General Mathematics Vol. 14, No. 1 (2006), 65–76
Certain class of p-valent Functions defined
by Dziok-Srivastava Linear Operator 1
Shahram Najafzadeh, S. R. Kulkarni and G.
Murugusundaramoorthy
Abstract
In this paper, we introduce a new class of multivalent functions defined by Dziok-Srivastava operator to study some of the
interesting properties like coefficient estimates, distortion bounds
and to prove the class is closed under convolution product and
integral representation.
2000 Mathematical Subject Classification: 30C45, 30C50.
Keywords: p-valent and hypergeometric functions, convolution,
distortion bounds, closure theorem
1
Received February 20, 2006
Accepted for publication (in revised form) March 11, 2006
65
66
1
Sh. Najafzadeh, S. R. Kulkarni and G.Murugusundaramoorthy
Introduction
Let Ap be the class of p-valent analytic functions with positive coefficients
of the form
p
(1)
f (z) = z +
∞
X
ak z k , z ∈ ∆ = {z : |z| < 1}.
k=p+1
For functions f (z) given by (1) and
p
(2)
g(z) = z +
∞
X
bk z k ,
k=p+1
the Hadamard product (or convolution) of f (z) and g(z) denoted by
(f ∗ g)(z) = (g ∗ f )(z) is defined by
p
(3)
(f ∗ g)(z) = z +
∞
X
ak bk z k .
k=p+1
For {α1 , α2 , · · · , αm } ⊆ C and {β1 , β2 , · · · , βn } ⊆ C − {0, −1, −2, · · ·}
the generalized hypergeometric function
m Fn (α1 , · · · , αm ; β1 , · · · , βn ; z)
is defined by
m Fn (α1 , · · · , αm ; β1 , · · · , βn ; z) =
∞
P
k=0
(α1 )k ···(αm )k z k
(β1 )k ···(βn )k k!
(4)
(m ≤ n + 1, m, n, ∈ IN0 = {0, 1, 2, · · ·})
where (λ)k is the pochhammer symbol defined by
(5)

k=0
Γ(λ + k)  1
(λ)k =
=
 λ(λ + 1) · · · (λ + k − 1) k ∈ IN
Γ(k)
Certain class of p-valent Functions...
67
Using Dziok - Srivastava operator [2] , f (z) ∈ Ap we have
DSpm,n = DSp(m,n) (α1 , · · · , αm ; β1 , · · · , βn )f (z)
(6)
= hp (α1 , · · · , αm ; β1 , · · · , βn ; z) ∗ f (z)
∞
X
(α1 )k−p · · · (αm )k−p ak z k
p
= z +
(β1 )k−p · · · (βn )k−p (k − p)!
k=p+1
where
hp (α1 , · · · , αm ; β1 , · · · , βn ; z) = z p
For 1 < γ < 1 +
1
,
2p
m Fn (α1 , · · · , αm ; β1 , · · · , βn ; z).
z ∈ ∆ and let g(z) given by (2) we define the
class
Ap (g(z), α1 , · · · , αm ; β1 , β2 , · · · , βn , γ) = Ag(z)
p (m, n, γ)
by
g(z)
Ap (m, n, γ)
(7)
2
n
n
= f (z) ∈ Ap : Re 1 +
z(DSpm,n (f ∗g)(z))00
(DSpm,n (f ∗g)(z))0
o
< pγ,
1
(1 < γ < 1 + , z ∈ ∆)
2p
Main Results
In this section we obtain a necessary and sufficient condition for functions
g(z)
to be in the class Ap (m, n, γ).
68
Sh. Najafzadeh, S. R. Kulkarni and G.Murugusundaramoorthy
g(z)
Theorem 2.1. f (z) ∈ Ap (m, n, γ) if and only if
∞
X
k(k − pγ)
θ(k, p) ak bk ≤ 1.
p2 (γ − 1)
k=p+1
(8)
where
θ(k, p) =
(α1 )k−p · · · (αm )k−p
.
