1. Introduction
Assume that
represents the analytic functions class in an open unit disk
;
Here
denotes the complex numbers set.
Similarly we consider the class
of those analytic functions that satisfies
The class
is normalized by
In the literature, the univalent functions class in
is expressed by
. According to [
1], the starlike functions class in
is represented by
, that include
with given condition
Furthermore, the convex functions class in
is represented by
, that consists the functions
with given condition
It can be deduced from conditions defined in Equations (
2) and (
3) (see [
2]) that
The analytic functions of the form
are denoted by the class
, for which
Subordination between any two analytic functions
f and
g in
may be represented as
In case of Schwarz function
w in
, if
w is analytic and satisfies
then
Similarly, if
g satisfies condition of univalent function in
. The equivalence transformed into
This implies
and
The conic domain was introduced by Kanas et al., described in [
3], denoted by
having the form
The extremal functions,
, family of conic domain
, having
with
are normalized univalent functions in the form
where
and we choose
such that
Here
is Legendre’s complete elliptic integral of the first kind and
given by
is the complementary integral of
.
Definition 1 ([
4])
. Assume that h be an analytic function with , then , iff The class of analytic functions represented by
was initially introduced by Janowski in 1973 (see [
4]). He demonstrated that if a function
exists then
. Mathematically, it takes the form
Definition 2. Assume that define the q-number , then Definition 3. According to [5,6], the q-derivative of a function f in a subset of is defined by It provided the existence
. Similarly from Definition 3, it is noticed that
which is differentiable in the subset of
. We also deduced from Equations (
1) and (
7) that
Recently, the usage of the
q-derivative operator is quite significant due to its applications in many diverse areas of mathematics, physics and other sciences. According to Srivastava et al. [
7], the
q-difference operator
in the context of Geometric Function Theory (GFT) was first utilized by Ismail et al., described in [
8]. They studied a
q-extension of starlike functions in
(see Definition 4 below). Afterwards many mathematicians continued their research highlighting the fundamental role in GFT. Mahmood et al. in [
9] presents a detail description of the
q-starlike functions class in conic domain, whereas in [
10], the authors provided the class of
q-starlike functions associated with Janowski functions. Moreover, the problems related to upper bound of third Hankel determinant
for the class of
q-starlike functions have been investigated, available in [
11]. Later on, Srivastava et al. [
12] have investigated the Hankel and Toeplitz determinants of a subclass of
q-starlike functions. Many other authors have studied and investigated a number of other new subclasses of
q-starlike,
q-convex and
q-close-to-convex functions. They obtained a number of useful results like, coefficient inequalities, sufficient conditions, partial sums results and results related to radius problems (see for example [
13,
14,
15,
16]).
Definition 4 ([
8])
. Let a function . Then ifand As
, it is clearly noticed that
Similarly the class of
q-starlike functions denoted by
decreases to the known class
. Likewise, using subordination principle among analytic functions, the conditions described in Equations (
9) and (
10), may be revised as follows (see also [
17])
Remark 1. For function , the idea of Alexander’s theorem [1] was used by Baricz and Swaminathan [18] to define the class of q-convex function in the following way In order to utilize the q-difference operator (), this study has introduced two new subclasses of , i.e., and .
Definition 5. For , and , an analytic function if it satisfieswhere is given by Equation (6). Definition 6. For , and , an analytic function if it satisfies According to [
4,
19,
20], it is noted that
.
.
.
.
.
.
Hence, from Equations (
12) and (
13) we can write
As far as we know, there is minimal work on q-calculus related with conic domain in the literatures. The major objective of this work is to define a new subclass of q-starlike functions associated with the conic type domain. We find a number of useful results for our define function class and present some special cases of our results, in form of corollaries and remarks.
2. Main Results
In this section, we assume , , , whereas .
