Generalized $q$-convex functions characterized by $q$-calculus

1.   Introduction

The mathematical study of $q$-calculus has been a subject of top importance for researchers due to its huge applications in unique fields. Few recognized work at the application of $q$-calculus firstly added through Jackson ^[4]. Later, $q$-analysis with geometrical interpretation become diagnosed. Currently, $q$-calculus has attained the attention researchers due to its massive applications in mathematics and physics. The in-intensity evaluation of $q$-calculus changed into first of all noted with the aid of Jackson ^[4,5], wherein he defined $q$-derivative and $q$-integral in a totally systematic way. Recently, authors are utilizing the $q$-integral and $q$-derivative to study some new sub-families of univalent functions and obtain certain new results, see for example Nadeem et al. ^[8], Obad et al. ^[11] and reference therein.

Assume that $f\in\mathbb C$. Furthermore, $f$ is normalized analytic, if $f$ along $f(0) = 0, \; f^{^{\prime }}(0) = 1$ and characterized as

We denote by $\mathcal{A}$, the family of all such functions. Let $f\in\mathcal A$ be presented as (1.1). Furthermore,

We present by $\mathcal S$ the family of all univalent functions. Let $\tilde p\in\mathbb C$ be analytic. Furthermore, $\tilde p\in\mathcal P$, iff $\mathfrak{R}(\tilde {p}(z)) > 0$, along $\tilde {p}(0) = 1$ and presented as follows:

Broadening the idea of $\mathcal P$, the family $\mathcal P(\alpha_{_{0}})$, $0\leq \alpha_{_{0}} < 1$ defined by

for further details one can see ^[2].

Assume that $\mathcal C$, $\mathcal K$ and $\mathcal S^{^{\ast}}$ signify the common sub-classes of $\mathcal A$, which contains convex, close-to-convex and star-like functions in $\mathcal E$. Furthermore, by $\mathcal S^{^{\ast}}\left(\alpha_{_{0}}\right)$, we meant the class of starlike functions of order $\alpha_{_{0}}$, $0\leq\alpha_{_{0}} < 1$, for details, see ^[1,2] and references therein. Main motivation behind this research work is to extend the concept of Kurki and Owa ^[6] into $q$-calculus.

The structure of this paper is organized as follows. For convenience, Section 2 give some material which will be used in upcoming sections along side some recent developments in $q$-calculus. In Section 3, we will introduce our main classes $\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$ and $\mathcal S^{\ast}_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$. In Section 4, we will discuss our main result which include, inclusion relations, $q$-limits on real parts and integral invariant properties. At the end, we conclude our work.

2.   Materials and methods

The concept of Hadamard product (convolution) is critical in GFT and it emerged from

and

is integral convolution. Let $f$ be presented as in (1.1), the convolution $\left(f\ast g\right)$ is characterize as

where

for details, see ^[2].

Let $h_{_{1}}$ and $h_{_{2}}$ be two functions. Then, $h_{1}\prec h_{_{2}}\Longleftrightarrow\exists$, $\varpi$ analytic such that $\varpi(0) = 0, \ \ \left\vert \varpi(z)\right\vert < 1$, with $h_{1}\left(z\right) = \left(h_{2}\circ\varpi\right)(z)$. It can be found in ^[1] that, if $h_{_{2}}\in\mathcal S$, then

for more information, see ^[7].

Assume that $q\in \left(0, 1\right)$. Furthermore, $q$-number is characterized as follows:

Utilizing the $q$-number defined by (2.2), we define the shifted $q$-factorial as the following:

Assume that $q\in \left(0, 1\right)$. Furthermore, the shifted $q$-factorial is denoted and given by

Let $f\in \mathbb{C}$. Then, utilizing $(2.2)$, the $q$-derivative of the function $f$ is denoted and defined in ^[4] as

provided that $f^{\prime }\left(0\right)$ exists.

That is

If $f\in\mathcal A$ defined by (1.1), then,

Also, the $q$-integral of $f\in\mathbb C$ is defined by

provided that the series converges, see ^[5].

The $q$-gamma function is defined by the following recurrence relation:

In recent years, researcher are utilizing the $q$-derivative defined by (2.3), in various branches of mathematics very effectively, especially in Geometric Function Theory (GFT). For further developments and discussion about $q$-derivative defined by (2.3), we can obtain excellent articles produced by famous mathematician like ^{[3,8,9,10,12,13,14]} and many more.

