1. Introduction
Quantum calculus is basically usual calculus without the notion of limits. It has wide applications in mathematics and physics. The
q-derivative and
q-integral are the main tools introduced by Jackson [
1,
2] in a systematic way. The linear
q-difference equation, and
q-differential equations, are studied in [
3,
4]. Mansour [
5] investigated linear sequential
q-differential equation of fractional order. Using the
q-derivative, Ismail [
6] introduced the class of
q-starlike functions. In the recent past, the theory of
q-calculus operators has been applied in general fractional calculus. Al-Salam [
7] and Agarwal [
8] introduced several types of fractional
q-integral operators and fractional
q-derivatives. Rajkovi’c [
9] investigated fractional integral and derivatives in
q-calculus. Additionally,
q-integral operators for certain analytic functions by using the concept and theory of fractional
q-calculus that was studied by Selvakumaran et al. [
10]. Recently, researchers proposed
q-version of well known operators like Baskakov Durrmeyer operator, Picard integral operator, Bernardi integral operator and the Gauss–Weierstrass integral operator, see [
11,
12,
13,
14,
15]. Furthermore, Purohit and Raina [
16] applied
q-operators on subclasses of analytic functions. Ismail [
6] introduced the well known class of
q-starlike functions related to
q-derivative operator [
16,
17]. Wingsaijai and Sukantmala [
18] presented the class
of
q-starlike functions of order
,
, Certain Coefficient Estimate Problems for Three-Leaf-Type Starlike Functions. Sahoo and Sharma [
17] defined and studied
analogue of a close-to-convex function.
The starlikeness of normalized bessel functions with symmetric points is studied in [
19]. Recently, certain generalized classes of
q-starlike functions have been investigated, see [
20,
21]. Zainab et al. [
22] defined a new class of
q-starlike functions by using
q-Ruscheweyh differential operator. The recent contributions on fractional derivatives by several researchers are also worth reading, see [
23,
24,
25]. Sokół [
26] introduced a one-parameter family of functions
, as shown in (
5). Using this family of functions, he defined a classes of starlike functions, and certain properties of these functions were investigated. However,
has not been studied under the
q-analogue of analytic and univalent functions of negative order, which has vital applications in different zones of mathematics. This was the main motivation behind Definitions 1 and 3 and their related results, and to keep in mind recent developments on starlike functions and their associated functions, we have categorized our main results into two sections. In the first section, we have investigated some interesting properties for our new class
,
of
q-starlike functions of order
,
, which is introduced in Definition 1. We primarily focus on
q-integral representation of functions belonging to this class, and its related results. Further, we have investigated distortion bounds and order of starlikeness in class of convex functions. In the second section, we have defined the class
of
q-starlike functions of order
(
) with negative coefficients. It is investigated that functions belonging to this class are preserved under
q-Bernardi integral operator and its radius of univalence is also determined. Several other properties such as coefficient inequities, radius of
q-convexity, growth and distortion theorem, covering result and some applications of fractional
q-calculus for the said class are presented. It is noted that obtained results are the advancement of several known results, proved by researchers in their research articles.
Let
A consist of the analytic functions of the form
Let
be the class of univalent functions in
. The classes
of starlike functions of order
and
of convex functions of order
, which are subclasses of
S are defined as:
When
, we have the well-known class
of starlike functions and the class
C of convex functions, see [
27]. Let
,
. Then, we say that
is subordinate to
, written as
, if there exists a function
w, analytic in
E with
and
,
, such that
If
, it is known that the above subordination is equivalent to
and
, see [
27].
Let
T denote the subclass of
S consisting of the analytic and univalent functions, whose functions can be expressed as
Silverman [
28] introduced and studied the classes
and
of starlike functions of order
and convex functions of order
,
in the open unit disc
. He defined these classes as follows:
When
, the above classes reduce to the classes
and
K, of starlike and convex functions of negative coefficients, respectively, see [
28].
Our work is related to a one-parameter family of functions defined and studied by Sokół [
26]. We recall the properties of these functions, which we shall need to derive our results.
Remark 1. LetThen, the following assertions are true. - 1.
is univalent in E.
- 2.
- 3.
When , is convex univalent function in E.
Now, we include some basic definitions and concepts of q-calculus, which are used in this work.
The
q-derivative of a function
is defined by
and
, where
, see [
2]. For a function
, the
q-derivative is
where
We note that as
,
, here
is ordinary derivative and
as
. From (
4), one can deduce that
Jackson [
1] introduced the
q-integral of a function
f, which is given by
provided that series converges.
In [
18], Wongsaijai and Sukantamala introduce the class
of
q-starlike functions of order
as follows:
The corresponding class
of
q-convex functions is defined as
By seting
in above definitions, we get
of
q-convex functions and
of
q-starlike functions introduced in [
6].
Then, we define a new class
, which is the refinement of the above known classes of starlike functions. Results related to this class will be derived in
Section 2.
Definition 1. A function from the class A is said to be in the class if and only if it satisfies the conditionwhere is given by (5). From the Remark 1, we have
when
.
Our aim is to investigate geometric properties of class of q-starlike functions of order . It deals with several ideas and techniques used in geometric function theory. The order of starlikeness in the class of convex functions of negative order and distortion bounds is also formulated. It provides an interesting connection of our above-defined class with well known classes in the form of following special cases.
