1. Introduction
The
q-calculus is classical calculus without the concept of limit. In recent years,
q-calculus has attracted great attention of scholars on account of its applications in the research field of physics and mathematics as, for example, in the study of quantum groups,
q-deformed superalgebras, fractals and multifractal measures, optimal control problems and in chaotic dynamical systems. The application of
q-calculus involving
q-derivatives and
q-integrals was initiated by Jackson [
1,
2]. Later, the
q-derivative operator (or
q-difference operator) was used to investigate the geometry of
q-starlike functions for the first time in [
3]. Moreover, Aral [
4] and Anastassiou and Gal [
5,
6] generalized some complex operators which are known as the
q-Picad and the
q-Gauss–Weierstrass singular integral operators. Recently, Srivastava et al. [
7] have written a series of articles [
8,
9,
10] in which they combined the
q-difference operator and the Janowski functions to define new function classes and studied their useful properties from different viewpoints. In addition, we choose to refer the interested reader to further developments on
q-theory in [
11,
12,
13,
14,
15,
16,
17,
18]. In paricular, in his recent survey-cum-expository review article, Srivastava [
18] exposed the trivial and inconsequential developments in the literature in which known
q-results are being routinely translated into the corresponding
-results by forcing an obviously redundant or superfluous parameter
into the known
q-results.
Let
denote the class of
p-valent analytic functions
given by the following Taylor-Maclaurin series expansion:
in the open unit disk
. For
, we write
. In the whole paper, we let
,
and
be the sets of positive integers, complex numbers and real numbers, respectively.
A function
is called to be a
p-valent starlike function of order
and is written as
, if it satisfies the following inequality:
for all
.
A function
is known as a
p-valent convex function of order
and is denoted by
, if it meets the following condition:
for all
.
From (1) and (2), we have the following equivalence:
Definition 1. Let and introduce the q-number by Let . In particular, when , we have .
Definition 2. (See [
1,
2])
. Let . Then the q-difference operator of a function is given byif exists.
One can observe from Definition 2 that provided that is a differentiable function in a set of . Furthermore, for , one can see that A function belonging to is called to be a p-valent q-starlike function of order σ and is written as , if it meets the condition: for all .
A function belonging to is referred to as a p-valent q-convex function of order σ and is written as , if it meets the condition: for all .
From (3) and (4), it is not difficult to verify that Definition 3. A function analytic in D with is called to belong to , if equivalently we can write For analytic functions and , the function is said to subordinate to the function and written , if there exists an analytic function with and so that . Suppose that is analytic univalent in D, then the following equivalence holds true: In q-calculus concept, we now define the following subclasses of in connection with the q-difference operator .
Definition 4. A function belonging to is called to be in , if it meets the condition
We note that:
- (1)
For , we obtain , the family of p-valent q-starlike functions associated with Janowski function;
- (2)
For , and , we obtain , the family of p-valent q-starlike functions;
- (3)
For , , and , we have , the family of p-valent starlike functions;
- (4)
For , , , and , we obtain , the family of starlike functions.
A well-known question in GFT is to discuss the functional composed of combinations of certain coefficients of functions. The class is made up of functions of the form . The Fekete–Szegö functional describes a specific relationship between coefficient and , i.e., , . Fekete and Szegö [19] found that is bounded by for and and the bound is sharp for every λ. In particular, if we let and , then . More recently, Srivastava et al. researched the Fekete–Szegö inequalities for several classes of q-convex and q-starlike functions in [20]. Let Ω
denote the family of functions of the form: in D with .
To derive the main results, we recall the following lemmas.
Lemma 1. (References [
21,
22,
23])
. Let . ThenFor or , the equality occurs when or . For , the equality is true when or . For , the equality occurs when For , the equality occurs when These above upper bounds are best possible, and they could be further extended. For : Lemma 2. (Reference [
24])
. Let . Then for , we have the following sharp estimates:The above are given as the following: Unless otherwise stated, we assume the entire paper that In this paper, we shall study some geometric properties of functions belonging to such as Fekete–Szegö inequality, necessary and sufficient conditions, distortion and growth theorems, coefficient estimates, radii of convexity and starlikeness, closure theorems and partial sums.