Ismail et al. ^[3] investigated and study the class $\mathcal C_{_{q}}$ as

If $q\longrightarrow 1^{^{-}}$, then $\mathcal C_{_{q}} = \mathcal C$.

Later, Ramachandran et al. ^[12] discussed the class $\mathcal C_{_{q}}(\alpha_{_{0}})$, $0\leq\alpha_{_{0}} < 1$, given by

For $\alpha_{_{0}} = 0$, $\mathcal C_{_{q}}(\alpha_{_{0}}) = \mathcal C_{_{q}}$.

2.1.   On the classes $\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$ and $\mathcal S^{\ast}_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$

Now, extending the idea of ^[13] and by utilizing the $q$-derivative defined by (2.3), we define the families $\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}} \right)$ and $\mathcal S^{\ast}_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$ as follows:

Definition 2.1. Let $f\in\mathcal A$ and $\alpha_{_{0}}, \beta_{_{0}}\in\mathbb R$ such that $0\leq \alpha_{_{0}} < 1 < \beta_{_{0}}$. Then,

It is obvious that if $q\longrightarrow 1^{^{-}}$, then $\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}} \right)\longrightarrow\mathcal C\left(\alpha_{_{0}}, \beta_{_{0}} \right)$, see ^[13]. This means that

Definition 2.2. Let $\alpha_{_{0}}, \beta_{_{0}}\in\mathbb R$ such that $0\leq \alpha_{_{0}} < 1 < \beta_{_{0}}$ and $f\in\mathcal A$ defined by (1.1). Then,

Or equivalently, we can write

Remark 2.1. From Definitions 2.1 and 2.2, it follows that $f\in\mathcal C_{_{q}}(\alpha_{_{0}}, \beta_{_{0}})$ or $f\in\mathcal S^{\ast}_{_{q}}(\alpha_{_{0}}, \beta_{_{0}})$ iff $f$ fulfills

or

for all $z\in\mathcal E$.

We now consider $q$-analogue of the function $p$ defined by ^[13] as

Firstly, we fined the series form of (2.9).

Consider

If we let $w = qe^{2\pi i\left(\frac{1-\alpha_{_{0}} }{\beta_{_{0}}-\alpha_{_{0}}}\right)}z$, then,

This implies that

and

Utilizing these, Eq $(2.11)$ can be written as

This shows that the $p_{_{q}}\in \mathcal P$.

Motivated by this work and other aforementioned articles, the aim in this paper is to keep with the research of a few interesting properties of $\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$ and $\mathcal S^{\ast}_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$.

3.   Results and discussion

Utilizing the meaning of subordination, we can acquire the accompanying Lemma, which sum up the known results in ^[6].

Lemma 3.1. Let $f\in\mathcal A$ be defined by (1.1), $0\leq \alpha_{_{0}} < 1 < \beta_{_{0}}$ and $0 < q < 1$. Then,

Proof. Assume that $\digamma$ be characterized as

At that point it can without much of a stretch seen that function $\digamma$ ia simple and analytic along $\digamma(0) = 1$ in $\mathcal E$. Furthermore, note

Therefore,

A simple calculation leads us to conclude that $\digamma$ maps $\mathcal E$ onto the domain $\Omega$ defined by

Therefore, it follows from the definition of subordination that the inequalities (2.7) and (3.1) are equivalent. This proves the assertion of Lemma 3.1.

Lemma 3.2. Let $f\in \mathcal A$ and $0\leq \alpha_{_{0}} < 1 < \beta_{_{0}}$. Then,

and if $p$ presented as in (2.9) has the structure

then,

Proof. Proof directly follows by utilizing (2.8), (2.12) and Lemma 3.1.

Example 3.1. Let $f$ be defined as

This implies that

According to the proof of Lemma 3.1, it can be observed that $f$ given by (3.5) satisfies (2.7), which means that $f\in S_{_{q}}^{\ast}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$. Similarly, it can be seen by utilizing Lemma 3.2 that

belongs to the class $\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$.

Inclusion relations:

In this segment, we study some inclusion relations and furthermore acquire some proved results as special cases. For this, we need below mentioned lemma which is the $q$-analogue of known result in ^[7].

