# Certain Properties of Some Families of Generalized Starlike Functions with respect toq-Calculus

Authors: Ben Wongsaijai; Nattakorn Sukantamala
Journal: Abstract and Applied Analysis
 (2016)

---

## Abstract

By making use of the concept ofq-calculus, various types of generalized starlike functions of order α were introduced and studied from different viewpoints. In this paper, we investigate the relation between various former types of q-starlike functions of order α. We also introduce and study a new subclass of q-starlike functions of order α. Moreover, we give some properties of those q-starlike functions with negative coefficient including the radius of univalency and starlikeness. Some illustrative examples are provided to verify the theoretical results in case of negative coefficient functions class.


---

## Body

## 1. Introduction and Preliminaries

The quantum calculus, so calledq-calculus and h-calculus, is the usual calculus without using the notion of limits. The letter h apparently stands for Planck’s constant and the letter q obviously stands for quantum. Here, quantum calculus is not the same as quantum physics. Due to the applications in various fields of mathematics and physics, the study of q-calculus has been very attractive for many researchers. Jackson [1, 2] was the first person in developing a q-derivative, also a q-integral, in a systematic mean. Afterward on quantum groups, the geometrical interpretation of q-analysis has been studied. The relation between q-analysis and integrable systems has been recognized. Based on q-analogue of beta function, Aral and Gupta [3–5] defined and studied the q-analogue of Baskakov Durrmeyer operator. Also, there are some discussions on q-Picard and q-Gauss-Weierstrass singular integral operators which are the other important q-generalization of complex operators (see [6–8]).In geometric function theory, there are many applications ofq-calculus on subclasses of analytic functions, especially subclasses of univalent functions. In [9], Ismail et al. first introduced the class of generalized functions via q-calculus. In [10], Raghavendar and Swaminathan have studied some basic properties of q-close-to-convex functions. In [11], Mohammed and Darus studied geometric properties and approximations of these q-operators in some subclasses of analytic functions in the disk. By using the convolution of normalized analytic functions and q-hypergeometric functions, these q-operators have been defined. The inclusive study on applications of q-calculus in operator theory could be seen in [12]. Recently, Esra Özkan Uçar [13] studied the coefficient inequality for q-closed-to-convex functions with respect to Janowski starlike functions. Here, many newsworthy results related to q-calculus and subclasses of analytic functions theory are studied by various authors (see [14–21]).LetD
r
=
z
∈
C
:
z
<
r be the open disk radius r centered at origin and the open unit disk is then defined by D
≡
D
1. We denote A by the class of functions f in the form(1)
f
z
=
z
+
∑
k
=
2
∞
a
k
z
k
z
∈
D
,which is analytic in D and satisfying the usual normalization condition f
(
0
)
=
f
′
(
0
)
-
1
=
0. We denote by S the subclass of A consisting of functions, which are univalent on D. A function f
∈
A is said to be starlike of order α
(
0
≤
α
<
1
) in D if f satisfies(2)
Re
⁡
z
f
′
z
f
z
>
α
z
∈
D
.We denote this class by S
∗
(
α
). In particular, we set S
∗
(
0
)
≡
S
∗ for a class of starlike functions on D. Class S
α
∗ is closely related to class S
∗
(
α). A function f
∈
A is said to belong to class S
α
∗ if f satisfies(3)
z
f
′
z
f
z
-
1
<
1
-
α
z
∈
D
.For the convenience, we provide some basic definitions and concept details ofq-calculus which are used in this paper. For any fixed complex number μ, a set A
⊂
C is called a μ-geometric set if for z
∈
A, μ
z
∈
A. Let f be a function defined on a q-geometric set. Jackson’s q-derivative and q-integral of a function on a subset of C are, respectively, given by (see Gasper and Rahman [22], pp. 19–22)(4)
D
q
f
z
=
f
z
-
f
z
q
z
1
-
q
,
z
≠
0
,
q
≠
0
,
∫
0
z
f
t
d
q
t
=
z
1
-
q
∑
k
=
0
∞
q
k
f
z
q
k
.In case f
(
z
)
=
z
n, the q-derivative and q-integral of f
(
z
), where n is a positive integer, are given by (5)
D
q
z
n
=
z
n
-
z
q
n
1
-
q
z
=
n
q
z
n
-
1
,
∫
0
z
t
n
d
q
t
=
z
1
-
q
∑
k
=
0
∞
q
k
z
q
k
n
=
z
n
+
1
n
+
1
q
.As q
→
1
- and n
∈
N, we have [
n
]
q
=
1
-
q
n
/
1
-
q
=
1
+
q
+
⋯
+
q
n
-
1
→
n.To generalize the class of starlike functions, it seems that replacing the derivative functionf
′, which appears in (2), by the q-difference operator D
q is an easily way to generalize the class of starlike functions. The definition turned out to be the following.Definition 1.
A functionf
∈
A is said to belong to class S
q
,
1
∗
(
α
), 0
≤
α
<
1, if(6)
Re
⁡
z
D
q
f
z
f
z
>
α
z
∈
D
.To put it in words, we call S
q
,
1
∗
(
α
) the class of q-starlike functions of order α type 1.Now we recall another way to generalize the class of starlike functions proposed by Ismail et al. [9]. In their works, the usual derivative was replaced by the q-difference operator D
q. Moreover, the right-half plane w
:
Re
⁡
w
>
α was substituted by an appropriate domain. Later, Agrawal and Sahoo in [14] extended the ideas in [9] to q-starlike function of order α. Then the definition turned out to be the following.Definition 2.
A functionf
∈
A is said to belong to class S
q
,
2
∗
(
α
), 0
≤
α
<
1, if(7)
z
D
q
f
z
/
f
z
-
α
1
-
α
-
1
1
-
q
<
1
1
-
q
z
∈
D
.To put it in words, we call S
q
,
2
∗
(
α
) the class of q-starlike functions of order α type 2.In addition, we now introduce new type ofq-starlike functions.Definition 3.
A functionf
∈
A is said to belong to class S
q
,
3
∗
(
α
), 0
≤
α
<
1, if(8)
z
D
q
f
z
f
z
-
1
<
1
-
α
z
∈
D
.To put it in words, we call S
q
,
3
∗
(
α
) the class of q-starlike functions of order α type 3.The main objective of this paper is to characterize in 4 sections. In Section2, we give some relations between such classes and a sufficient condition via coefficient inequality. In Section 3, we study some properties of those q-starlike functions of order α with negative coefficient. Here, some results on the radius of univalent and starlikeness order α on the class of q-starlike functions with negative coefficient are obtained. Some illustrative examples of radius of univalent and starlikeness on some functions with negative coefficient are demonstrated in Section 4.