Special Cases
- 1.
- 2.
For
, we obtain
- 3.
Let
, and taking
, we get the class of starlike functions, which is univalent in
E, see [
27].
- 4.
If
, then
f belongs to the class
of
q-starlike functions, which is defined and studied in [
6].
- 5.
For
, we have the known class
of
q-starlike functions of order
, see [
27].
- 6.
When , then f belongs to the class of starlike function with order .
Now we define another class , a subclass of . This class will be used in derivation of Theorems 1 and 8.
Definition 2. For , , the class is defined as follows. We note that, for , then the function maps the unit disc E onto half plane and onto the disc with center and radius , for .
Next, we define the class
of
q-starlike functions of order
with negative coefficients. Results regarding this class are presented in
Section 3.
Definition 3. A function from the class T is said to be in the class if and only if it satisfies the condition Special Cases
- 1.
If
, we have
which contains
q-starlike functions with negative coefficients, and taking
, we obtain the known class
introduced in [
28].
- 2.
For
, we have
of
q-starlike functions of order
with negative coefficients, and taking
, we obtain the known class
K of convex functions defined and studied in [
28].
- 3.
When
, we obtain
of starlike function of order
with negative coefficients.
Next, we define the corresponding class of q-convex functions of order and having negative coefficients. An application of this class will be shown in investigating the radius problem as given in Theorem 10.
Definition 4. A function from the class T is said to be in the class if and only if it satisfies the condition We require following lemma to obtain our main results.
Lemma 1 (
q-Jack’s Lemma, [
29]).
Let be analytic in E with . Then, if attains its maximum value on the circle at a point , then we have real number. 2. The Class
In this section, we obtain some results related to newly defined class of q-starlike functions of order . For the following results, we consider , , , unless otherwise stated. To prove our main results, we first prove the following lemma.
Lemma 2. Let h be analytic in E with . A function G is in the classif and only if there exists an analytic function , , such that Proof. Consider
where
is analytic and
in
E. Using
q-integral properties, we get
It follows that
which is (
19). Now conversely, let (
19) hold, that is
The Logarithmic
q-differentiation of (
21) gives us
Using the formulation
, and the fundamental theorem of
q-calculus, see [
30], we get
which implies that
it follows that
, and this implies that
in
E. This completes the proof. □
By taking
in Lemma 2, we obtain the known result proved by Sokół [
26].
Then, by using the class
given by (
15), we derive the following theorem for the function
,
.
Theorem 1. Let , . If , then there exists a function and the function , such that(We note that, if ; then, Proof. Let
. Then, by Lemma 2, there exists an analytic function
with
and
,
, such that
and from (
5), we have
which implies that
Using the Lemma 2, we have
which shows that the functions
and
satisfy
and
, and this completes the proof. □
Theorem 2. Let , . If there exists a function and , such thatandfor analytic function , and , , then the function Proof. From (
23) and (
24), we find that
and
are generated by (
19) wih the same function
w, so by using Theorem 1 and Lemma 2, we have
Hence, we have
which is the required result. □
Next, we obtain distortion result for our class , by using Theorems 1 and 2.
Theorem 3. If , , and , , then Proof. Let
. Then, by Theorem 1, there exist
and
, such that (
25) holds.
Let
. Then, we have
which implies that
Using
q-differential properties and partial
q-derivatives, we get
The
q-integration gives us
Raising (
26) to power
, we get
Now, we suppose that
, so we can write
It follows that
as the linear transformation
maps
onto disc of center
and radius
. Additionally, we know that
so, we have
The
q-Integration on both sides gives us
Raising (
28) to the power
, we get
because
, when
Multiplying (
29) and (
27), we obtain our required result. □
Next, we will obtain the order of starlikeness in the class of convex functions.
Theorem 4. Let , and let , for ,Then, Proof. Consider
where
is analytic with
in
E. The
q-logarithmic differentiation of (
31) gives us
On contrary, we assume
, such that
,
and
,
; thus, we have
Using (
31), we have
where
If
, we have
Re-substituting
, and since
, so we have
where
which is contraction to our given hypothesis. Thus, the required result follows. □
We note that by substituting various values to the parameters involved in above result, we get known and new results, as shown in the following corollaries.
Corollary 1. Let , and . Then, ifthenBy further taking , we obtain the well known result that a convex function of order zero is starlike of order one-half, see [27]. Corollary 2. Let , and . Then,implies that Corollary 3. Let and . Then, ifthenThe following theorem shows the coefficients inequality for the functions of the class Theorem 5. Let , . Ifwhere is given by (8), then . Proof. It is sufficient to prove that the values for
lie in a circle centered at 1 and radius
. For this, consider
As (
34) is bounded by
if
which is equivalent to
However, (
35) is true by hypothesis. Thus, we have
, and this gives us the required result. □
Taking and in Theorem 5, we get the following known result.
Corollary 4 ([
31]).
Let , . Then, ifthen .Taking , and in Theorem 5, we obtain a result proved by Schild [32], as shown in the following corollary. Corollary 5 ([
32]).
Let Then, ifthen .Set in Theorem 5, this gives us the following result.
Corollary 6. Let Then, ifthen .