2. Main Results
Theorem 1. If and then Proof of Theorem 1. If
, by Definition 4, there is a function
such that
Therefore, we obtain
where
By applying Lemmas 1 and 2 to (8) and (9), respectively, we obtain (6) and (7). Now the proof of the Theorem is completed. □
Corollary 1. Let and . If and , then Theorem 2. Let . Additionally, let Then the function belongs to if and only if Proof of Theorem 2. Assuming that the inequality (10) holds true, we need to show the inequality (5). Now we have
which shows that the function
belongs to
.
On the other hand, we let the function
. Then from (5), one can see that
The inequality (11) is correct for . By choosing , we obtain (10). Thus, the Theorem is proved. □
Corollary 2. Let . If , then The result is best possible for defined as Theorem 3. Let . If then, for , The bounds are best possible for given as Proof of Theorem 3. Then, by applying the triangle inequality, we have
Since
, we can see that
. Thus, we have
and
Considering
, we know from Theorem 2 that
Since the sequence
is increasing regarding
n, we have
Hence by transitivity we obtain
which implies that
Substituting (14) into (12) and (13), we obtain the required results. The proof of Theorem 3 is completed. □
Theorem 4. Let and . If , then, for , we have The results are best possible for the following function Proof of Theorem 4. Then, from Definition 2, we can write
By applying the triangle inequality, we obtain
Because of the function
belonging to the class
, we find from Theorem 2 that
As we know that
is an increasing sequence regarding
n, so
Thus, by transitivity, we have
which implies that
Now, by putting (17) in (15) and (16), we complete the proof of Theorem 4. □
Theorem 5. Additionally, let . If then, for , is p-valent starlike function of order δ, where Proof of Theorem 5. Let
. In order to prove
, we need to show that
The subordination above is equivalent to
. After some calculations and simplifications, we obtain
From the inequality (10), we can obviously find that
Inequality (18) can be seen to be true if it satisfies the following inequality:
The above inequality indicates that
or
Let
then we obtain the required result. The proof of Theorem 5 is completed.
With the aid of the method in the proof of Theorem 5, we also obtain the following theorems for the classes , and , respectively. □
Theorem 6. Let , and . If then, for , is p-valent convex function of order δ, where Theorem 7. Let , and . If then, for , is p-valent q-starlike function of order σ, where Theorem 8. Let , and . If the function then, for , is p-valent q-convex function of order σ, where Taking and in Theorems 5 to 8, we obtain the following corollaries, respectively.
Corollary 3. Let and . If the function then, for is p-valent starlike function of order where Corollary 4. Let and . If the function then, for is p-valent convex function of order where Corollary 5. Let and . If the function then, for is p-valent q-starlike function of order where Corollary 6. Let and . If the function then, for is p-valent q-convex function of order where Next, we will study the ratio of a function to its sequence of partial sums given by Theorem 9. Let . If Proof of Theorem 9. After some simplifications, we have
and
Thus, we find that
if and only if
It is not difficult to see that the sequence
is increasing regarding n. Additionally, we can see that
. Thus, we obtain
Thus, the inequality (22) is true. This proves (19).
Next, in order to prove the inequality (20), we consider
By simple calculating, we find that
and
Thus, we obtain
if it satisfies the following condition:
The remaining part of the proof is similar to that of (19) and we omit it. Now we complete The proof of Theorem 9. □
Theorem 10. Let . If then, for , the function Proof of Theorem 10. For
, the function
can be written as follows:
For functions
, by Theorem 2, we obtain
This indicates that the function belongs to . Now we complete the proof of Theorem 10. □
Corollary 7. Let and Then and also belongs to .
Theorem 11. Let . If belongs to then also belongs to
Proof of Theorem 11. Theorem 11 is proved using the similar arguments as in Theorem 10. □