## 2. Main Results

We first show the inclusion theorem via geometric properties of each type ofq-starlike functions.Theorem 4.
For0
<
α
<
1, then (9)
S
q
,
3
∗
α
⊂
S
q
,
2
∗
α
⊂
S
q
,
1
∗
α
.Proof.
Assuming thatf
∈
S
q
,
3
∗
(
α
), by using triangle inequality and (8), we have(10)
z
D
q
f
z
/
f
z
-
α
1
-
α
-
1
1
-
q
=
1
1
-
α
z
D
q
f
z
f
z
-
α
-
1
-
α
1
-
q
≤
1
1
-
α
z
D
q
f
z
f
z
-
1
+
q
1
-
q
≤
1
+
q
1
-
q
≔
1
1
-
q
.Then f
∈
S
q
,
2
∗
(
α
); that is, S
q
,
3
∗
(
α
)
⊂
S
q
,
2
∗
(
α
). Next, we let f
∈
S
q
,
2
∗
(
α
). Since (11)
f
∈
S
q
,
2
∗
α
⟺
z
D
q
f
z
f
z
-
1
-
α
q
1
-
q
<
1
-
α
1
-
q
,that is, z
D
q
f
(
z
)
/
f
(
z
) lies in the circle of radius 1
-
α
/
1
-
q with a center at 1
-
α
q
/
1
-
q, and we observe that (12)
1
-
α
q
1
-
q
-
1
-
α
1
-
q
=
α
,which means that Re
⁡
z
D
q
f
(
z
)
/
f
(
z
)
>
α, then f
∈
S
q
,
1
∗
(
α
); that is, S
q
,
2
∗
(
α
)
⊂
S
q
,
1
∗
(
α
). This completes the proof.Geometrically, forf
∈
S
q
,
k
∗
(
α
), k
=
1,2
,
3, z
D
q
f
(
z
)
/
f
(
z
) lied in the difference domains:(13)
Ω
1
=
w
∈
C
:
Re
⁡
w
>
α
,
Ω
2
=
w
∈
C
:
w
-
1
-
α
q
1
-
α
<
1
-
α
1
-
q
,
Ω
1
=
w
∈
C
:
w
-
1
<
1
-
α
,respectively; see Figure 1.Figure 1
Boundary of each domain.The next result is directly obtained by using Theorem4 and the result in [14].Corollary 5.
ClassesS
q
,
1
∗
(
α
), S
q
,
2
∗
(
α
), and S
q
,
3
∗
(
α
) satisfy the following properties: (14)
⋂
0
<
q
<
1
S
q
,
1
∗
α
=
⋂
0
<
q
<
1
S
q
,
2
∗
α
=
S
∗
α
,
⋂
0
<
q
<
1
S
q
,
1
∗
α
=
⋂
0
<
q
<
1
S
q
,
3
∗
α
⊂
S
∗
α
.Next, we give a sufficient condition ofS
q
,
3
∗ via coefficient inequality which guarantees a sufficient condition for S
q
,
1
∗ and S
q
,
2
∗.Theorem 6.
Iff
∈
A satisfies the inequality(15)
∑
k
=
2
∞
k
q
-
α
a
k
≤
1
-
α
,then f
(
z
) is a q-starlike function of order α type 3; that is, f
∈
S
q
,
3
∗
(
α
).Proof.
Suppose that inequality (15) holds. We obtain(16)
z
D
q
f
z
-
f
z
-
1
-
α
f
z
=
∑
k
=
2
∞
k
q
-
1
a
k
z
k
-
1
-
α
z
+
∑
k
=
2
∞
a
k
z
k
≤
∑
k
=
2
∞
k
q
-
1
a
k
-
1
-
α
1
-
∑
k
=
2
∞
a
k
=
∑
k
=
2
∞
k
q
-
1
a
k
-
1
-
α
.Then f
∈
S
q
,
3
∗
(
α
) as desired.Remark 7.
In Theorem6, if q
→
1
-, we obtain Theorem  1 in [23].


