# Convolution Properties for Certain Classes of Analytic Functions Defined byq-Derivative Operator

Authors: T. M. Seoudy; M. K. Aouf
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We investigate convolution properties and coefficients estimates for two classes of analytic functions involving theq-derivative operator defined in the open unit disc. Some of our results improve previously known results.


---

## Body

## 1. Introduction

Simply,h-calculus or q-calculus is ordinary classical calculus without the notion of limits. Here h ostensibly stands for Planck’s constant, while q stands for quantum. Recently, the area of q-calculus has attracted the serious attention of researchers. This great interest is due to its application in various branches of mathematics and physics. The application of q-calculus was initiated by Jackson [1, 2]. He was the first to develop q-integral and q-derivative in a systematic way. Later, geometrical interpretation of q-analysis has been recognized through studies on quantum groups. It also suggests a relation between integrable systems and q-analysis. Aral and Gupta [3–5] defined and studied the q-analogue of Baskakov Durrmeyer operator which is based on q-analogue of beta function. Another important q-generalization of complex operators is q-Picard and q-Gauss-Weierstrass singular integral operators discussed in [6–8]. Mohammed and Darus [9] studied approximation and geometric properties of these q-operators in some subclasses of analytic functions in compact disk. These q-operators are defined by using convolution of normalized analytic functions and q-hypergeometric functions, where several interesting results are obtained (see also [10, 11]). A comprehensive study on applications of q-calculus in operator theory may be found in [12].LetA denote the class of functions of the form:
(1)
f
(
z
)
=
z
+
∑
k
=
2
∞
‍
a
k
z
k
which are analytic in the open unit disk U
=
{
z
∈
C
:
|
z
|
<
1
}. Let S
(
α
) and K
(
α
)
(
0
≤
α
<
1
) denote the subclasses of A that consists, respectively, of starlike of order α and convex of order α in U (see [13]). If f
(
z
) and g
(
z
) are analytic in U, we say that f
(
z
) is subordinate to g
(
z
), written f
(
z
)
≺
g
(
z
) if there exists a Schwarz function ω, which (by definition) is analytic in U with ω
(
0
)
=
0 and |
ω
(
z
)
|
<
1for all z
∈
U, such that f
(
z
)
=
g
(
ω
(
z
)
), z
∈
U. Furthermore, if the function gis univalent in U, then we have the following equivalence (see [14–16]):
(2)
f
(
z
)
≺
g
(
z
)
⟺
f
(
0
)
=
g
(
0
)
,
f
(
U
)
⊂
g
(
U
)
.For functionsf given by (1) and g given by
(3)
g
(
z
)
=
z
+
∑
k
=
2
∞
‍
b
k
z
k
the Hadamard product or convolution of f and g is defined by
(4)
(
f
*
g
)
(
z
)
=
z
+
∑
k
=
2
∞
‍
a
k
b
k
z
k
=
(
g
*
f
)
(
z
)
.
Let S
[
A
,
B
] and K
[
A
,
B
] denote the subclasses of the class A for -
1
≤
B
<
A
≤
1 which are defined by (see [17–22])
(5)
S
[
A
,
B
]
=
{
f
∈
A
:
z
f
′
(
z
)
f
(
z
)
≺
1
+
A
z
1
+
B
z
,
z
∈
U
}
,
K
[
A
,
B
]
=
{
f
∈
A
:
(
z
f
′
(
z
)
)
′
f
′
(
z
)
≺
1
+
A
z
1
+
B
z
,
z
∈
U
}
.
We note that
(6)
S
[
1
-
2
α
,
-
1
]
=
S
(
α
)
,
K
[
1
-
2
α
,
-
1
]
=
K
(
α
)
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
(
0
≤
α
<
1
)
.