Lemma 3.3. Let $\mathbf{u}, \mathbf{v}\in\mathbb C$, such that $\mathbf{u}\neq 0$ and what's more, $\hbar\in\mathcal H$ such that $\mathfrak{R}\left[\mathbf{u}\hbar(z)+\mathbf{v}\right] > 0$. Assume that $\wp\in\mathcal P$, fulfill

Theorem 3.1. For $0\leq\alpha_{_{0}} < 1 < \beta_{_{0}}$ and $0 < q < 1$,

Proof. Let $f\in\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$. Consider

Differentiating $q$-logarithamically furthermore, after some simplifications, we get

Note that by utilizing Lemma 3.3 with $u = 1$ and $v = 0$, we have

Consequently,

This completes the proof.

Note for distinct values of parameters in Theorem 3.1, we obtain some notable results, see ^[2,6,13].

Corollary 3.1. For $q\longrightarrow 1^{^{-}}$, $0\leq\alpha_{_{0}} < 1 < \beta_{_{0}}$, we have

Corollary 3.2. For $q\longrightarrow 1^{^{-}}$, $\alpha_{_{0}} = 0$ and $\beta_{_{0}} > 1$, we have

where

$q$-limits on real parts:

In this section, we discuss some $q$-bounds on real parts for the function $f$ in $\mathcal C_{_{q}}(\alpha_{_{0}}, \beta_{_{0}})$ and following lemma will be utilize which is the $q$-analogue of known result of ^[7].

Lemma 3.4. Let $U\subset{\mathbb C\times\mathbb C}$ and let $c\in\mathbb C$ along $\mathfrak{R}(b) > 0$. Assume that $\mho:\mathbb{C}^{2}\times E\longrightarrow\mathbb{C}$ fulfills

If $p\left(z\right) = c+c_{1}z+c_{2}z^{2}+...$ is in $\mathcal P$ along

Lemma 3.5. Let $p\left(z\right) = \sum\limits_{j = 1}^{\infty }C_{j}z^{j}$ and assume that $p(E)$ is a convex domain. Furthermore, let $q\left(z\right) = \sum\limits_{j = 1}^{\infty }A_{j}z^{j}$ is analytic and if $q\prec p$ in $E$. Then,

Theorem 3.2. Suppose $f\in\mathcal A$, $0\leq \alpha_{_{0}} < 1$ and

Then,

Proof. Let $\gamma = \frac{1}{2-\alpha_{_{0}} }$ and for $0\leq \alpha_{_{0}} < 1$ implies $\frac{1}{2}\leq \gamma < 1$. Let

Differentiating $q$-logrithmically, we obtain

Let us construct the functional $\mho$ such that

Utilizing (3.7), we can write

Now, $\rho, \delta \in \mathbb R$ with $\delta \leq -\frac{\left(1+\rho ^{2}\right)}{2}$, we have

This implies that

Utilizing $\delta \leq -\frac{\left(1+\rho ^{2}\right)}{2}$, we can write

Let

Then, $g\left(-\rho \right) = g\left(\rho \right)$, which shows that $g$ is even continuous function. Thus,

and $D_{_{q}}\left(g\left(0\right)\right) = 0$. Also, it can be seen that $g$ is increasing function on $(0, \infty)$. Since $\frac{1}{2}\leq \gamma < 1$, therefore,

Now by utilizing (3.9) and (3.10), we have

This means that $\mathfrak{R}\left(\mho \left(i\rho, \delta; z\right) \right)\notin\Omega_{\alpha_{_{0}}}$ for all $\rho, \delta\in\mathbb R$ with $\delta \leq -\frac{\left(1+\rho ^{2}\right)}{2}$. Thus, by utilizing Lemma 3.4, we conclude that $\mathfrak{R} p(z) > 0$, $\forall z\in\mathcal E$.

Theorem 3.3. Suppose $f\in\mathcal A$ be defined by (1.1) and $1 < \beta_{_{0}} < 2$,

Then,

Proof. Continuing as in Theorem 3.2, we have the result.

Combining Theorems $3.2$ and $3.3$, we obtain the following result.

