# Meromorphic Parabolic Starlike Functions Associated withq-Hypergeometric Series

Authors: G. Murugusundaramoorthy; T. Janani
Journal: ISRN Mathematical Analysis
 (2014)

---

## Abstract

We introduce a new class of meromorphic parabolic starlike functions with a fixed point defined in the punctured unit diskΔ
*
≔
z
∈
ℂ
:
0
<
z
<
1 involving the q-hypergeometric functions. We obtained coefficient inequalities, growth and distortion inequalities, and closure results for functions f
∈
ℳ
m
l
λ
,
β
,
γ. We further established some results concerning convolution and the partial sums.


---

## Body

## 1. Introduction

Letξ be a fixed point in the unit disc Δ
∶
=
{
z
∈
ℂ
:
|
z
|
<
1
}. Denote by ℋ
(
Δ
) the class of functions which are regular and
(1)
𝒜
(
ξ
)
=
{
f
∈
H
(
Δ
)
:
f
(
ξ
)
=
f
′
(
ξ
)
-
1
=
0
}
.
Also denote by 𝒮
ξ
=
{
f
∈
𝒜
(
ξ
)
:
f
is
univalent
in
Δ
}, the subclass of 𝒜
(
ξ
) consisting of the functions of the form
(2)
f
(
z
)
=
(
z
-
ξ
)
+
∑
n
=
2
∞
‍
a
n
(
z
-
ξ
)
n
which are analytic in Δ. Note that 𝒮
0
=
𝒮 is subclasses of 𝒜 consisting of univalent functions in Δ. By 𝒮
w
*
(
β
) and 𝒦
w
(
β
), respectively, we mean the classes of analytic functions that satisfy the analytic conditions ℜ
{
(
z
-
ξ
)
f
′
(
z
)
/
f
(
z
)
}
>
β, and ℜ
{
1
+
(
(
z
-
ξ
)
f
′′
(
z
)
/
f
′
(
z
)
)
}
>
β, (
z
-
w
)
∈
Δ for 0
≦
β
<
1 introduced and studied by Kanas and Ronning [1]. The class 𝒮
ξ
*
(
0
) is defined by geometric property that the image of any circular arc centered at ξ is starlike with respect to f
(
ξ
) and the corresponding class 𝒦
ξ
*
(
0
) is defined by the property that the image of any circular arc centered at ξ is convex. We observe that the definitions are somewhat similar to the ones introduced by Goodman in [2, 3] for uniformly starlike and convex functions, except that in this case the point ξ is fixed. In particular, 𝒦
=
𝒦
0
(
0
) and 𝒮
0
*
=
𝒮
*
(
0
), respectively, are the well-known standard classes of convex and starlike functions.LetΣ denote the class of meromorphic functions f of the form
(3)
f
(
z
)
=
1
z
+
∑
n
=
1
∞
‍
a
n
z
n
,
defined on the punctured unit disk Δ
*
∶
=
{
z
∈
ℂ
:
0
<
|
z
|
<
1
}.Denote byΣ
ξ the subclass of Σ consisting of the functions of the form
(4)
f
(
z
)
=
1
z
-
ξ
+
∑
n
=
1
∞
‍
a
n
(
z
-
ξ
)
n
,
a
n
≥
0
;
z
≠
ξ
.A functionf of the form (4) is in the class ofmeromorphic starlike of order
γ (0
≤
γ
<
1) denoted by Σ
ξ
*
(
γ
), if
(5)
-
ℜ
(
(
z
-
ξ
)
f
′
(
z
)
f
(
z
)
)
>
γ
,
z
-
ξ
∈
Δ
∶
=
Δ
*
∪
{
0
}
,
and is in the class ofmeromorphic convex of order
γ (0
≤
γ
<
1) denoted by Σ
ξ
K
(
γ
), if
(6)
-
ℜ
(
1
+
(
z
-
ξ
)
f
′′
(
z
)
f
′
(
z
)
)
>
γ
,
z
-
ξ
∈
Δ
∶
=
Δ
*
∪
{
0
}
.For functionsf
(
z
) given by (4) and g
(
z
)
=
(
1
/
(
z
-
ξ
)
)
+
∑
n
=
1
∞
‍
b
n
(
z
-
ξ
)
n,  (
b
n
≥
0
) we define the Hadamard product or convolution of f and g by
(7)
(
f
*
g
)
(
z
)
∶
=
1
z
-
ξ
+
∑
n
=
1
∞
‍
a
n
b
n
(
z
-
ξ
)
n
.More recently, Purohit and Raina [4] introduced a generalized q-Taylor’s formula in fractional q-calculus and derived certain q-generating functions for q-hypergeometric functions. In this work we proceed to derive a generalized differential operator on meromorphic functions in Δ
*
=
{
z
∈
ℂ
:
0
<
|
z
|
<
1
} involving these functions and discuss some of their properties.For complex parametersa
1
,
…
,
a
l and b
1
,
…
,
b
m  (
b
j
≠
0
,
-
1
,
…
;
j
=
1,2
,
…
,
m
) the q-hypergeometric function l
Ψ
m
(
z
) is defined by
(8)
l
Ψ
m
(
a
1
,
…
a
l
;
b
1
,
…
,
b
m
;
q
,
z
)
∶
=
∑
n
=
0
∞
‍
(
a
1
,
q
)
n
⋯
(
a
l
,
q
)
n
(
q
,
q
)
n
(
b
1
,
q
)
n
⋯
(
b
m
,
q
)
n
×
[
(
-
1
)
n
q
(
n
2
)
]
1
+
m
-
l
z
n
,
with (
n
2
)
=
n
(
n
-
1
)
/
2 where q
≠
0 when l
>
m
+
1
(
l
,
m
∈
ℕ
0
=
ℕ
∪
{
0
}
;
z
∈
𝕌
).Theq-shifted factorial is defined for a
,
q
∈
ℂ as a product of n factors by
(9)
(
a
;
q
)
n
=
{
1
n
=
0
(
1
-
a
)
(
1
-
a
q
)
⋯
(
1
-
a
q
n
-
1
)
n
∈
ℕ
,
and in terms of basic analogue of the gamma function
(10)
(
q
a
;
q
)
n
=
Γ
q
(
a
+
n
)
(
1
-
q
)
n
Γ
q
(
a
)
,
n
>
0
.
It is of interest to note that lim