For function f
∈
A given by (1) and 0
<
q
<
1, the q-derivative of a function f is defined by (see [1])
(7)
D
q
f
(
z
)
=
f
(
q
z
)
-
f
(
z
)
(
q
-
1
)
z
(
z
≠
0
)
,
and D
q
f
(
0
)
=
f
′
(
0
). From (7), we deduce that
(8)
D
q
f
(
z
)
=
1
+
∑
k
=
2
∞
‍
[
k
]
q
a
k
z
k
-
1
,
z
≠
0
,
where
(9)
[
k
]
q
=
1
-
q
k
1
-
q
.
As q
→
1, [
k
]
q
→
k. For a function h
(
z
)
=
z
k, we observe that
(10)
D
q
h
(
z
)
=
D
q
(
z
k
)
=
1
-
q
k
1
-
q
z
k
-
1
=
[
k
]
q
z
k
-
1
,
lim
⁡
q
→
1
D
q
h
(
z
)
=
lim
⁡
q
→
1
[
k
]
q
z
k
-
1
=
k
z
k
-
1
=
h
′
(
z
)
,
where h
′ is the ordinary derivative.Making use of theq-derivative D
q
f
(
z
), we introduce the subclasses S
q
[
A
,
B
] and K
q
[
A
,
B
] of A for 0
<
q
<
1 and -
1
≤
B
<
A
≤
1 as follows:
(11)
S
q
[
A
,
B
]
=
{
f
∈
A
:
z
D
q
f
(
z
)
f
(
z
)
≺
1
+
A
z
1
+
B
z
,
z
∈
U
}
,
K
q
[
A
,
B
]
=
{
f
∈
A
:
D
q
(
z
D
q
f
(
z
)
)
D
q
f
(
z
)
≺
1
+
A
z
1
+
B
z
,
z
∈
U
}
.
We note that(i)
S
q
[
1
-
2
α
,
-
1
]
=
S
q
(
α
)
(
0
≤
α
<
1
)
(12)
S
q
(
α
)
=
{
f
∈
A
:
Re
z
D
q
f
(
z
)
f
(
z
)
>
α
,
z
∈
U
}
;
(ii)
K
q
[
1
-
2
α
,
-
1
]
=
K
q
(
α
)
(
0
≤
α
<
1
)
(13)
K
q
(
α
)
=
{
f
∈
A
:
Re
D
q
(
z
D
q
f
(
z
)
)
D
q
f
(
z
)
>
α
,
z
∈
U
}
;
(iii)
S
q
[
(
1
-
2
α
)
β
,
-
β
]
=
S
q
(
α
,
β
)
(
0
≤
α
<
1
,
0
<
β
≤
1
)
(14)
S
q
(
α
,
β
)
=
{
f
∈
A
:
|
(
z
D
q
f
(
z
)
/
f
(
z
)
)
-
1
(
z
D
q
f
(
z
)
/
f
(
z
)
)
+
1
-
2
α
|
<
β
,
z
∈
U
}
,
(iv)
K
q
[
(
1
-
2
α
)
β
,
-
β
]
=
K
q
(
α
,
β
)
(
0
≤
α
<
1
,
0
<
β
≤
1
)
(15)
K
q
(
α
,
β
)
=
{
f
∈
A
:
|
(
D
q
(
z
D
q
f
(
z
)
)
/
D
q
f
(
z
)
)
-
1
(
D
q
(
z
D
q
f
(
z
)
)
/
D
q
f
(
z
)
)
+
1
-
2
α
|
<
β
,
h
h
h
h
h
h
h
h
h
h
h
z
∈
U
|
(
D
q
(
z
D
q
f
(
z
)
)
/
D
q
f
(
z
)
)
-
1
(
D
q
(
z
D
q
f
(
z
)
)
/
D
q
f
(
z
)
)
+
1
-
2
α
|
}
,
(v)
(16)
lim
⁡
q
→
1
S
q
[
A
,
B
]
=
{
f
∈
A
:
lim
⁡
q
→
1
z
D
q
f
(
z
)
f
(
z
)
≺
1
+
A
z
1
+
B
z
}
=
S
[
A
,
B
]
,
lim
⁡
q
→
1
K
q
[
A
,
B
]
=
{
f
∈
A
:
lim
⁡
q
→
1
D
q
(
z
D
q
f
(
z
)
)
D
q
f
(
z
)
≺
1
+
A
z
1
+
B
z
}
=
K
[
A
,
B
]
.From (11), we have
(17)
f
∈
K
q
[
A
,
B
]
⟺
z
D
q
f
∈
S
q
[
A
,
B
]
.