Theorem 3.4. Suppose $f\in\mathcal A$, $0\leq \alpha_{_{0}} < 1 < \beta_{_{0}} < 2$ and

Theorem 3.5. Let $f\in\mathcal A$ be defined by (1.1) and $\alpha_{_{0}}, \beta_{_{0}} \in \mathbb{R}$ such that $0\leq\alpha_{_{0}} < 1 < \beta_{_{0}}$. If $f\in\mathcal C_{_{q}}(\alpha_{_{0}}, \beta_{_{0}})$, then,

where $\left\vert B_{1}\right\vert$ is given by

Proof. Assume that

Then, by definition of $\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$, we obtain

Let $p_{_{q}}$ be defined by (3.3) and $B_n(q)$ is given as in (3.4). If

by (3.12), we have

Note that by utilizing (1.1), (2.4) and (3.14), one can obtain

Comparing the coefficient of of $z^{j-1}$ on both sides, we have

This implies that by utilizing Lemma 3.5 with (3.13), we can write

Now by utilizing (3.16) in (3.15) and after some simplifications, we have

Furthermore, for $j = 2, 3, 4$,

By utilizing mathematical induction for $q$-calculus, it can be observed that

which is required.

Remark 3.1. Note that by taking $q\longrightarrow 1^{^{-}}$ in Theorems 3.2–3.4, we attain remarkable results in ordinary calculus discussed in ^[6].

Integral invariant properties:

In this portion, we show that the family $\mathcal C_{_{q}}(\alpha_{_{0}}, \beta_{_{0}})$ is invariant under the $q$-Bernardi integral operator defined and discussed in ^[9] is given by

Making use of (1.1) and (2.5), we can write

Finally, we obtain

For $c = 1$, we obtain

It is well known ^[9] that the radius of convergence $R$ of

is $q$ and the function given by

belong to the class $\mathcal C_{_{q}}$ of $q$-convex function introduced by ^[3].

Theorem 3.6. Let $f\in\mathcal A$. If $f\in\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$, then $F_{_{c, q}}\in C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$, where $F_{_{c, q}}$ is defined by (3.17).

Proof. Let $f\in\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$ and set

$q$-differentiation of (3.17) yields

Again $q$-differentiating and utilizing (3.20), we obtain

Now, logarithmic $q$-differentiation of this yields

By utilizing the definition of the class $C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$, we have

Therefore,

Consequently, utilizing Lemma 3.3, we have

The proof is complete.

Remark 3.2. Letting $q\longrightarrow 1^{^{-}}$, in Theorem $3.6$, we obtain a known result from ^[13].

Corollary 3.3. Let $f\in\mathcal A$. If $f\in\mathcal C\left(\alpha_{_{0}}, \beta_{_{0}}\right)$, then $F_{_{c}}\in\mathcal C\left(\alpha_{_{0}}, \beta_{_{0}}\right)$, where $F_{_{c}}$ is Bernardi integral operator defined in ^[1].

Also, for $q\longrightarrow 1^{^{-}}$, $\alpha_{_{0}} = 0$ and $\beta_{_{0}} = 0$, we obtain the well known result proved by ^[1]. It is well known ^[9] that for $0\leq\alpha_{_{0}} < 1 < \beta_{_{0}}$, $0 < q < 1$ and $c\in\mathbb N$, the function (3.19) belong to the class $\mathcal C_{_{q}}$. Utilizing this, we can prove

Remark 3.3. As an example consider the function $f\in\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$ defined by (3.6) and $\phi_{_{q}}\in\mathcal C_{_{q}}$ given by (3.19), implies $\left(f\ast\phi_{_{q}}\right)\in\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$.

4.   Conclusions

In this article, we mainly focused on $q$-calculus and utilized this is to study new generalized sub-classes $\mathcal C_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$ and $\mathcal S^{\ast}_{_{q}}\left(\alpha_{_{0}}, \beta_{_{0}}\right)$ of $q$-convex and $q$-star-like functions. We discussed and study some fundamental properties, for example, inclusion relation, $q$-coefficient limits on real part, integral preserving properties. We have utilized traditional strategies alongside convolution and differential subordination to demonstrate main results. This work can be extended in post quantum calculus. The path is open for researchers to investigate more on this discipline and associated regions.

Acknowledgments

The work was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R52), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that they have no conflicts of interest.