⁡
q
→
1
-
(
(
q
a
;
q
)
n
/
(
1
-
q
)
n
)
=
(
a
)
n
=
a
(
a
+
1
)
⋯
(
a
+
n
-
1
) is the familiar Pochhammer symbol and
(11)
l
Ψ
m
(
a
1
,
…
,
a
l
;
b
1
,
…
,
b
m
;
z
)
=
∑
n
=
0
∞
‍
(
a
1
)
n
⋯
(
a
l
)
n
(
b
1
)
n
⋯
(
b
m
)
n
z
n
n
!
.
Now for z
∈
𝕌, 0
<
|
q
|
<
1, and l
=
m
+
1, the basic hypergeometric function defined in (8) takes the form
(12)
l
Ψ
m
(
a
1
;
…
a
l
;
b
1
,
…
,
b
m
;
q
,
z
)
=
∑
n
=
0
∞
‍
(
a
1
,
q
)
n
⋯
(
a
l
,
q
)
n
(
q
,
q
)
n
(
b
1
,
q
)
n
⋯
(
b
m
,
q
)
n
z
n
,
which converges absolutely in the open unit disk 𝕌.Corresponding to the functionl
Ψ
m
(
a
1
;
…
a
l
;
b
1
,
…
,
b
m
;
q
,
z
) recently for meromorphic functions f
∈
Σ
0 consisting functions of the form (3), Huda and Darus [5] introduce q-analogue of Liu-Srivastava operator as below:
(13)
l
Ψ
m
(
a
1
;
…
a
l
;
b
1
,
…
,
b
m
;
q
,
z
)
*
f
(
z
)
=
1
z
Ψ
l
m
(
a
1
;
…
a
l
;
b
1
,
…
,
b
m
;
q
,
z
)
*
f
(
z
)
=
1
z
+
∑
n
=
1
∞
‍
(
a
1
;
q
)
n
+
1
⋯
(
a
l
;
q
)
n
+
1
(
q
;
q
)
n
+
1
(
b
1
;
q
)
n
+
1
⋯
(
b
m
,
q
)
n
+
1
a
n
z
n
,
where z
∈
Δ
*
∶
=
{
z
∈
ℂ
:
0
<
|
z
|
<
1
}.In this paper for functionsf
∈
Σ
ξ and for real parameters a
1
,
…
,
a
l and b
1
,
…
,
b
m  (
b
j
≠
0
,
-
1
,
…
;
j
=
1,2
,
…
,
m
) we define the following new linear operator:
(14)
ℐ
m
l
(
a
1
;
…
a
l
;
b
1
,
…
,
b
m
;
q
,
z
-
ξ
)
:
Σ
ξ
⟶
Σ
ξ
,
as
(15)
ℐ
m
l
(
a
1
;
…
a
l
;
b
1
,
…
,
b
m
;
q
,
z
-
ξ
)
=
1
z
-
ξ
Ψ
l
m
(
a
1
;
…
a
l
;
b
1
,
…
,
b
m
;
q
,
z
-
ξ
)
ℐ
m
l
[
a
l
,
q
]
=
1
z
-
ξ
+
∑
n
=
1
∞
‍
Υ
n
l
,
m
[
a
1
,
q
]
(
z
-
ξ
)
n
,
where
(16)
Υ
n
l
,
m
[
a
1
,
q
]
=
(
a
1
;
q
)
n
+
1
⋯
(
a
l
;
q
)
n
+
1
(
q
;
q
)
n
+
1
(
b
1
;
q
)
n
+
1
⋯
(
b
m
,
q
)
n
+
1
.Throughout our study forf
∈
Σ
ξ, we let
(17)
ℐ
m
l
f
(
z
)
=
ℐ
m
l
[
a
l
,
q
]
*
f
(
z
)
=
1
z
-
ξ
+
∑
n
=
1
∞
‍
Υ
m
l
(
n
)
a
n
(
z
-
ξ
)
n
,
(18)
Υ
m
l
(
n
)
=
Υ
n
l
,
m
[
a
1
,
q
]
=
(
a
1
;
q
)
n
+
1
⋯
(
a
l
;
q
)
n
+
1
(
q
;
q
)
n
+
1
(
b
1
;
q
)
n
+
1
⋯
(
b
m
,
q
)
n
+
1
,
unless otherwise stated.Motivated by earlier works on meromorphic functions by function theorists (see [6–14]), we define the following new subclass of functions in Σ
ξ by making use of the generalized operator ℐ
m
l.For0
≤
γ
<
1 and 0
≤
λ
<
1
/
2, we let ℳ
m
l
(
λ
,
β
,
γ
) denote a subclass of Σ
ξ consisting functions of the form (4) satisfying the condition that
(19)
-
ℜ
(
(
z
-
ξ
)
(
ℐ
m
l
f
(
z
)
)
′
+
λ
(
z
-
ξ
)
2
(
ℐ
m
l
f
(
z
)
)
′′
(
1
-
λ
)
ℐ
m
l
f
(
z
)
+
λ
(
z
-
ξ
)
(
ℐ
m
l
f
(
z
)
)
′
)
>
β
|
(
z
-
ξ
)
(
ℐ
m
l
f
(
z
)
)
′
+
λ
(
z
-
ξ
)
2
(
ℐ
m
l
f
(
z
)
)
′′
(
1
-
λ
)
ℐ
m
l
f
(
z
)
+
λ
(
z
-
ξ
)
(
ℐ
m
l
f
(
z
)
)
′
+
1
|
+
γ
,
where ℐ
m
l
f is given by (17).Further, shortly we can state this condition by(20)
-
ℜ
(
(
z
-
ξ
)
G
′
(
z
)
G
(
z
)
)
>
β
|
(
z
-
ξ
)
G
′
(
z
)
G
(
z
)
+
1
|
+
γ
,
where
(21)
G
(
z
)
=
(
1
-
λ
)
ℐ
m
l
f
(
z
)
+
λ
(
z
-
ξ
)
(
ℐ
m
l
f
(
z
)
)
′
=
1
-
2
λ
z
-
ξ
+
∑
n
=
1
∞
‍
(
n
λ
-
λ
+
1
)
Υ
m
l
(
n
)
a
n
(
z
-
ξ
)
n
,
1
-
2
λ
z
-
ξ
+
∑
n
=
1
∞
‍
(
n
λ
-
λ
+
1
)
Υ
m
l
(
n
)
a
n
000
i
a
n
≥
0
.It is of interest to note that, on specializing the parametersλ, β and l, m, we can define various new subclasses of Σ
ξ. We illustrate two important subclasses in the following examples.Example 1.
Forλ
=
0, we let ℳ
m
l
(
0
,
β
,
γ
)
=
ℳ
m
l
(
β
,
γ
) denote a subclass of Σ
ξ consisting functions of the form (4) satisfying the condition that
(22)
-
ℜ
(
(
z
-
ξ
)
(
ℐ
m
l
f
(
z
)
)
′
ℐ
m
l
f
(
z
)
)
>
β
|
(
z
-
ξ
)
(
ℐ
m
l
f
(
z
)
)
′
ℐ
m
l
f
(
z
)
+
1
|
+
γ
,
where ℐ
m
l
f
(
z
) is given by (17).Example 2.
Forλ
=
0, β
=
0 we let ℳ
m
l
(
0,0
,
γ
)
=
ℳ
m
l
(
γ
) denote a subclass of Σ
ξ consisting functions of the form (4) satisfying the condition that
(23)
-
ℜ
(
(
z
-
ξ
)
(
ℐ
m
l
f
(
z
)
)
′
ℐ
m
l
f
(
z
)
)
>
γ
,
where ℐ
m
l
f
(
z
) is given by (17).In this paper, we obtain the coefficient inequalities, growth and distortion inequalities, and closure results for the function classℳ
m
l
(
λ
,
β
,
γ
). Properties of certain integral operator and convolution properties of the new class ℳ
m
l
(
λ
,
β
,
γ
) are also discussed.