In this paper, we investigate convolution properties, the necessary and sufficient condition and coefficient estimates for the classes S
q
[
A
,
B
] and K
q
[
A
,
B
] associated with the q-derivative D
q
f
(
z
). The motivation of this paper is to improve and generalize previously known results.


## 2. Convolution Properties

Unless otherwise mentioned, we assume throughout this section thatθ
∈
[
0,2
π
), 0
<
q
<
1 and -
1
≤
B
<
A
≤
1.Theorem 1.
The functionf defined by (1) is in the class S
q
[
A
,
B
] if and only if
(18)
1
z
[
f
(
z
)
*
z
-
L
q
z
2
(
1
-
z
)
(
1
-
q
z
)
]
≠
0
(
z
∈
U
)
for all L
=
L
θ
=
(
e
-
i
θ
+
A
)
/
(
A
-
B
) and also L
=
1.Proof.
First supposef defined by (1) is in the class S
q
[
A
,
B
]; we have
(19)
z
D
q
f
(
z
)
f
(
z
)
≺
1
+
A
z
1
+
B
z
.
Since the function from the left-hand side of the subordination is analytic in U, it follows f
(
z
)
≠
0, z
∈
U
*
=
U
∖
{
0
}; that is, (
1
/
z
)
f
(
z
)
≠
0
,
z
∈
U, and this is equivalent to the fact that (18) holds for L
=
1. From (19) according to the subordination of two analytic functions we say that there exists a function w
(
z
) analytic in U with w
(
0
)
=
0, |
w
(
z
)
|
<
1 such that
(20)
z
D
q
f
(
z
)
f
(
z
)
=
1
+
A
w
(
z
)
1
+
B
w
(
z
)
(
z
∈
U
)
which is equivalent to
(21)
z
D
q
f
(
z
)
f
(
z
)
≠
1
+
A
e
i
θ
1
+
B
e
i
θ
(
z
∈
U
;
0
≤
θ
<
2
π
)
,
or
(22)
1
z
[
(
1
+
B
e
i
θ
)
z
D
q
f
(
z
)
-
(
1
+
A
e
i
θ
)
f
(
z
)
]
≠
0
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
(
z
∈
U
;
0
≤
θ
<
2
π
)
.
Since
(23)
f
(
z
)
*
z
1
-
z
=
f
(
z
)
,
f
(
z
)
*
z
(
1
-
z
)
(
1
-
q
z
)
=
z
D
q
f
(
z
)
.
Now from (23), we may write (22) as
(24)
1
z
[
f
(
z
)
*
(
(
1
+
B
e
i
θ
)
z
(
1
-
z
)
(
1
-
q
z
)
-
(
1
+
A
e
i
θ
)
z
1
-
z
)
]
=
(
B
-
A
)
e
i
θ
z
×
[
f
(
z
)
*
z
-
(
(
e
-
i
θ
+
A
)
/
(
A
-
B
)
)
q
z
2
(
1
-
z
)
(
1
-
q
z
)
]
≠
0
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
i
(
z
∈
U
;
0
≤
θ
<
2
π
)
,
which leads to (18), which proves the necessary part of Theorem 1.