(β1 )k−p · · · (βn )k−p (k − p)!
g(z)
Proof. If f (z) ∈ Ap (m, n, γ), then by using (6) and (7) we obtain




Re



z(p(p − 1)z
p−2
+
1+
pz p−1
∞
P
+
θ(k, , p)k(k − 1)ak bk z
k=p+1
∞
P
θ(k, p)kak bk
z k−1



k−2 



< pγ.
k=p+1
Choosing values of z on real axis and letting z → 1− we have
p2 +
p+
∞
P
k=p+1
∞
P
θ(k, p)k 2 ak bk
< pγ
θ(k, p)kak bk
k=p+1
or equivalently
∞
X
k(k − pγ)θ(k, p)ak bk ≤ p2 (γ − 1).
k=p+1
To prove the “if” part, let (8) holds true, so
z(DSpm,n (f ∗ g)(z))00 − (p − 1)(DSpm,n (f ∗ g)(z))0
m,n
m,n
z(DSp (f ∗ g)(z)00 − [2p(1 − γ) − 1 + p](DSp (f ∗ g)(z))0 ∞
P
k(k − p)ak bk
k=p+1
≤
≤1
∞
P
2
2p (γ − 1) −
[k(k − p)(1 − 2(1 − γ)))]ak bk
k=p+1
Certain class of p-valent Functions...
69
g(z)
or equivalently f (z) ∈ Ap (m, n, γ).
g(z)
Theorem 2.2. If f (z) ∈ Ap (m, n, γ), then
(9)
ak ≤
p2 (γ − 1)
k(k − pγ)bk θ(k, p)
the result is sharp for functions of the form
fk (z) = z p +
p2 (γ − 1)
z k k = p + 1, p + 2, · · · .
k(k − pγ)bk θ(k, p)
g(z)
Proof. Since f (z) ∈ Ap (m, n, γ), by (8) we have
∞
X
k(k − pγ)θ(k, p)ak bk ≤
k(k − pγ)θ(k, p)ak bk ≤ p2 (γ − 1)
k=p+1
or
ak ≤
p2 (γ − 1)
.
k(k − pγ)θ(k, p)bk
The sharpness is trivial and so omitted.
3
Distortion Bounds
g(z)
In this section we obtain the distortion bounds for f (z) ∈ Ap (m, n, γ).
g(z)
Theorem 3.1. If f (z) ∈ Ap (m, n, γ), then
(10)
p2 (γ − 1)
rp+1 ≤ |f (z)|
(p + 1)(p + 1 − pγ)θ(p + 1, p)bp+1
p2 (γ − 1)
≤ rp +
rp+1
(p + 1)(p + 1 − pγ)θ(p + 1, p)bp+1
rp −
where
m
Q
θ(p + 1, p) =
i=1
n
Q
j=1
αi
,
βj
|z| = r < 1.
70
Sh. Najafzadeh, S. R. Kulkarni and G.Murugusundaramoorthy
The result is sharp for the function
(11)
p2 (γ − 1)
f (z) = z +
z p+1 .
(p + 1)(p + 1 − pγ)θ(p + 1, p)bp+1
p
Proof. By using (8), (9) we obtain
∞
X
bp+1 θ(p+1, p)(p+1)(p+1−pγ)
ak ≤
k=p+1
∞
X
k(k−pγ)θ(k, p)ak bk ≤ p2 (γ−1)
k=p+1
or
(12)
∞
X
ak ≤
k=p+1
p2 (γ − 1)
.
(p + 1)(p + 1 − pγ)θ(p + 1, p)bp+1
For the function f (z) = z p +
∞
P
ak z k and using (12) and |z| = r we
k=p+1
have
p
|f (z)| ≤ r +
∞
X
ak rk
k=p+1
p
< r +r
∞
X
p+1
ak
k=p+1
≤ rp +
p2 (γ − 1)
rp+1 ,
(p + 1)(p + 1 − pγ)θ(p + 1, p)bp+1
also
p
|f (z)| ≥ r −
∞
X
ak r k
k=p+1
p2 (γ − 1)
≥ r −
rp+1 .
