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Abstract
This article presents new classes of ζ-uniformly q-starlike and q-convex analytic functions of order υ using the principle of q-calculus. The renowned classes of starlike and convex functions are utilized to establish these classes. The investigation of these uniformly classes leads to the study of some geometric notions, including coefficient estimates, sharpness, distortion and growth theorems, and convex linear combinations. Furthermore, some subordination properties involving integral means inequalities and subordinate sequences are examined.
1. Introduction
We represent the set of analytic functions f in as
, where
(1)
(1) and
is the open unit disc. The set
comprises of all functions that are both normalized and univalent.
We also consider the subset of the set
of the following analytic function:
(2)
(2) If the functions
and
belong to the class of holomorphic functions (i.e. they are both analytic), then
is subordinate to
(
), if a Schwarz function ϑ exists and satisfying certain conditions
such that
Equivalently, if the inequality below holds:
then the functions
and
will be subordinated.
The convolution of any analytic functions , expressed by
, is given as
where
have been aforementioned in (Equation1
(1)
(1) ) and
The renowned starlike function class, denoted by
, is formulated by
(3)
(3) Moreover, the convex function class, denoted by
, is formulated by
(4)
(4) Noting that from inequalities (Equation3
(3)
(3) ) and (Equation4
(4)
(4) ) (see [Citation1]), we have
For
Nasr and Aouf [Citation2] established the subsequent class of starlike functions
of order ρ
(5)
(5) while Wiatrowski [Citation3] defined and verified the class of convex functions
of order ρ as follows:
(6)
(6) The ζ-uniformly of starlike functions
of order υ is given by
(7)
(7) and the ζ-uniformly of convex functions
of order υ is given by
(8)
(8) where the classes
and
with their generalizations provided by Shams et al. [Citation4] (also see [Citation5–10], for related findings).
2. Quantum calculus
Quantum calculus is a fascinating mathematical framework that extends traditional calculus by incorporating principles from quantum mechanics. It introduces a new perspective on differentiation and integration, allowing for the analysis of functions and operators in a quantum-like context. The principle of q-calculus has profoundly influenced the investigation of Geometric Function Theory (GFT) and its significant applications in various areas, such as mathematical science and quantum physics. This principle presents a resemble framework to ordinary calculus but eliminates the need for limits. The notion of q-calculus, involving the derivative and integral, was primarily provided by Jackson [Citation11]. The investigation of q-calculus has led to the exploration of other pertinent aspects, such as the examination of special functions (for a more comprehensive review, see [Citation12–21]).
Ismail et al. [Citation22] proposed the notion of q-calculus in the context of GFT. This work has led to the identification of various classes of analytic functions in the disc , known as the Ma and Minda classes, which exhibit a strong connection to the concept of subordination. Furthermore, the use of q-calculus, specifically fractional q-integral operators, has been employed to build a multitude of analytic functions. Moreover, a considerable body of literature has been dedicated to the study of certain categories of analytic functions in
through the utilization of q-calculus (e.g. see [Citation23–32]).
More recently, numerous studies have examined certain uniformly classes of analytic functions by employing the notion of q-calculus (for instance, see [Citation33–37]).
This investigation begins with the essential notions and consequently, the analysis for our proposed study of the q-calculus. This article makes the assumption that 0<q<1. The next definitions are necessary for a univalent function :
Definition 2.1
[Citation11]
The q-number can be represented by the following expression:
Definition 2.2
[Citation11]
The operator denoted as represents the q-derivative and given by
Then the q-derivative of
in (Equation1
(1)
(1) ) is supplied by
In addition, the notion of q-generalized Pochhammer is supplied by
and the definition of the q-Gamma function is
It follows that
Given that the q-derivative is a generalized version of the ordinary derivative. Thus, considering the classes (Equation5(5)
(5) )–(Equation8
(8)
(8) ), we introduce novel classes of ζ-uniformly analytic functions that exhibit q-starlikeness and q-convexity in
. We extensively investigate various aspects including coefficient estimates (
), growth and distortion theorems, Hadamard products property, convex linear combinations, and several subordination properties encompassing integral means inequalities and subordinate factor sequences as well.
3. The uniformly classes ![](//:0)
and ![](//:0)
![](//:0)
The following definitions introduce classes of ζ-uniformly q-starlike and q-convex
univalent functions.
Definition 3.1
The function belongs to
if it fulfils the subsequent condition
(9)
(9) where
and
Definition 3.2
The function belongs to
if it fulfils the subsequent condition
(10)
(10) where
and
We also define
and
Example 3.3
The classes and
reduce to various subclasses as follows:
If
, then
and
as follows:
and
respectively.
and
, which have been established by Mahmood et al. [Citation38], which are formulated by
and
If
, then
and
, which have been defined by Seoudy and Aouf [Citation39], which are formulated by
and
4. A set of main results
The following theorems determine coefficient estimates for the classes
and
as outlined below:
Theorem 4.1
For and
Then
if and only if
(11)
(11)
Proof.
Let then from (Equation9
(9)
(9) ) we have
From the fact
, we attain
Using theorem of maximum modulus, we conclude that
Conversely, let the inequality (Equation11
(11)
(11) ) be correct for
, then
then
If we let
along the real axis, select the values of z for which
is real, and then eliminate the denominator from the above inequality, we can observe that
Corollary 4.2
Let , then
(12)
(12) for
, and
with the sharpness of
given by
Theorem 4.3
For and
Then
if and only if
(13)
(13) and we find that the sharpness of
as below
Proof.