Reversely, because assumption (18) holds for L
=
1, it follows that (
1
/
z
)
f
(
z
)
≠
0 for all z
∈
U; hence, the function φ
(
z
)
=
z
D
q
f
(
z
)
/
f
(
z
) is analytic in U (i.e., it is regular at z
0
=
0, with φ
(
0
)
=
0). Since it was shown in the first part of the proof that assumption (18) is equivalent to (21), we obtain that
(25)
z
D
q
f
(
z
)
f
(
z
)
≠
1
+
A
e
i
θ
1
+
B
e
i
θ
(
z
∈
U
;
0
≤
θ
<
2
π
)
,
and if we denote
(26)
ψ
(
z
)
=
1
+
A
z
1
+
B
z
(
z
∈
U
)
,
relation (25) shows that φ
(
U
)
∩
ψ
(
U
)
=
∅. Thus, the simply connected domain φ
(
U
) is included in a connected component of C
∖
ψ
(
∂
U
). From here, using the fact that φ
(
0
)
=
ψ
(
0
) together with the univalence of the function ψ, it follows that φ
(
z
)
≺
ψ
(
z
), which represents in fact subordination (19); that is, f
∈
S
q
[
A
,
B
]. This completes the proof of Theorem 1.Takingq
→
1
- in Theorem 1, we obtain the following result which improves the convolution result of Aouf and Seoudy [23, Theorem 1] and also the result of Silverman and Silvia [21, Theorem 7].Corollary 2.
The functionf defined by (1) is in the class S
[
A
,
B
] if and only if
(27)
1
z
[
f
(
z
)
*
z
-
L
z
2
(
1
-
z
)
2
]
≠
0
(
z
∈
U
)
for all L
=
L
θ
=
(
e
-
i
θ
+
A
)
/
(
A
-
B
) and also L
=
1.PuttingA
=
1
-
2
α
(
0
≤
α
<
1
) and B
=
-
1 in Theorem 1, we obtain the following corollary.Corollary 3.
The functionf defined by (1) is in the class S
q
(
α
)
(
0
≤
α
<
1
) if and only if
(28)
1
z
[
f
(
z
)
*
z
-
M
q
z
2
(
1
-
z
)
(
1
-
q
z
)
]
≠
0
(
z
∈
U
)
for all M
=
M
θ
=
(
e
-
i
θ
+
1
-
2
α
)
/
2
(
1
-
α
)
,
0
≤
α
<
1, and also M
=
1.Takingq
→
1
- in Corollary 3, we obtain the following result which improves the convolution result of Silverman et al. [22, Theorems 1].Corollary 4.
The functionf defined by (1) is in the class S
(
α
)
(
0
≤
α
<
1
) if and only if
(29)
1
z
[
f
(
z
)
*
z
-
M
z
2
(
1
-
z
)
2
]
≠
0
(
z
∈
U
)
for all M
=
M
θ
=
(
e
-
i
θ
+
1
-
2
α
)
/
2
(
1
-
α
), 0
≤
α
<
1, and also M
=
1.Theorem 5.
The functionf defined by (1) is in the class K
q
[
A
,
B
] if and only if
(30)
1
z
[
f
(
z
)
*
z
+
[
1
-
(
q
+
1
)
L
]
q
z
2
(
1
-
z
)
(
1
-
q
z
)
(
1
-
q
2
z
)
]
≠
0
(
z
∈
U
)
for all L
=
L
θ
=
(
e
-
i
θ
+
A
)
/
(
A
-
B
) and also L
=
1.Proof.
Set(31)
g
(
z
)
=
z
-
L
q
z
2
(
1
-
z
)
(
1
-
q
z
)
,
and we note that
(32)
z
D
q
g
(
z
)
=
z
+
[
1
-
(
q
+
1
)
L
]
q
z
2
(
1
-
z
)
(
1
-
q
z
)
(
1
-
q
2
z
)
.
From the identity z
D
q
f
(
z
)
*
g
(
z
)
=
f
(
z
)
*
z
D
q
g
(
z
)
(
f
,
g
∈
A
) and the fact that
(33)
f
∈
K
q
[
A
,
B
]
⟺
z
D
q
f
(
z
)
∈
S
q
[
A
,
B
]
the result follows from Theorem 1.Takingq
→
1
- in Theorem 1, we obtain the following result which improves the result of Aouf and Seoudy [23, Theorem 2].Corollary 6.
