ Φ(γ, k, n)
¯P
¯
P
Φ(γ,k,n)
n−1 ¯
|w(z)| = ¯¯
a
z
an ≤ |z|.
n
1−γ
¯ ≤ |z|
1−γ
n=2
n=2
This completes the proof.
5. Neighbourhood results
In this section we discuss neighbourhood results of the class T W pq (k, γ). Following [5, 13], we define the δ-neighbourhood of function f (z) ∈ T by
½
¾
∞
∞
P
P
Nδ (f ) := h ∈ T : h(z) = z −
dn z n and
n|an − dn | ≤ δ .
(35)
n=2
n=2
282
J. Dziok, G. Murugusundaramoorthy
Particulary for the identity function e(z) = z, we have
½
¾
∞
∞
P
P
Nδ (e) := h ∈ T : g(z) = z −
dn z n and
n|dn | ≤ δ .
n=2
Theorem 11. If
δ :=
(36)
n=2
2(1 − γ)
,
Φ(λ, γ, k, 2)
(37)
where Φ(λ, γ, k, 2) is defined in (30), then T W pq (k, γ) ⊂ Nδ (e).
Proof. For f ∈ T W pq (k, γ), Theorem 2 immediately yields Φ(λ, γ, k, 2)
1 − γ, so that
∞
P
n=2
an ≤
(1 − γ)
.
Φ(λ, γ, k, 2)
∞
P
n=2
an ≤
(38)
On the other hand, from (11) and (38) that
σ2 (1 + λ)
∞
P
n=2
nan ≤ (1 − γ) − (1 + λ)(k − γ)σ2
∞
P
n=2
≤ (1 − γ) − (1 + λ)(k − γ)σ2
that is
∞
P
n=2
nan ≤
an
(1 − γ)
2(1 − γ)
≤
,
Φ(λ, γ, k, 2)
[2 + k − γ]
2(1 − γ)
:= δ
Φ(λ, γ, k, 2)
(39)
which, in view of the definition (36) proves Theorem 11.
Remark. We observe that, if we specialize the parameters of the class
T W pq (k, γ), we obtain the analogous results for the classes RS(γ, k), LS(γ, k) and
GS(γ, k) defined in this paper.
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(received 18.07.2009; in revised form 07.11.2009)
J. Dziok, Institute of Mathematics, University of Rzeszów ul. Rejtana 16A, PL-35-310 Rzeszów, Poland
E-mail: jdziok@univ.rzeszow.pl
G. Murugusundaramoorthy, School of Science and Humanities, V I T University, Vellore - 632014,
T.N., India
E-mail: gmsmoorthy@yahoo.com