## 2. Coefficients Inequalities

In order to obtain the necessary and sufficient condition for a function,f
∈
ℳ
m
l
(
λ
,
β
,
γ
), we recall the following lemmas.Lemma 3.
Ifγ is a real number and w is a complex number, then ℜ
(
w
)
≥
γ
⇔
|
w
+
(
1
-
γ
)
|
-
|
w
-
(
1
+
γ
)
|
≥
0.Lemma 4.
Ifw is a complex number and γ, k are real numbers, then
(24)
ℜ
(
w
)
≥
k
|
w
-
1
|
+
γ
⟺
ℜ
{
w
(
1
+
k
e
i
θ
)
-
k
e
i
θ
}
≥
γ
,
ℜ
(
w
)
≥
0
i
00000000
k
|
w
-
1
|
+
γ
⟺
ℜ
-
π
≤
θ
≤
π
.Analogous to the lemma proved by Dziok et al. [8], we state the following lemma without proof.Lemma 5.
Suppose thatγ
∈
[
0,1
), r
∈
(
0,1
], and the function H
∈
Σ
ξ
(
γ
) is of the form H
(
z
)
=
(
1
/
(
z
-
ξ
)
)
+
∑
n
=
1
∞
‍
b
n
(
z
-
ξ
)
n, 0
<
|
z
-
ξ
|
<
r
<
1, with b
n
≥
0, then
(25)
∑
n
=
1
∞
‍
(
n
+
γ
)
b
n
r
n
+
1
≤
1
-
γ
.Theorem 6.
Letf
∈
Σ
ξ be given by (4). Then f
∈
ℳ
m
l
(
λ
,
β
,
γ
) if and only if
(26)
∑
n
=
1
∞
‍
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
(
1
+
n
λ
-
λ
)
Υ
m
l
(
n
)
a
n
≤
(
1
-
2
λ
)
(
1
-
γ
)
.Proof.
Iff
∈
ℳ
m
l
(
λ
,
β
,
γ
), then by (20) we have
(27)
-
ℜ
(
(
z
-
ξ
)
G
′
(
z
)
G
(
z
)
)
>
β
|
(
z
-
ξ
)
G
′
(
z
)
G
(
z
)
+
1
|
+
γ
.
Making use of Lemma 4,
(28)
-
ℜ
(
(
z
-
ξ
)
(
1
+
β
e
i
θ
)
G
′
(
z
)
+
β
e
i
θ
G
(
z
)
G
(
z
)
)
>
γ
,
where G
(
z
) is given by (21). Substituting G
(
z
), G
′
(
z
) and letting |
z
-
ξ
|
<
r
→
1
-, we have
(29)
{
(
(
1
-
2
λ
)
-
∑
n
=
1
∞
‍
(
1
+
n
λ
-
λ
)
Υ
m
l
(
n
)
a
n
)
-
1
(
(
1
-
2
λ
)
(
1
-
γ
)
-
∑
n
=
1
∞
‍
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
×
(
1
+
n
λ
-
λ
)
Υ
m
l
(
n
)
a
n
∑
n
=
1
∞
‍
)
×
(
(
1
-
2
λ
)
-
∑
n
=
1
∞
‍
(
1
+
n
λ
-
λ
)
Υ
m
l
(
n
)
a
n
)
-
1
}
>
0
.
This shows that (26) holds.
Conversely, assume that (26) holds. Since -
ℜ
(
w
)
>
γ, if and only if |
w
+
1
|
<
|
w
-
(
1
-
2
γ
)
|, it is sufficient to show that
(30)
|
w
+
1
w
-
(
1
-
2
γ
)
|
<
1
,
|
w
-
(
1
-
2
γ
)
|
≠
0
000000000
for
|
z
-
ξ
|
<
r
≤
1
,
(
z
-
ξ
)
∈
Δ
.
Using (26) and taking w
(
z
)
=
(
(
z
-
ξ
)
(
1
+
β
e
i
θ
)
G
′
(
z
)
+
β
e
i
θ
G
(
z
)
)
/
G
(
z
), we get
(31)
|
w
+
1
w
-
(
1
-
2
γ
)
|
≤
(
(
∑
n
=
1
∞
‍
(
1
+
n
λ
-
λ
)
[
(
n
+
1
)
(
1
+
β
)
]
Υ
m
l
(
n
)
a
n
)
×
(
2
(
1
-
γ
)
(
1
-
2
λ
)
-
∑
n
=
1
∞
‍
(
1
+
n
λ
-
λ
)
×
[
n
(
1
+
β
)
+
(
β
+
2
γ
-
1
)
]
Υ
m
l
(
n
)
a
n
∑
n
=
1
∞
‍
)
-
1
)
≤
1
.
Thus we have f
∈
ℳ
m
l
(
λ
,
β