The functionf defined by (1) is in the class K
[
A
,
B
] if and only if
(34)
1
z
[
f
(
z
)
*
z
+
[
1
-
2
L
]
z
2
(
1
-
z
)
3
]
≠
0
(
z
∈
U
)
for all L
=
L
θ
=
(
e
-
i
θ
+
A
)
/
(
A
-
B
) and also L
=
1.PuttingA
=
1
-
2
α
(
0
≤
α
<
1
) and B
=
-
1 in Theorem 5, we obtain the following corollary.Corollary 7.
The functionf defined by (1) is in the class K
q
(
α
)
(
0
≤
α
<
1
) if and only if
(35)
1
z
[
f
(
z
)
*
z
+
[
1
-
(
q
+
1
)
L
]
q
z
2
(
1
-
z
)
(
1
-
q
z
)
(
1
-
q
2
z
)
]
≠
0
(
z
∈
U
)
for all M
=
M
θ
=
(
e
-
i
θ
+
1
-
2
α
)
/
2
(
1
-
α
), 0
≤
α
<
1, and also L
=
1.Takingq
→
1
- in Corollary 7, we obtain the following result which improves the convolution result of Silverman et al. [22, Theorem 2].Corollary 8.
The functionf defined by (1) is in the class K
(
α
)
(
0
≤
α
<
1
) if and only if
(36)
1
z
[
f
(
z
)
*
z
+
[
1
-
2
L
]
q
z
2
(
1
-
z
)
3
]
≠
0
(
z
∈
U
)
for all M
=
M
θ
=
(
e
-
i
θ
+
1
-
2
α
)
/
2
(
1
-
α
), 0
≤
α
<
1, and also L
=
1.Theorem 9.
A necessary and sufficient condition for the functionf defined by (1) to be in the class S
q
[
A
,
B
] is that
(37)
1
-
∑
k
=
2
∞
‍
[
k
]
q
(
e
-
i
θ
+
B
)
-
e
-
i
θ
-
A
A
-
B
a
k
z
k
-
1
≠
0
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
(
z
∈
U
)
.Proof.
From Theorem1, we find that f
∈
S
q
[
A
,
B
] if and only if
(38)
1
z
[
f
(
z
)
*
z
-
L
q
z
2
(
1
-
z
)
(
1
-
q
z
)
]
≠
0
(
z
∈
U
)
for all L
=
L
θ
=
(
e
-
i
θ
+
A
)
/
(
A
-
B
) and also for L
=
1. The left-hand side of (38) can be written as
(39)
1
z
[
f
(
z
)
*
(
z
(
1
-
z
)
(
1
-
q
z
)
-
L
q
z
2
(
1
-
z
)
(
1
-
q
z
)
)
]
=
1
z
{
z
D
q
f
(
z
)
-
L
[
z
D
q
f
(
z
)
-
f
(
z
)
]
}
=
1
-
∑
k
=
2
∞
‍
(
[
k
]
q
(
L
-
1
)
-
L
)
a
k
z
k
-
1
.
Thus, the proof of The Theorem 9 is completed.Takingq
→
1
- in Theorem 9, we obtain the following result.Corollary 10.
A necessary and sufficient condition for the functionf defined by (1) to be in the class S
[
A
,
B
] is that
(40)
1
-
∑
k
=
2
∞
‍
k
(
e
-
i
θ
+
B
)
-
e
-
i
θ
-
A
A
-
B
a
k
z
k
-
1
≠
0
(
z
∈
U
)
.PuttingA
=
1
-
2
α
(
0
≤
α
<
1
) and B
=
-
1 in Theorem 9, we obtain the following corollary.Corollary 11.
A necessary and sufficient condition for the functionf defined by (1) to be in the class S
q
(
α
) is that
(41)
1
-
∑
k
=
2
∞
‍
[
k
]
q
(
e
-
i
θ
-
1
)
-
e
-
i
θ
-
1
+
2
α
2
(
1
-
α
)
a
k
z
k
-
1
≠
0
(
z
∈
U
)
.Takingq
→
1
- in Corollary 11, we obtain the following corollary which improves the result of Ahuja [17, Corollary 1 when n
=
0].Corollary 12.