Theorem 1. For any analytic function, ifffor all , also . Proof. If
, then
The LHS of Equation (
15) in
is holomorphic, which follows
and
. This means
, because Equation (
14) holds for
. From Equation (
15) we can say that there must exist a function
in
, which should be analytic having
with
. This is because of the property of subordination between the two holomorphic functions, such that
and is equivalent to
or
Since,
From Equation (
18), we may write Equation (
17) in the form
which gives similar result as presented in Equation (
14). This proves the necessary part of Theorem 1.
Conversely: As we know that for
, Equation (
14) holds, it obeys the condition that
is not equal to zero for all
. Therefore, the function
is analytic in
. In the earlier part of our proof, it was shown that our supposition Equation (
14) can also be written in the form of Equation (
16). So
If we write
then Equation (
19) shows that
. Thus, the connected component of
includes the simply-connected domain
. As we know that
along with univalence of function giving
, as mentioned in the subordination fact (Equation (
15)), i.e.,
, which gives the desired result. □
Corollary 1 ([
20])
. For any function f represented by Equation (1), iff∀, also . Corollary 2 ([
20])
. For any function f represented by Equation (1), iff∀, as well as . Corollary 3 ([
20])
. For any function f represented by Equation (1), iff∀ with , as well as .By putting in corollary 2, we will get the desired corollary.
Theorem 2. For any function f represented by Equation (1). The function iff, also . Proof. Let us consider
and
From identity
. By using the relation
Theorem 1 also gives this result.
By setting
in Theorem 2, one may get the result obtained by Aouf and Seoudy, presented in [
20]. □
Corollary 4 ([
20])
. For any function f represented by (1), iff∀, also .Substituting in Theorem 2, one may get the desired corollary.
Corollary 5 ([
20])
. For any function f represented by Equation (1), iff∀, as well as . Corollary 6 ([
20])
. For any function f represented by Equation (1), with iff∀ and .By taking and in corollary 5, we will get the desired corollary.
Theorem 3. For any function f represented by Equation (1), has a necessary and sufficient condition Proof. Keeping in view Theorem 1, we could found
iff
for all
and
.
Using Equation (
21) we may write
which completes the desired proof. □
Corollary 7 ([
20])
. For any function f represented by Equation (1), has a necessary and sufficient conditionSubstituting in Theorem 3, one may get the desired corollary. Corollary 8 ([
20])
. For any function f represented by Equation (1), has a necessary and sufficient condition Corollary 9 ([
20])
. For any function f represented by Equation (1), has a necessary and sufficient conditionBy taking and in Theorem 3, one may get the desired corollary. Theorem 4. For any function f represented by Equation (1). The function has a necessary and sufficient condition Proof. From Theorem 2, we could found
iff
∀
and
.
After simplification, the LHS of Equation (
23) may takes the form
This proves our required result. □
Corollary 10 ([
20])
. For any function f represented by Equation (1), has a necessary and sufficient conditionSubstituting in Theorem 4, one may get the desired corollary. Corollary 11 ([
20])
. For any function f represented by Equation (1), has a necessary and sufficient condition Corollary 12 ([
20])
. For any function f represented by Equation (1), has a necessary and sufficient condition As an application of Theorems 3 and 4, one may determine inclusion property and coefficient estimates for a function of the form Equation (
1) in subclasses defined by
and
.
Theorem 5. If any function f represented by Equation (1) satisfiesThen . Proof. Assume that Equation (
24) holds, it is sufficient that
Now let us consider
which is bounded by 1 if
□
Corollary 13 ([
20])
. If any function f represented by Equation (1) satisfiesThen .Substituting in Theorem 5, we will get the desired corollary.
Theorem 6. If any function f represented by Equation (1) satisfiesThen . Theorem 5 and Equation (12) give the immediate proof of the desired theorem. Corollary 14 ([
20])
. If any function f represented by Equation (1) satisfiesThen .Substituting in Theorem 6, one may get the desired corollary.
Corollary 15 ([
19])
. If any function f represented by Equation (1) satisfiesThen .