(p + 1)(p + 1 − pγ)θ(p + 1, p)bp+1
p
Hence the proof is complete.
Certain class of p-valent Functions...
71
g(z)
Corollary. If f (z) ∈ Ap (m, n, γ), then
p2 (γ − 1)
rp ≤ |f 0 (z)|
(p + 1 − pγ)θ(p + 1, p)bp+1
p2 (γ − 1)
rp .
≤ prp−1 +
(p + 1 − pγ)θ(p + 1, p)bp+1
prp−1 −
The result is sharp for the function given by (11).
4
Integral Representation
In this section we obtain integral representation for DSpm,n (f ∗ g)(z).
g(z)
Theorem 4.1. If f (z) ∈ Ap (m, n, γ) then
Z z R
z Q(t)
m,n
e 0 t dt dt.
DSp (f ∗ g)(z) = (pγ − 1)
0
Proof. By letting DSpm,n (f ∗ g)(z) = M (z) in (7) we have
zM 00 (z)
< pγ.
Re 1 +
M 0 (z)
Thus
zM 00 (z)
< pγ − 1
M 0 (z)
or
zM 00 (z)
= Q(z)(pγ − 1)
M 0 (z)
where |Q(z)| < 1, z ∈ ∆.
So
M 00 (z)
M 0 (z)
=
Q(z)
(pγ
z
− 1), after integration we obtain
Z
z
0
log(M (z)) =
0
Q(t)
(pγ − 1)dt
t
72
Sh. Najafzadeh, S. R. Kulkarni and G.Murugusundaramoorthy
thus
Z
z
0
M (z) = exp
0
Q(t)
(pγ − 1)dt .
t
After integration we have
Z
M (z) =
DSpm,n (f
Z
z
∗ g) =
z
exp
0
0
Q(t)
(pγ − 1)dt dt
t
and this gives the result.
5
Closure Theorems
g(z)
In this section, we discuss certain inclusion properties of the class Ap (m, n, γ).
∞
P
ak,j z k (j = 1, 2, · · · , q) be in the
Theorem 5.1. Let Fj (z) = z p +
k=p+1
g(z)
class Ap (m, n, γ) and ηj ≥ 0 for j = 1, 2, · · · , q and
q
P
ηj ≤ 1 then the
j=1
function
f (z) = z p +
belongs to
∞
X
q
X
k=p+1
j=1
!
ηj ak,j
zk
g(z)
Ap (m, n, γ).
g(z)
Proof. Since Fj (z) ∈ Ap (m, n, γ), then from Theorem 2.1 for every
j = 1, 2, · · · , q we have
∞
X
k(k − pγ)θ(k, p)bk ak,j ≤ p2 (γ − 1).
k=p+1
Also
∞
X
k(k − pγ)θ(k, p)bk
X
j=1
!
ηj ak,j
j=1
k=p+1
q
=
q
X
ηj (
∞
X
k=p+1
k(k − pγ)θ(k, p)bk ak,j )
Certain class of p-valent Functions...
≤
q
X
73
ηj p2 (γ − 1)
j=1
2
≤ p (γ − 1).
g(z)
So by Theorem 2.1 f (z) ∈ Ap (m, n, γ).
g(z)
Corollary. The class Ap (m, n, γ) is closed under convex linear combination.
Theorem 5.2. Let Fp (z) = z p and
p2 (γ − 1)
z k , (k = p + 1, · · ·).
Fk (z) = z +
k(k − pγ)θ(k, p)bk
p
g(z)
Then f (z) ∈ Ap (m, n, γ) if and only if
∞
X
p
f (z) = ηp z +
ηk Fk (z)
k=p+1
where
∞
P
ηk = 1 and ηk ≥ 0.
k=p
g(z)
Proof. Let f (z) ∈ Ap (m, n, γ), then from Theorem 2.2, we have
ak ≤
p2 (γ − 1)
k(k − pγ)θ(k, p)bk
(k = p + 1, p + 2, · · ·)
therefore by letting
ηk =
and ηp = 1 −
k(k − pγ)θ(k, p)bk ak
p2 (γ − 1)
∞
P
(k = p + 1, p + 2, · · ·)
ηk .
k=p+1
We conclude the required result.