A necessary and sufficient condition for the functionf defined by (1) to be in the class S
(
α
) is that
(42)
1
-
∑
k
=
2
∞
‍
k
(
e
-
i
θ
-
1
)
-
e
-
i
θ
-
1
+
2
α
2
(
1
-
α
)
a
k
z
k
-
1
≠
0
(
z
∈
U
)
.Theorem 13.
A necessary and sufficient condition for the functionf
(
z
) defined by (1) to be in the class K
q
[
A
,
B
] is that
(43)
1
-
∑
k
=
2
∞
‍
[
k
]
q
[
k
]
q
(
e
-
i
θ
+
B
)
-
e
-
i
θ
-
A
A
-
B
a
k
z
k
-
1
≠
0
(
z
∈
U
)
.Proof.
From Theorem5, we find that f
∈
K
q
[
A
,
B
] if and only if
(44)
1
z
{
f
(
z
)
*
z
+
[
1
-
(
q
+
1
)
L
]
q
z
2
(
1
-
z
)
(
1
-
q
z
)
(
1
-
q
2
z
)
}
≠
0
(
z
∈
U
)
,
for all L
=
L
θ
=
(
e
-
i
θ
+
A
)
/
(
A
-
B
) and also for L
=
1. The left-hand side of (44) may be written as
(45)
1
z
{
f
(
z
)
*
(
z
(
1
-
z
)
(
1
-
q
z
)
(
1
-
q
2
z
)
h
h
h
h
h
h
h
h
h
+
[
1
-
(
q
+
1
)
L
]
q
z
2
(
1
-
z
)
(
1
-
q
z
)
(
1
-
q
2
z
)
)
}
=
1
z
{
q
z
2
D
q
(
D
q
f
(
z
)
)
+
z
D
q
f
(
z
)
-
L
[
q
z
2
D
q
(
D
q
f
(
z
)
)
]
}
=
1
-
∑
k
=
2
∞
‍
[
k
]
q
[
k
-
1
]
q
q
e
-
i
θ
-
A
+
[
k
]
q
B
A
-
B
a
k
z
k
-
1
,
and this proves Theorem 13.Takingq
→
1
- in Theorem 13, we obtain the following result.Corollary 14.
A necessary and sufficient condition for the functionf
(
z
) defined by (1) to be in the class K
[
A
,
B
] is that
(46)
1
-
∑
k
=
2
∞
‍
k
k
(
e
-
i
θ
+
B
)
-
e
-
i
θ
-
A
A
-
B
a
k
z
k
-
1
≠
0
(
z
∈
U
)
.PuttingA
=
1
-
2
α
(
0
≤
α
<
1
) and B
=
-
1 in Theorem 13, we obtain the following corollary.Corollary 15.
A necessary and sufficient condition for the functionf defined by (1) to be in the class K
q
(
α
)
(
0
≤
α
<
1
) is that
(47)
1
-
∑
k
=
2
∞
‍
[
k
]
q
[
k
]
q
(
e
-
i
θ
-
1
)
-
e
-
i
θ
-
1
+
2
α
2
(
1
-
α
)
a
k
z
k
-
1
≠
0
(
z
∈
U
)
.Takingq
→
1
- in Corollary 15, we obtain the following corollary which improves the result of Ahuja [17, Corollary 1 when n
=
1].Corollary 16.
A necessary and sufficient condition for the functionf defined by (1) to be in the class K
(
α
)
(
0
≤
α
<
1
) is that
(48)
1
-
∑
k
=
2
∞
‍
k
k
(
e
-
i
θ
-
1
)
-
e
-
i
θ
-
1
+
2
α
2
(
1
-
α
)
a
k
z
k
-
1
≠
0
(
z
∈
U
)
.