,
γ
).For the sake of brevity throughout this paper we let(32)
d
n
(
λ
,
β
,
γ
)
=
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
(
1
+
n
λ
-
λ
)
,
d
1
(
λ
,
β
,
γ
)
=
(
1
+
γ
+
2
β
)
,
unless otherwise stated.Our next result gives the coefficient estimates for functions inℳ
m
l
(
λ
,
β
,
γ
).Theorem 7.
Iff
∈
ℳ
m
l
(
λ
,
β
,
γ
), then
(33)
a
n
≤
(
1
-
γ
)
(
1
-
2
λ
)
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
,
n
=
1,2
,
3
,
…
.
The result is sharp for the functions f
n
(
z
) given by
(34)
f
n
(
z
)
=
1
z
-
ξ
+
1
-
γ
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
(
z
-
ξ
)
n
,
00000000000000000000000
n
=
1,2
,
3
,
…
.Proof.
Iff
∈
ℳ
m
l
(
λ
,
β
,
γ
), then we have, for each n,
(35)
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
a
n
≤
∑
n
=
1
∞
‍
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
a
n
≤
(
1
-
γ
)
(
1
-
2
λ
)
.
Therefore we have
(36)
a
n
≤
(
1
-
γ
)
(
1
-
2
λ
)
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
.
Since
(37)
f
n
(
z
)
=
1
z
-
ξ
+
(
1
-
γ
)
(
1
-
2
λ
)
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
(
z
-
ξ
)
n
satisfies the conditions of Theorem 6, f
n
(
z
)
∈
ℳ
m
l
(
λ
,
β
,
γ
) and the equality is attained for this function.Theorem 8.
Suppose that there exists a positive numberν:
(38)
ν
=
inf
⁡
n
∈
ℕ
{
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
}
.
If f
∈
ℳ
m
l
(
λ
,
β
,
γ
), then
(39)
|
1
r
-
(
1
-
γ
)
(
1
-
2
λ
)
ν
r
|
≤
|
f
(
z
)
|
≤
1
r
+
(
1
-
γ
)
(
1
-
2
λ
)
ν
r
,
(
|
z
-
ξ
|
=
r
)
,
|
1
r
2
-
(
1
-
γ
)
(
1
-
2
λ
)
ν
|
≤
|
f
′
(
z
)
|
≤
1
r
2
+
(
1
-
γ
)
(
1
-
2
λ
)
ν
(
|
z
-
ξ
|
=
r
)
.
If ν
=
d
1
(
λ
,
β
,
γ
)
Υ
m
l
(
1
)
=
(
1
+
γ
+
2
β
)
Υ
m
l
(
1
), then the result is sharp for
(40)
f
(
z
)
=
1
z
-
ξ
+
(
1
-
γ
)
(
1
-
2
λ
)
(
1
+
γ
+
2
β
)
r
2
Υ
m
l
(
1
)
(
z
-
ξ
)
.Proof.
Letf
∈
∑
ξ and be given by (4)
(41)
|
f
(
z
)
|
≤
1
r
+
∑
n
=
1
∞
‍
a
n
r
n
≤
1
r
+
r
∑
n
=
1
∞
‍
a
n
.
Since f
∈
ℳ
m
l
(
λ
,
β
,
γ
), and by Theorem 6,
(42)
∑
n
=
1
∞
‍
a
n
≤
(
1
-
γ
)
(
1
-
2
λ
)
ν
.
Using this, we have
(43)
|
f
(
z
)
|
≤
1
r
+
(
1
-
γ
)
(
1
-
2
λ
)
ν
r
.
Similarly
(44)
|
f
(
z
)
|
≥
|
1
r
-
(
1
-
γ
)
(
1
-
2
λ
)
ν
r
|
.
The result is sharp for function (40) with
(45)
ν
=
d
1
(
λ
,
β
,
γ
)
Υ
m
l
(
1
)
=
(
1
+
γ
+
2
β
)
Υ
m
l
(
1
)
.
Similarly we can prove the other inequality |
f
′
(
z
)
|.