∞
P
Conversely, let f (z) = ηp z p +
ηk Fk (z), then
k=p+1
p
f (z) = ηp z +
∞
X
k=p+1
ηk
p2 (γ − 1)
zk
z +
k(k − pγ)θ(k, p)bk
p
74
Sh. Najafzadeh, S. R. Kulkarni and G.Murugusundaramoorthy
p
= z +
∞
X
ηk p2 (γ − 1)
zk .
k(k − pγ)θ(k, p)bk
k=p+1
Therefore
∞
X
k(k − pγ)
ηk p2 (γ − 1)
θ(k, p)bk
2 (γ − 1)
k(k
−
pγ)θ(k,
p)b
p
k
k=p+1
=
∞
X
ηk = 1 − ηp ≤ 1.
k=p+1
g(z)
Hence by Theorem 2.1, we have f (z) ∈ Ap (m, n, γ).
6
Convolution Property and Integral Operator
g(z)
In this section we show that the class Ap (m, n, γ) is closed under convolution and integral operator.
Theorem 6.1.
∞
P
Let h(z) = z p +
and 0 ≤ ck ≤ 1. If f (z) ∈
ck z k be analytic in unit disk ∆
k=p+1
g(z)
Ap (m, n, γ),
then (f ∗ h)(z) is also in the
g(z)
class Ap (m, n, γ).
g(z)
Proof. Since f (z) ∈ Ap (m, n, γ) then by Theorem 2.1 we have
∞
X
k(k − pγ)θ(k, p)ak bk ≤ p2 (γ − 1).
k=p+1
By using the last inequality and the fact that
p
(f ∗ h)(z) = z +
∞
X
k=p+1
a k ck z k
Certain class of p-valent Functions...
75
we have
∞
X
k(k − pγ)θ(k, p)ak ck bk
k=p+1
∞
X
≤
k(k − pγ)θ(k, p)ak bk ≤ p2 (γ − 1)
k=p+1
and hence by Theorem 2.1 result follows.
g(z)
Theorem 6.2. If f (z) ∈ Ap (m, n, γ), then
λ+p
F (z) =
zλ
Z
z
tλ−1 f (t)dt (λ > −1; z ∈ ∆)
0
g(z)
is also in the class Ap (m, n, γ).
Proof.
Since F (z) = f (z) ∗
zp +
∞
P
k=p+1
!
λ+p k
z
λ+k
and
λ+p
λ+k
≤ 1, by
Theorem 6.1, the proof is trivial.
References
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A class of multivalent functions with positive coefficients defined by
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[2] J. Dziok and H. M. Srivastava, Certain subclass of analytic functions associated with the generalized hypergeometric functions, Integral Transforms Spec. Funct. 14 (1), (2003), 7-18.
76
Sh. Najafzadeh, S. R. Kulkarni and G.Murugusundaramoorthy
[3] R. J. Libera, Some classes of regular univalent functions, Proc.
Amer. Math. Soc. 16 (1965), 755-758.
[4] J. L. Liu, On a class of p-valent analytic functions, Chinese Quar.
J. Math., 15(4) (2000), 27-32.
[5] J. L. Liu, Some applications of certain integral operator. Kyungpook
Math. J. 43 (2003), 211-219.
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Department of Mathematics,
Fergusson College, Pune University,
Pune - 411004, India
Department of Mathematics,
Vellore Institute of Technology,Deemed University,
Vellore-632 014 ,T.N., India
Shahram Najafzadeh : Najafzadeh1234@yahoo.ie
S. R. Kulkarni : kulkarni− ferg@yahoo.com
G.Murugusundaramoorthy: gmsmoorthy@yahoo.com