## 3. Coefficient Estimates

As an application of Theorems9 and 13, we next determine coefficient estimate and inclusion property for a function of form (1) to be in the classes S
q
[
A
,
B
] and K
q
[
A
,
B
].Theorem 17.
If the functionf defined by (1) satisfies the following inequality:
(49)
∑
k
=
2
∞
‍
{
[
k
]
q
(
1
-
B
)
-
1
+
A
}
|
a
k
|
≤
A
-
B
,
then f
∈
S
q
[
A
,
B
].Proof.
Since(50)
|
1
-
∑
k
=
2
∞
‍
[
k
]
q
(
e
-
i
θ
+
B
)
-
e
-
i
θ
-
A
A
-
B
a
k
z
k
-
1
|
>
1
-
∑
k
=
2
∞
‍
|
[
k
]
q
(
e
-
i
θ
+
B
)
-
e
-
i
θ
-
A
A
-
B
|
|
a
k
|
=
1
-
∑
k
=
2
∞
‍
|
[
k
]
q
(
e
-
i
θ
+
B
)
-
e
-
i
θ
-
A
|
A
-
B
|
a
k
|
>
1
-
∑
k
=
2
∞
‍
[
k
]
q
(
1
-
B
)
-
1
+
A
A
-
B
|
a
k
|
>
0
the result follows from Theorem 9.Takingq
→
1
- in Theorem 17, we obtain the result of Ahuja [17, Theorem 3 when n
=
0].Corollary 18.
If the functionf defined by (1) satisfies the following inequality:
(51)
∑
k
=
2
∞
‍
[
k
(
1
-
B
)
-
1
+
A
]
|
a
k
|
≤
A
-
B
,
then f
∈
S
[
A
,
B
].PuttingA
=
1
-
2
α
(
0
≤
α
<
1
) and B
=
-
1 in Theorem 21, we obtain the following corollary.Corollary 19.
If the functionf defined by (1) satisfies the following inequality:
(52)
∑
k
=
2
∞
‍
(
[
k
]
q
-
α
)
|
a
k
|
≤
1
-
α
,
then f
∈
S
q
(
α
).Takingq
→
1
- in Corollary 19, we obtain the following corollary obtained by Silverman [24].Corollary 20.
If the functionf defined by (1) satisfies the following inequality:
(53)
∑
k
=
2
∞
‍
(
k
-
α
)
|
a
k
|
≤
1
-
α
,
then f
∈
S
(
α
).Similarly, we can prove the following theorem.Theorem 21.
If the functionf defined by (1) satisfies the following inequality:
(54)
∑
k
=
2
∞
‍
[
k
]
q
{
[
k
]
q
(
1
-
B
)
-
1
+
A
}
|
a
k
|
≤
A
-
B
,
then f
∈
K
q
[
A
,
B
].Takingq
→
1
- in Theorem 21, we obtain the result of Ahuja [17, Theorem 3 when n
=
1].Corollary 22.
If the functionf defined by (1) satisfies the following inequality:
(55)
∑
k
=
2
∞
‍
k
[
k
(
1
-
B
)
-
1
+
A
]
|
a
k
|
≤
A
-
B
,
then f
∈
K
[
A
,
B
].PuttingA
=
1
-
2
α
(
0
≤
α
<
1
) and B
=
-
1 in Theorem 21, we obtain the following corollary.Corollary 23.
The functionf defined by (1) belongs to the class K
q
(
α
)
(
0
≤
α
<
1
) if
(56)
∑
k
=
2
∞
‍
[
k
]
q
(
[
k
]
q
-
α
)
|
a
k
|
≤
1
-
α
.Takingq
→
1
- in Corollary 23, we obtain the following corollary obtained by Silverman [24].Corollary 24.
The functionf defined by (1) belongs to the class K
(
α
)
(
0
≤
α
<
1
) if
(57)
∑
k
=
2
∞
‍
k
(
k
-
α
)
|
a
k
|
≤
1
-
α
.



---

*Source: 846719-2014-10-14.xml*