## 3. Order of Starlikeness

In the following theorem we obtain the order of starlikeness for the classℳ
m
l
(
λ
,
β
,
γ
). We say that f given by (4) is meromorphically starlike of order ρ, (
0
≤
ρ
<
1
), in |
z
-
ξ
|
<
r when it satisfies condition (5) in |
z
-
ξ
|
<
r.Theorem 9.
Let the functionf given by (4) be in the class ℳ
m
l
(
λ
,
β
,
γ
). Then, if there exists
(46)
r
=
r
1
(
λ
,
γ
,
ρ
)
=
inf
⁡
n
≥
1
[
(
1
-
ρ
)
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
(
n
+
ρ
)
(
1
-
γ
)
(
1
-
2
λ
)
]
1
/
(
n
+
1
)
and it is positive, then f is meromorphically starlike of order ρ in |
z
-
ξ
|
<
r
≤
r
1
(
λ
,
γ
,
ρ
).Proof.
Let the functionf
∈
ℳ
m
l
(
λ
,
β
,
γ
) be of the form (4). If 0
<
r
≤
r
1
(
λ
,
γ
,
ρ
), then by (46)
(47)
r
n
+
1
≤
(
1
-
ρ
)
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
(
n
+
ρ
)
(
1
-
γ
)
(
1
-
2
λ
)
,
for all n
∈
ℕ. From (47) we get
(48)
n
+
ρ
1
-
ρ
r
n
+
1
≤
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
(
1
-
γ
)
(
1
-
2
λ
)
,
for all n
∈
ℕ, and thus
(49)
∑
n
=
1
∞
‍
n
+
ρ
1
-
ρ
a
n
r
n
+
1
≤
∑
n
=
1
∞
‍
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
(
1
-
γ
)
(
1
-
2
λ
)
a
n
≤
1
,
because of (26). Hence, from (49) and (25), f is meromorphically starlike of order ρ in |
z
-
ξ
|
<
r
≤
r
1
(
λ
,
γ
,
ρ
)
=
r.Suppose that there exists a numberr
~, r
~
>
r
1
(
λ
,
γ
,
ρ
), such that each f
∈
ℳ
m
l
(
λ
,
β
,
γ
) is meromorphically starlike of order ρ in |
z
-
ξ
|
<
r
~
≤
1. The function
(50)
f
(
z
)
=
1
z
-
ξ
+
(
1
-
γ
)
(
1
-
2
λ
)
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
(
z
-
ξ
)
n
is in the class ℳ
m
l
(
λ
,
β
,
γ
); thus it should satisfy (25) with r
~
:
(51)
∑
n
=
1
∞
‍
(
n
+
ρ
)
a
n
r
~
n
+
1
≤
1
-
ρ
,
while the left–hand side of (51) becomes
(52)
(
n
+
ρ
)
(
1
-
γ
)
(
1
-
2
λ
)
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
r
~
n
+
1
>
(
n
+
ρ
)
(
1
-
γ
)
(
1
-
2
λ
)
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
(
1
-
ρ
)
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
(
n
+
ρ
)
(
1
-
γ
)
(
1
-
2
λ
)
=
1
-
ρ
,
which contradicts (51). Therefore the number r
1
(
λ
,
γ
,
ρ
) in Theorem 9 cannot be replaced with a greater number. This means that r
1
(
λ
,
γ
,
ρ
) is called radius of meromorphically starlikeness of order ρ for the class ℳ
m
l
(
λ
,
β
,
γ
).