## 3. Functions with Negative Coefficients

Now, we introduce new subclasses ofq-starlike functions with negative coefficients. Let T be a subset of A containing negative coefficient functions; that is, (17)
f
z
=
z
-
∑
k
=
2
∞
a
k
z
k
.Next, we let (18)
T
S
q
,
k
∗
α
≡
S
q
,
k
∗
α
∩
T
,
k
=
1,2
,
3
.Theorem 8.
For0
<
α
<
1, then(19)
T
S
q
,
1
∗
α
≡
T
S
q
,
2
∗
α
≡
T
S
q
,
3
∗
α
.Proof.
By using Theorem4, it is sufficient to show that T
S
q
,
1
∗
(
α
)
⊂
T
S
q
,
3
∗
(
α
). Assuming that f
∈
T
S
q
,
1
∗
(
α
), we have(20)
Re
⁡
z
D
q
f
z
f
z
=
Re
⁡
1
-
∑
k
=
2
∞
k
q
a
k
z
k
-
1
1
-
∑
k
=
2
∞
a
k
z
k
-
1
>
α
.Take z on the real axis so that the value of z
D
q
f
(
z
)
/
f
(
z
) is real. Letting z approach 1
- on the real line, we have (21)
1
-
∑
k
=
2
∞
k
q
a
k
>
α
1
-
∑
k
=
2
∞
a
k
,which satisfies (15). Theorem 6 implies the proof of this theorem.By using the result of Theorem8, all types of q-starlike functions are exactly the same. For convenience, we introduce a new notation for each class of q-starlike functions T
S
q
,
k
∗
(
α
)
≡
T
S
q
∗
(
α
), for k
=
1,2, and 3
.By using Theorem6, it is easy to see that function (22)
f
0
z
=
z
-
1
-
α
-
ϵ
n
q
-
α
z
n
∈
T
S
q
∗
α
,where 0
<
ϵ
<
n
(
1
-
α
)
-
[
n
]
q
+
α
/
n and [
n
]
q
-
α
<
n
(
1
-
α
-
ϵ
), but f
0
′
(
z
)
=
0 at z
0
=
[
n
]
q
-
α
/
n
(
1
-
α
-
ϵ
)
1
/
n
cos
⁡
2
k
π
/
n
+
i
sin
⁡
2
k
π
/
n
∈
D. That is, f
0
(
z
)
∉
S and also f
0
(
z
)
∉
S
∗
(
α
). So, it is interesting to study the radius of univalency and starlikeness of class T
S
q
∗
(
α
).Lemma9 is required to prove the radius of univalency and starlikeness. By using the same techniques of Theorem  1 in [24] and Theorem  1 in [25], we can easily prove Lemma 9. So, the proof is omitted.Lemma 9.
Iff
∈
T, then f is univalent on D
r if and only if f is starlike on D
r.Theorem 10.
Iff
∈
T
S
q
∗
(
α
) then f is univalent and starlike in z
<
r
0, where(23)
r
0
=
min
2
≤
k
≤
M
0
⁡
k
q
-
α
k
1
-
α
1
/
k
-
1
,and M
0 satisfies M
0
>
e
1
+
ln
⁡
(
1
-
q
)
(
1
-
α
)
/
q
+
(
1
-
q
)
(
1
-
α
).Proof.
To prove this, we need to find0
<
r
0
≤
1 such that Re
⁡
f
′
z
>
0 on D
r
0, where D
r
0
=
z
∈
C
:
z
<
r
0 due to the following formula:(24)
Re