Let then from (Equation9
(9)
(9) ) we have
we can obtain the desired result utilizing an approach resembling that of Theorem 4.1.
Theorems 4.1 and 4.3 lead to the subsequent corollaries, when .
Corollary 4.4
For and
Then
if and only if
Corollary 4.5
For and
Then
if and only if
Theorem 4.6
Distortion Theorem
If , then
Proof.
Let , we note from Theorem 4.1 that
From the fact
, we get
Similarly, we obtain
Here, the proof completes.
Theorem 4.7
Distortion Theorem
If , then
Proof.
Our argument of this theorem is similar to that of Theorem 4.6.
Theorem 4.8
Growth Theorem
Let , then
Proof.
Let , we note from Theorem 4.1 that
From the fact
, we get
Similarly, we obtain
Hence, the desired outcome can be inferred.
Similarly, we can establish the subsequent growth bounds:
Theorem 4.9
Let , then
Theorem 4.10
Put and
(14)
(14) then
if and only if
is given by
where
and
Proof.
Assume that , then Corollary 4.2 gives
Setting
with
Then
Conversely, let
be defined as
Then
Consequently, it may be inferred that
Hence
.
Theorem 4.11
Put and
(15)
(15) then
if and only if
is defined by
where
and
Proof.
By employing the identical technique as demonstrated in Theorem 4.10, we can attain the necessary finding of Theorem 4.11.
4.1. Integral means inequalities
To investigate our main findings, we must mention the following Littlewood's Lemma:
Lemma 4.12
[Citation40]
Suppose and
are two analytic functions and
then
Equality holds if and only if r = 0 or if
but
constant, or
.
Theorem 4.13
Suppose that and
(16)
(16) For
, we deduce that
Proof.
Since then
By Lemma 4.12, we have
Setting
We get
and
is analytic in
with
.
It suffices to demonstrate that satisfies
That is,
Hence,
Theorem 4.14
Suppose that and
(17)
(17) For
, we deduce that
The proof of the desired result readily follows from the application of Theorem 4.13.
With the same approach used in the proofs of Theorems 4.13 and 4.14, we find Theorem 4.15.
Theorem 4.15
Suppose that and
(18)
(18) For
, we deduce that
If in Theorems 4.13, 4.14 and 4.15, we get
Corollary 4.16
Suppose that and
For
, we deduce that
Corollary 4.17
Suppose that and
For
, we deduce that
4.2. Subordinate sequence properties
This section explores interesting subordination results within the classes of ζ-uniformly of q-starlike and q-convex
of analytic functions.
Definition 4.18
[Citation41]
A complex sequence is named a subordinating factor sequence if
is univalent and convex in
, we obtain
Lemma 4.19
[Citation41]
A complex sequence is said to be subordinating factor sequence iff
Theorem 4.20
Suppose and
are any other function in the class of convex functions, then
(19)
(19) where
and
(20)
(20) The constant
is the best estimate.
Proof.
Assume and
then
Thus, by Definition 4.18, Theorem 4.20 is correct if the following sequence is a subordinating factor sequence, with
By Lemma 4.19, equivalently
(21)
(21) Now, for
then
Since
is an increasing function of
, then
This establishes the validity of inequality (Equation21
(21)
(21) ) and confirms the subordinating result (Equation19
(19)
(19) ) as proven in Theorem 4.20. The inequality (Equation20
(20)
(20) ) follows from (Equation19
(19)
(19) ) by taking
Now, we express the function
as below
Then from (Equation19
(19)
(19) ), we have
We observe that
This demonstrates the best possible of the following constant:
Theorem 4.21
Suppose and
be any other function in the class of convex functions, then
(22)
(22) where
and
(23)
(23) The constant
is the best estimate.
Proof.
The proof is similar to that of Theorem 4.20.
Theorems 4.20 and 4.21 yield to the following outcomes, when .
Corollary 4.22
Suppose and
are any other function in the class convex of functions, then
where
and
The constant
is the best estimate.
Corollary 4.23
Suppose and
be any other function in the class of convex functions, then
(24)
(24) where
and
(25)
(25) The constant
is the best estimate.
5. Conclusion
Having as inspiration the results obtained by applying q-calculus on certain analytic functions, we have established in this investigation new classes of ζ-uniformly q-starlike and q-convex functions of order υ. Additionally, we have derived a range of geometric properties, including estimates for coefficients, the sharpness, distortion and growth theorems, and convex linear combinations. Furthermore, we have leveraged the classes of uniformly q-starlike and q-convex functions to investigate various subordination findings, such as integral means inequalities and subordinate sequences. This study serves as a valuable reference for future works and sheds light on new concepts within geometric function theory. Moreover, numerous other geometrical consequences such as the convolution, radius theorems, closer property, partial sums and neighbourhood, which have not yet been addressed and are still open to other researchers.
Acknowledgments
The authors would like to express their heartfelt gratitude to the dedicated editorial teams and thoughtful reviewers for their invaluable contributions to our work.
Disclosure statement
No potential conflict of interest was reported by the author(s).
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