## 4. Results Involving Modified Hadamard Products

For functions(53)
f
j
(
z
)
=
1
z
-
ξ
+
∑
n
=
1
∞
‍
a
n
,
j
(
z
-
ξ
)
n
,
a
n
,
j
≥
0
,
we define the Hadamard product or convolution of f
1 and f
2 by
(54)
(
f
1
*
f
2
)
(
z
)
∶
=
1
z
-
ξ
+
∑
n
=
1
∞
‍
a
n
,
1
a
n
,
2
(
z
-
ξ
)
n
.
Let
(55)
Ψ
(
n
,
λ
)
=
(
n
λ
-
λ
+
1
)
(
1
-
2
λ
)
Υ
m
l
(
n
)
.Theorem 10.
For functionsf
j
(
j
=
1,2
) defined by (53), let f
1
∈
ℳ
m
l
(
λ
,
β
,
γ
) and f
2
∈
ℳ
m
l
(
λ
,
β
,
δ
). Then f
1
*
f
2
∈
ℳ
m
l
(
λ
,
β
,
η
) where
(56)
η
=
1
-
(
1
-
γ
)
(
1
-
δ
)
(
3
+
β
)
(
1
+
γ
+
2
β
)
(
1
+
δ
+
2
β
)
Ψ
(
1
,
λ
)
-
2
(
1
-
γ
)
(
1
-
δ
)
,
and Ψ
(
1
,
λ
)
=
Υ
m
l
(
1
)
/
(
1
-
2
λ
). The results are the best possible for
(57)
f
1
(
z
)
=
1
z
-
ξ
+
1
-
γ
(
1
+
γ
+
2
β
)
Ψ
(
1
,
λ
)
(
z
-
ξ
)
,
f
2
(
z
)
=
1
z
-
ξ
+
1
-
δ
(
1
+
δ
+
2
β
)
Ψ
(
1
,
λ
)
(
z
-
ξ
)
,
where Ψ
(
1
,
λ
)
=
Υ
m
l
(
1
)
/
(
1
-
2
λ
).Proof.
In view of Theorem6, it suffices to prove that
(58)
∑
n
=
1
∞
‍
[
n
(
1
+
β
)
+
(
η
+
β
)
]
(
1
-
η
)
Ψ
(
n
,
λ
)
a
n
,
1
a
n
,
2
≤
1
,
where η is defined by (56) under the hypothesis. It follows from (26) and the Cauchy-Schwarz inequality that
(59)
∑
n
=
1
∞
‍
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
1
/
2
[
n
(
1
+
β
)
+
(
δ
+
β
)
]
1
/
2
(
1
-
γ
)
(
1
-
δ
)
×
Ψ
(
n
,
λ
)
a
n
,
1
a
n
,
2
≤
1
.
Thus we need to find the largest η such that
(60)
∑
n
=
1
∞
‍
[
n
(
1
+
β
)
+
(
η
+
β
)
]
(
1
-
η
)
Ψ
(
n
,
λ
)
a
n
,
1
a
n
,
2
≤
∑
n
=
1
∞
‍
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
1
/
2
[
n
(
1
+
β
)
+
(
δ
+
β
)
]
1
/
2
(
1
-
γ
)
(
1
-
δ
)
×
Ψ
(
n
,
λ
)
a
n
,
1
a
n
,
2
≤
1
.
By virtue of (59) it is sufficient to find the largest η, such that
(61)
(
1
-
γ
)
(
1
-
δ
)
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
1
/
2
[
n
(
1
+
β
)
+
(
δ
+
β
)
]
1
/
2
Ψ
(
n
,
λ
)
≤
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
1
/
2
[
n
(
1
+
β
)
+
(
δ
+
β
)
]
1
/
2
(
1
-
γ
)
(
1
-
δ
)
×
1
-
η
[
n
(
1
+
β
)
+
(
η
+
β
)
]
,
which yields
(62)
η
≤
1
-
(
Ψ
(
n
,
λ
)
-
(
1
-
γ
)
(
1
-
δ
)
(
n
+
1
)
)
-
1
(
(
1
-
γ
)
(
1
-
δ
)
(
2
n
+
1
+
β
)
)
×
(
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
[
n
(
1
+
β
)
+
(
δ
+
β
)
]
×
Ψ
(
n
,
λ
)
-
(
1
-
γ
)
(
1
-
δ
)
(
n
+
1
)
)
-
1
)
,
for n
≥
1 where Ψ
(
n
,
λ
) is given by (55) and, since Ψ
(
n
,
λ
) is a decreasing function of n
(
n
≥
1
), we have
(63)
η
=
1
-
(
1
-
γ
)
(
1
-
δ
)
(
3
+
β
)
(
1
+
γ
+
2
β
)
(
1
+
δ
+
2
β
)
Ψ
(
1
,
λ
)
-
2
(
1
-
γ
)
(
1
-
δ
)
,
and Ψ
(
1
,
λ
)
=
Υ
m
l
(
1
)
/
(
1
-
2
λ
), which completes the proof.Theorem 11.
Let the functionsf
j, (
j
=
1,2
), defined by (53) be in the class ℳ
m
l
(
λ
,
β
,
γ
). Then (
f
1
*
f
2
)
(
z
)
∈
ℳ
m
l
(
λ
,
β
,
η
) where
(64)
η
=
1
-
(
1
-
γ
)
2
(
3
+
β
)
(
1
+
γ
+
2
β
)
2
Ψ
(
1
,
λ
)
-
2
(
1
-
γ
)
2
with Ψ
(
1
,
λ
)
=
Υ
m
l
(
1
)
/
(
1
-
2
λ
).Proof.
By takingδ
=
γ in the above theorem, the results follow.For functions in the classℳ
m
l
(
λ
,
β
,
γ
), we can prove the following inclusion property.Theorem 12.
Let the functionsf
j
(
j
=
1,2
) defined by (53) be in the class ℳ
m
l
(
λ
,
β
,
γ
). Then the function h, defined by
(65)
h
(
z
)
=
1
z
-
ξ
+
∑
n
=
1
∞
‍
(
a
n
,
1
2
+
a
n
,
2
2
)
(
z
-
ξ
)
n
,
is in the class ℳ
m
l
(
λ
,
β
,
δ
) where
(66)
δ
≤
1
-
4
(
1
-
γ
)
2
(
1
+
β
)
[
1
+
γ
+
2
β
]
2
Ψ
(
1
,
λ
)
+
2
(
1
-
γ
)
2
,
and Ψ
(
1
,
λ
)
=
Υ
m
l
(
1
)
/
(
1
-
2
λ
).Proof.
In view of Theorem6, it is sufficient to prove that
(67)
∑
n
=
2
∞
‍
Ψ
(
n
,
λ
)
[
n
(
1
+
β
)
+
(
δ
+
β
)
]
(
1
-
δ
)
(
a
n
,
1
2
+
a
n
,
2
2
)
≤
1
,
where f
j
∈
ℳ
m
l
(
λ
,
β
,
γ
)
(
j
=
1,2
); we find from (53) and Theorem 6 that
(68)
∑
n
=
1
∞
‍
[
Ψ
(
n
,
λ
)
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
1
-
γ
]
2
a
n
,
j
2
≤
∑
n
=
1
∞
‍
[
Ψ
(
n
,
λ
)
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
1
-
γ
a
n
,
j
]
2
≤
1
,
which would yield
(69)
∑
n
=
2
∞
‍
1
2
[
Ψ
(
n
,
λ
)
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
1
-
γ
]
2
(
a
n
,
1
2
+
a
n
,
2
2
)
≤
1
.
On comparing (67) and (69) it can be seen that inequality (66) will be satisfied if
(70)
Ψ
(
n
,
λ
)
[
n
(
1
+
β
)
+
(
δ
+
β
)
]
1
-
δ
(
a
n
,
1
2
+
a
n
,
2
2
)
≤
1
2
[
Ψ
(
n
,
λ
)
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
1
-
γ
]
2
×
(
a
n
,
1
2
+
a
n
,
2
2
)
.
That is, if
(71)
δ
≤
1
-
2
(
1
-
γ
)
2
[
(
n
+
1
)
(
1
+
β
)
]
[
n
(
1
+
β
)
+
(
γ
+
β
)
]
2
Ψ
(
n
,
λ
)
+
2
(
1
-
γ
)
2
,
where Ψ
(
n
,
λ
) is given by (55) and Ψ
(
n
,
λ
) is a decreasing function of n
(
n
≥
1
), we get (66), which completes the proof.