⁡
f
z
1
-
f
z
2
z
1
-
z
2
=
∫
0
1
Re
⁡
f
′
z
1
+
t
z
2
-
z
1
d
t
,which implies the univalency. Consider(25)
Re
⁡
f
′
z
=
Re
⁡
1
-
∑
k
=
2
∞
k
a
k
z
k
-
1
>
1
-
∑
k
=
2
∞
k
a
k
r
0
k
-
1
,for all z
<
r
0. By the application of Theorem 6 and (25), the inequality Re
⁡
f
′
z
>
0 holds on D
r
0, where (26)
r
0
=
inf
k
≥
2
⁡
k
q
-
α
k
1
-
α
1
/
k
-
1
.
Next, we need to findM
0
∈
N satisfying (23). Let f
:
[
2
,
∞
)
→
R
+ be the function defined by(27)
f
x
=
x
q
-
α
x
1
-
α
1
/
x
-
1
.Differentiating on both sides of (27) logarithmically, we have(28)
f
′
x
=
f
x
x
-
1
2
ln
⁡
x
-
x
-
1
q
x
ln
⁡
q
q
+
A
-
q
x
+
ln
⁡
A
q
+
A
-
q
x
-
x
-
1
x
,where A
=
(
1
-
q
)
(
1
-
α
). It is easy to see that the second term of (28) is positive. Since (29)
sup
x
≥
2
⁡
ln
⁡
A
q
+
A
-
q
x
=
ln
⁡
A
q
+
A
,
sup
x
≥
2
⁡
x
-
1
x
=
1
,then the third and the last term in (28) can be dominated by ln
⁡
x when x is sufficiently large. That implies that f is an increasing function on [
M
0
,
∞
], where M
0
>
e
1
+
ln
⁡
A
/
q
+
A. Therefore, the radius of univalency can be defined by (30)
r
0
=
inf
k
≥
2
⁡
k
q
-
α
k
1
-
α
1
/
k
-
1
=
min
2
≤
k
≤
M
0
⁡
k
q
-
α
k
1
-
α
1
/
k
-
1
.Finally, we complete the proof of this theorem by applying Lemma 9 to obtain the radius of starlikeness.Theorem11 guarantees the radius of starlike function of order α.Theorem 11.
Iff
∈
T
S
q
∗
(
α
) then f is starlike order α in z
<
r
1, where(31)
r
1
=
min
2
≤
k
≤
M
1
⁡
k
q
-
α
k
-
α
1
/
k
-
1
,and M
1 satisfies M
1
>
e
1
+
ln
⁡
1
-
q
/
1
-
α
(
1
-
q
).Proof.
We have to show thatz
f
′
(
z
)
/
z
-
1
<
1
-
α. That is,(32)
z
f
′
z
f
z
-
1
=
∑
k
=
2
∞
k
-
1
a
k
z
k
-
1
1
-
∑
k
=
2
∞
a
k
z
k
-
1
≤
∑
k
=
2
∞
k
-
1
a
k
z
k
-
1
1
-
∑
k
=
2
∞
a
k
z
k
-
1
≤
1
-
α
.Hence, (32) is true if (33)
∑
k
=
2
∞
k
-
α
a
k
z
k
-
1
≤
1
-
α
.By an application of Theorem 6, the above inequality holds on D
r
1, where (34)
r
1
=
inf
k
≥
2
⁡
k
q
-
α
k
-
α
1
/
k
-
1
.Finally, by using the same technique of Theorem 10, we obtain that function f
(
x
)
=
[
(
[
k
]
q
-
α
)
/
(
k
-
α
)
]
1
/
(
k
-
1
) is an increasing function on [
M
1
,
∞
), where M
1 satisfies M
1
>
e
1
+
ln
⁡
1
-
q
/
1
-
α
(
1
-
q
). This completes the proof.


