1. Introduction
Let
ξ
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 wellknown standard classes of convex and starlike functions.
Let
Σ
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 by
Σ
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 function
f
of the form (4) is in the class of meromorphic 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 of meromorphic 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 functions
f
(
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 parameters
a
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
)
.
The
q
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 function
l
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 LiuSrivastava 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 functions
f
∈
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 for
f
∈
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
.
For
0
≤
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 parameters
λ
,
β
and
l
,
m
, we can define various new subclasses of
Σ
ξ
. We illustrate two important subclasses in the following examples.
Example 1.
For
λ
=
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.
For
λ
=
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 class
ℳ
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.
If
γ
is a real number and
w
is a complex number, then
ℜ
(
w
)
≥
γ
⇔

w
+
(
1

γ
)



w

(
1
+
γ
)

≥
0
.
Lemma 4.
If
w
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 that
γ
∈
[
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.
Let
f
∈
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.
If
f
∈
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 in
ℳ
m
l
(
λ
,
β
,
γ
)
.
Theorem 7.
If
f
∈
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.
If
f
∈
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 number
ν
:
(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.
Let
f
∈
∑
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 class
ℳ
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 function
f
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 function
f
∈
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 number
r
~
,
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 functions
f
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 Theorem 6, 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 CauchySchwarz 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 functions
f
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 taking
δ
=
γ
in the above theorem, the results follow.
For functions in the class
ℳ
m
l
(
λ
,
β
,
γ
)
, we can prove the following inclusion property.
Theorem 12.
Let the functions
f
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 Theorem 6, 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 for
f
∈
ℳ
m
l
(
λ
,
β
,
γ
)
without proof (see [8–10]).
Theorem 13.
Let the function
f
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.
Let
f
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 class
ℳ
m
l
(
λ
,
β
,
γ
)
is closed under convex linear combination.
Now, we prove that the class is
ℳ
m
l
(
λ
,
β
,
γ
)
closed under integral transforms.
Theorem 16.
Let the function
f
(
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 function
f
(
z
)
=
(
1
/
(
z

ξ
)
)
+
(
(
1

γ
)
(
1

2
λ
)
/
(
1
+
γ
+
2
β
)
Υ
m
l
(
1
)
)
(
z

ξ
)
.
Proof.
Let
f
(
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 wellknown 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 that
f
is of the form (4) and its sequence of partial sums is denoted by (80).
Theorem 17.
Let
f
(
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 function
w
(
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 for
f
k
(
z
)
/
f
(
z
)
.
Theorem 18.
Under the assumptions of Theorem 17, 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) For
a
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) For
l
=
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) For
l
=
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].