## 5. Closure Theorems

We state the following closure theorems forf
∈
ℳ
m
l
(
λ
,
β
,
γ
) without proof (see [8–10]).Theorem 13.
Let the functionf
k
(
z
)
=
(
1
/
(
z
-
ξ
)
)
+
∑
n
=
1
∞
‍
a
n
,
k
(
z
-
ξ
)
n be in the class ℳ
m
l
(
λ
,
β
,
γ
) for every k
=
1,2
,
…
,
m. Then the function f defined by
(72)
f
(
z
)
=
1
z
-
ξ
+
∑
n
=
1
∞
‍
a
n
,
k
(
z
-
ξ
)
n
,
(
a
n
,
k
≥
0
)
belongs to the class ℳ
m
l
(
λ
,
β
,
γ
), where a
n
,
k
=
(
1
/
m
)
∑
k
=
1
m
‍
a
n
,
k, (
n
=
1,2
,
…
).Theorem 14.
Letf
0
(
z
)
=
1
/
(
z
-
ξ
) and f
n
(
z
)
=
(
1
/
(
z
-
ξ
)
)
+
(
(
1
-
γ
)
(
1
-
2
λ
)
/
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
)
(
z
-
ξ
)
n for n
=
1,2
,
…. Then f
∈
ℳ
m
l
(
λ
,
β
,
γ
) if and only if f can be expressed in the form f
(
z
)
=
∑
n
=
0
∞
‍
η
n
f
n
(
z
) where η
n
≥
0 and ∑
n
=
0
∞
‍
η
n
=
1.Theorem 15.
The classℳ
m
l
(
λ
,
β
,
γ
) is closed under convex linear combination.Now, we prove that the class isℳ
m
l
(
λ
,
β
,
γ
) closed under integral transforms.Theorem 16.
Let the functionf
(
z
) given by (4) be in ℳ
m
l
(
λ
,
β
,
γ
). Then the integral operator
(73)
F
(
z
)
=
c
∫
0
1
‍
u
c
f
(
u
z
)
d
u
(
0
<
u
≤
1,0
<
c
<
∞
)
is in ℳ
m
l
(
λ
,
β
,
δ
), where
(74)
δ
≤
(
+
(
c
+
1
)
(
γ
+
β
)
+
c
β
(
1
-
γ
)
n
2
(
1
+
β
)
+
n
[
(
γ
+
β
)
+
(
1
+
β
)
(
1
+
c
γ
)
]
+
(
c
+
1
)
(
γ
+
β
)
+
c
β
(
1
-
γ
)
n
2
(
1
+
β
)
+
n
[
(
γ
+
β
)
+
(
1
+
β
)
(
1
+
c
γ
)
]
)
×
(
+
(
1
+
c
)
(
γ
+
β
)
+
c
(
1
-
γ
)
n
2
(
1
+
β
)
+
n
[
(
γ
+
β
)
+
(
1
+
c
)
(
1
+
β
)
]
+
(
1
+
c
)
(
γ
+
β
)
+
c
(
1
-
γ
)
n
2
(
1
+
β
)
+
n
[
(
γ
+
β
)
+
(
1
+
c
)
(
1
+
β
)
]
)
-
1
.
The result is sharp for the functionf
(
z
)
=
(
1
/
(
z
-
ξ
)
)
+
(
(
1
-
γ
)
(
1
-
2
λ
)
/
(
1
+
γ
+
2
β
)
Υ
m
l
(
1
)
)
(
z
-
ξ
).Proof.
Letf
(
z
)
∈
ℳ
m
l
(
λ
,
β
,
γ
). Then
(75)
F
(
z
)
=
c
∫
0
1
‍
u
c
f
(
u
z
)
d
u
=
1
z
-
w
+
∑
n
=
1
∞
‍
c
c
+
n
+
1
a
n
(
z
-
ξ
)
n
.
It is sufficient to show that
(76)
∑
n
=
1
∞
‍
c
d
n
(
λ
,
β
,
δ
)
Υ
m
l
(
n
)
(
c
+
n
+
1
)
(
1
-
δ
)
a
n
≤
1
.
Since f
∈
ℳ
m
l
(
λ
,
β
,
γ
), we have
(77)
∑
n
=
1
∞
‍
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
(
1
-
γ
)
(
1
-
2
λ
)
a
n
≤
1
.
Note that (76) is satisfied if
(78)
c
d
n
(
λ
,
β
,
δ
)
Υ
m
l
(
n
)
(
c
+
n
+
1
)
(
1
-
δ
)
≤
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
(
1
-
γ
)
(
1
-
2
λ
)
.
From (78), we have
(79)
δ
≤
(
+
(
1
+
c
)
(
γ
+
β
)
+
c
(
1
-
γ
)
n
2
)
-
1
(
n
2
(
1
+
β
)
+
n
[
(
γ
+
β
)
+
(
1
+
β
)
(
1
+
c
γ
)
]
+
(
c
+
1
)
(
γ
+
β
)
+
c
β
(
1
-
γ
)
n
2
)
×
(
n
2
(
1
+
β
)
+
n
[
(
γ
+
β
)
+
(
1
+
c
)
(
1
+
β
)
]
+
(
1
+
c
)
(
γ
+
β
)
+
c
(
1
-
γ
)
n
2
)
-
1
)
=
Φ
(
n
)
.
A simple computation will show that Φ
(
n
) is increasing and Φ
(
n
)
≥
Φ
(
1
). Using this, the results follow.