## 4. Examples and Applications

In this section, we give some examples to verify the radius of univalency and starlikeness obtained by Theorems10 and 11.Example 1.
Consider classT
S
q
∗ with q
=
0.75.By Theorem10, we obtain the radius of univalency of class T
S
q
∗ given by (35)
r
0
=
min
2
≤
k
≤
e
1
+
ln
⁡
0.25
⁡
k
0.75
k
1
/
k
-
1
=
min
2
≤
k
≤
11
⁡
k
0.75
k
1
/
k
-
1
=
0.875
.Now, we consider the sharpness example function f
0
(
z
) defined in (22) with n
=
2 and ϵ
=
0.001; that is,(36)
f
0
z
=
z
-
0.999
1.75
z
2
.Obviously, f
z
(
z
) is locally univalent on D
0.875 because f
′
(
z
0
)
=
0 at z
0
≈
0.87587
… outside the open disk D
0.875. By applying Theorem 10, function f
0
(
z
) is univalent on D
0.875. Moreover, Figure 2 shows the image of ∂
D
r with maximum circumferences r
=
0.875 and r
=
1. Figure 2(a) demonstrates that function f
0
(
z
) is a univalent and starlike function on D
0.875. On the other hand, f
0
(
z
) is not a univalent on D (see Figure 2(b)).Figure 2
The image of∂
D
r with maximum circumferences r
=
0.875 (a) and r
=
1 (b) on the polynomial f
0
(
z
) defined in (36).
(a)
(b)Another example is in casen
=
5 with ϵ
=
0.001; that is,(37)
f
0
z
=
z
-
0.999
5
0.75
z
5
.We see that f is not locally univalent at z
0
=
5
0.75
/
4.995
1
/
4
cos
⁡
k
π
/
2
+
i
sin
⁡
k
π
/
2, for k
=
0,1
,
2,3 with z
0
=
0.88403
…. Figure 3 shows that function f
0 defined in (37) is univalent and starlike on D
0.88403 which contains the open disk D
0.875 from Theorem 10. That is, the example shows that radius r
0 in Theorem 10 is only the sufficient condition for univalency and starlikeness but it is not necessary condition due to function f
0
(
z
) defined in (37).Figure 3
The image of∂
D
r with maximum circumferences r
=
0.884 under the polynomial f
0
(
z
) defined in (37).The next example is the class ofq-starlike functions of order α.Example 2.
Consider classT
S
q
∗
(
α
) with q
=
0.75.In this example, we also setq
=
0.75. For α
=
0.5, by Theorem 10, we obtain the radius of univalency of class T
S
q
∗
(
0.5
) given by(38)
r
0
=
min
2
≤
k
≤
e
1
+
ln
⁡
1
-
q
1
-
α
/
q
+
1
-
q
1
-
α
⁡
k
0.75
k
1
/
k
-
1
=
min
2
≤
k
≤
19
⁡
k
0.75
k
1
/
k
-
1
≈
0.94554
.However, function f
0
(
z
) defined in (22) with n
=
2 and ϵ
=
0.001, that is,(39)
f
0
z
=
z
-
0.499
1.25
z
2
,is locally univalent on D
1.2525 which contains the open disk D
0.925. Then it seems that function f
0
(
z
) is univalent and starlike on D as demonstrated by Figure 4(a). Also function f
0
(
z
) defined in (22) with n
=
5 and ϵ
=
0.001, that is,(40)
f
0
z
=
z
-
0.499
5
0.75
-
0.5
z
5
,is locally univalent on D
1.0055 and it seems that function f
0
(
z
) is univalent and starlike on D as demonstrated by Figure 4(b).Figure 4
The image of∂
D
r with maximum circumferences r
=
1.2525 (a) and r
=
1.005 (b) on the polynomial f
0
(
z
) defined in (39) and (40), respectively.
(a)
(b)



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*Source: 6180140-2016-09-25.xml*