## 6. Partial Sums

Silverman [15] determined sharp lower bounds on the real part of the quotients between the normalized starlike or convex functions and their sequences of partial sums. As a natural extension, one is interested in searching results analogous to those of Silverman for meromorphic univalent functions. In this section, motivated essentially by the work of Silverman [15] and Cho and Owa [16], we will investigate the ratio of a function of the form (4) to its sequence of partial sums. Consider
(80)
f
k
(
z
)
=
1
z
-
ξ
+
∑
n
=
1
k
‍
a
n
(
z
-
ξ
)
n
,
when the coefficients are sufficiently small to satisfy the condition analogous to
(81)
∑
n
=
1
∞
‍
d
n
(
λ
,
β
,
γ
)
Υ
m
l
(
n
)
a
n
≤
(
1
-
γ
)
(
1
-
2
λ
)
.
More precisely we will determine sharp lower bounds for ℜ
(
f
(
z
)
/
f
k
(
z
)
) and ℜ
(
f
k
(
z
)
/
f
(
z
)
). In this connection we make use of the well-known results that ℜ
(
(
1
+
w
(
z
)
)
/
(
1
-
w
(
z
)
)
)
>
0, (
z
-
ξ
∈
Δ
), if and only if w
(
z
)
=
∑
n
=
1
∞
c
n
(
z
-
ξ
)
n satisfies the inequality |
w
(
z
)
|
≤
|
z
-
ξ
|.Unless otherwise stated, we will assume thatf is of the form (4) and its sequence of partial sums is denoted by (80).Theorem 17.
Letf
(
z
)
∈
ℳ
m
l
(
λ
,
β
,
γ
) be given by (4) which satisfies condition (26) and suppose that all of its partial sums (80) do not vanish in Δ. Moreover, suppose that
(82)
2
-
2
∑
n
=
1
k
‍
|
a
n
|
-
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
(
1
-
γ
)
(
1
-
2
λ
)
∑
n
=
k
+
1
∞
‍
|
a
n
|
>
0
,
00000000000000000000000000
i
000000000
∀
k
∈
ℕ
.
Then,
(83)
ℜ
(
f
(
z
)
f
k
(
z
)
)
≥
1
-
(
1
-
γ
)
(
1
-
2
λ
)
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
(
z
-
ξ
∈
Δ
)
,
where
(84)
d
n
(
λ
,
β
,
γ
)
≥
{
(
1
-
γ
)
(
1
-
2
λ
)
,
i
f
n
=
1,2
,
3
,
…
,
k
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
,
i
f
n
=
k
+
1
,
k
+
2
,
…
.
The result (83) is sharp with the function given by
(85)
f
(
z
)
=
1
z
-
ξ
+
(
1
-
γ
)
(
1
-
2
λ
)
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
(
z
-
ξ
)
k
+
1
.Proof.
Define the functionw
(
z
) by
(86)
w
(
z
)
=
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
(
1
-
γ
)
(
1
-
2
λ
)
×
[
f
(
z
)
f
k
(
z
)
-
(
1
-
(
1
-
γ
)
(
1
-
2
λ
)
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
)
]
=
1
+
(
(
1
+
∑
n
=
1
k
‍
a
n
(
z
-
ξ
)
n
+
1
)
-
1
(
∑
n
=
k
+
1
∞
‍
(
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
)
×
(
(
1
-
γ
)
(
1
-
2
λ
)
)
-
1
×
∑
n
=
k
+
1
∞
‍
a
n
(
z
-
ξ
)
n
+
1
)
×
(
1
+
∑
n
=
1
k
‍
a
n
(
z
-
ξ
)
n
+
1
)
-
1
)
.
It suffices to show that ℜ
(
w
(
z
)
)
>
0; hence we find that
(87)
|
1
+
w
(
z
)
1
-
w
(
z
)
|
≤
(
∑
n
=
k
+
1
∞
‍
|
a
n
|
)
-
1
(
∑
n
=
k
+
1
∞
‍
|
a
n
|
(
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
)
×
(
(
1
-
γ
)
(
1
-
2
λ
)
)
-
1
×
∑
n
=
k
+
1
∞
‍
|
a
n
|
)
×
(
2
-
2
∑
n
=
1
k
‍
|
a
n
|
-
(
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
)
×
(
(
1
-
γ
)
(
1
-
2
λ
)
)
-
1
×
∑
n
=
k
+
1
∞
‍
|
a
n
|
)
-
1
)
≤
1
.
From condition (26), it readily yields the assertion (83) of Theorem 17.
To see that the function given by (85) gives the sharp result, we observe that for z
=
r
e
i
π
/
(
k
+
2
)
(88)
f
(
z
)
f
k
(
z
)
=
1
+
(
1
-
γ
)
(
1
-
2
λ
)
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
(
z
-
ξ
)
n
⟶
1
-
(
1
-
γ
)
(
1
-
2
λ
)
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
,
when r
→
1
- which shows that the bound (83) is the best possible for each k
∈
ℕ.We next determine bounds forf
k
(
z
)
/
f
(
z
).Theorem 18.
Under the assumptions of Theorem17, we have
(89)
ℜ
(
f
k
(
z
)
f
(
z
)
)
≥
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
+
(
1
-
γ
)
(
1
-
2
λ
)
00000000000000000000000000000000000
(
z
-
w
∈
Δ
)
,
The result (89) is sharp with the function given by (85).Proof.
By setting(90)
w
(
z
)
=
(
1
+
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
(
1
-
γ
)
(
1
-
2
λ
)
)
×
[
(
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
/
(
1
-
γ
)
(
1
-
2
λ
)
)
1
+
(
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
/
(
1
-
γ
)
(
1
-
2
λ
)
)
f
k
(
z
)
f
(
z
)
-
(
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
/
(
1
-
γ
)
(
1
-
2
λ
)
)
1
+
(
d
k
+
1
(
λ
,
β
,
γ
)
Υ
m
l
(
k
+
1
)
/
(
1
-
γ
)
(
1
-
2
λ
)
)
]
and proceeding as in Theorem 17, we get the desired result and so we omit the details.Concluding Remark. We observe that, if we specialize the parameters λ and β as mentioned in Examples 1 and 2, we obtain the analogous results for the classes ℳ
m
l
(
β
,
γ
) and ℳ
m
l
(
γ
). Further specializing the parameters l, m various other interesting results (as in Theorems 6–18) can be derived easily for the function class based on interesting differential operators as illustrated below.(1) Fora
i
=
q
a
i, b
j
=
q
b
j, a
i
>
0, b
j
>
0, (
i
=
1
,
…
,
l
;
j
=
1
,
…
,
m
,
l
=
m
+
1
), q
→
1, the operator ℐ
m
l
f
(
z
)
=
ℋ
m
l
[
a
1
]
f
(
z
) defined by Liu and Srivastava [10].(2) Forl
=
2, m
=
1, a
2
=
q, q
→
1, the operator ℒ
1
2
[
a
1
,
q
,
b
1
,
q
]
f
(
z
)
=
ℒ
[
a
1
;
b
1
]
f
(
z
) was introduced and studied by Liu and Srivastava [9].(3) Forl
=
1, m
=
0, a
1
=
δ
+
1, q
→
1, the operator ℒ
[
a
1
;
b
1
]
f
(
z
)
=
D
δ
f
(
z
)
=
(
1
/
z
(
1
-
z
)
δ
+
1
)
*
f
(
z
), (
δ
>
-
1
) where D
δ is the differential operator which was introduced by Ganigi and Uralegaddi [17].



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*Source: 923607-2014-04-07.xml*