Sharp inequalities for $q$-starlike functions associated with differential subordination and $q$-calculus

1.   Introductions and research background

Quantum calculus (known as $q$-calculus) extends traditional calculus by introducing the parameter $q$, offering flexible and powerful tools for analyzing mathematical functions. This framework has attracted growing attention due to its applications in various fields, including mathematics, engineering, and physics. Recent advancements have amplified its relevance, with notable applications in $q$-transform analysis, $q$-difference and $q$-integral equations, optimal control, and fractional calculus.

The $q$-derivative operator plays a central role in the theory of special functions, enabling the definition and in-depth analysis of various subclasses of analytic functions. For instance, Mahmood et al. ^[1] investigated the coefficients of $q$-starlike functions associated with conic domains, Ahmad et al. ^[2] explored the Hankel determinants and Zalcman conjecture for $q$-starlike functions associated with the balloon-shaped domain, and Shi et al. ^[3] studied the properties of generalized integral operators in the lemniscate domain of Bernoulli. Ahmad et al. ^[4] extended these ideas by exploring Janowski functions to establish $q$-analogues of differential subordination results. The $q$-derivative operator also underpins the $q$-extension of starlike functions, as discussed in ^[5].

Recent research highlights the significant role of $q$-calculus in advancing contemporary mathematical theory. Studies such as ^[6,7,8] deepen our understanding of analytic and symmetric functions through $q$-analytic techniques. This body of work demonstrates the broad applicability of $q$-calculus, with potential implications across quantum theory, number theory, and statistical mechanics. For further exploration of fractional derivatives, including Caputo and conformable fractional derivatives, readers may consult ^[9,10] and their references.

We present some basic definitions related to our work before moving on to our primary findings.

Let $\mathcal{H}\left(\mathfrak{D}\right)$ denote the class of all analytic functions $f$ defined in the open unit disk

where $\mathbb{C}$ represents the set of complex numbers. Let $\mathcal{A}\subseteq \mathcal{H}\left(\mathfrak{D}\right)$ be the subclass of all analytic functions $f$ having the Taylor series representation:

Suppose that $\mathcal{P}$ represents the Carathéodory class of analytic functions $h$ normalized by

such that the real part is positive:

For each given analytic function $f \in \mathcal{A}$, the $r^{th}$ Hankel determinant $\mathcal{H}_{r, n}$ is defined in ^[11] as follows,

where $r, n \in \mathbb{N}\ \text{and}\ \xi _{1} = 1$. The following are two second Hankel determinants.

In recent years, considerable attention has been devoted to investigating the upper bounds of the expression $\left| H_{2, 2}(f) \right|$ across various subclasses of analytic functions. Key contributions in this domain have been made by researchers such as Noonan and Thomas ^[12], Hayman ^[13], Ohran et al. ^[14], and Shi et al. ^[15]. Babalola ^[16] initiated the study of bounds for the third Hankel determinant, further enriching the field.

For a deeper exploration of this subject, recent studies provide valuable insights and are discussed in references ^[17,18,19]. This growing body of research highlights the importance of Hankel determinants in analytic function theory, offering the potential for discoveries and advancing our understanding of this mathematical area.

For the functions $f$ and $g$ $\in \mathcal{H}\left(\mathfrak{D}\right)$, we say that $f$ is subordinated to $g,$ written as

if there exists a Schwarz function $\omega (\varepsilon)$, which is an analytic function in $\mathfrak{D}$ with $\omega \left(0\right) = 0$ and $\left \vert \omega (\varepsilon)\right \vert \leq 1$, such that

Furthermore, if two functions $f$ and $g$ are analytic in $\mathfrak{D}$ and $g$ is univalent, then $f$ is subordinated to $g$ if and only if\

There are some applications of subordination below; see ^[20] for more applications.

Definition 1.1. A function $p\in \mathcal{P}\left[ A, B\right] \ $if

Equivalently,

In particular, $f$ is a Janowski starlike function if $\frac{ \varepsilon f^{\prime }\left(\varepsilon \right) }{f(\varepsilon)} \in \mathcal{P}\left[ A, B\right]$. See ^[21] for more details about Janowski starlike functions.

We take into consideration a class of functions in the domain bounded by a tangent function. All functions $h$ will belong to such a class if they fulfill

The images of these functions lie in the right-half plane and the geometrical representation is like an eight-shape domain. Simple computations allow the above (1.4) to be expressed as

In parallel comparison to starlike functions, Khan et al. ^[22] introduced the following class of Janowski-type starlike functions along with some properties.

Thus, by the relation of (1.4) and (1.5),

Definition 1.2. ^[23] For a function $f$, the $q$-derivative (also known as the $q$-difference operator) is defined by

where $\varepsilon \neq 0$ and $0 < q < 1.$

For example, for $n\in\mathbb{N}$ and $\varepsilon \in \mathfrak{D}$, we have

where the ${q}$-number is defined by

Now, we introduce a new class $\mathcal{S}_{\tan }^{\ast}\left(q\right)$ of Janowski-type $q$-starlike functions associated with the eight-shaped image domain. Several classes of Janowski-type $q$-starlike functions have been investigated previously (see ^[24]).

Definition 1.3. A function $f$ in $\mathcal{A}$ is said to belong to $\mathcal{S}_{\tan }^{\ast }\left(q\right)$ if the following holds

Remark 1.1. One can see that

The graphological representation for the class $\mathcal{S}_{\tan }^{\ast }\left(q\right)$ is given in the following Figure 1.

In our research, we set out to explore and characterize a novel class of $q$-starlike functions associated with an eight-like image domain. We employed differential subordination and quantum calculus techniques to achieve this goal. Our primary aim is to establish a fundamental understanding of the behavior of these $q$-starlike functions, focusing on deriving first-order differential subordination results. Additionally, we sought to determine sharp inequalities for initial Taylor coefficients and provide optimal estimates for the Fekete-Szegö problem and a second-order Hankel determinant applicable to all $q$-starlike functions within this newly defined class. Through this work, we intend to contribute to the broader field of geometric function theory, demonstrating the wider implications of our findings through a series of corollaries.

2.   A set of lemmas

The following lemmas are essential to investigate our main results.

Lemma 2.1. ^[25] $\left(q\mathit{\text{-Jack's Lemma}}\right)$ Let $\omega \left(\varepsilon \right) \in\mathcal{H}\left(\mathfrak{D}\right)$ with $\omega \left(0\right) = 0.$ If $\left \vert \omega \left(\varepsilon \right) \right \vert$ achieves the maximum value on the circle $\left \vert \varepsilon \right \vert = r$ at a point $\varepsilon _{0}$, then

where $0 < q < 1$ and $m\geq 1.$

Lemma 2.2. ^[26] Let $h\in \mathcal{P}$ have the series of the form (1.2). Then the following inequalities hold true:

Lemma 2.3. ^[27] Let $h\in \mathcal{P}$ be represented by (1.2), and then the following inequality holds.

Lemma 2.4. Let $h\in \mathcal{P}$ be given in (1.2), and then there exist $k$ and $\delta \in \mathfrak{D}$ such that

The values given in (2.3) and (2.4) are due to ^[26] and ^[28], respectively.

Lemma 2.5. ^[28] If $h\in {\mathcal{P}}$ has the form (1.2), then

where

Lemma 2.6. ^[29] Suppose that

where $P,$ $Q,$ and $R$ are real numbers, and $x\in\overline{\mathfrak{D}} = \left\{ \varepsilon \in \mathbb{C}:\left\vert \varepsilon \right\vert \leq 1\right\}$.

If $PR\geq 0,$ then

3.   First-order differential subordination results

In this section, we start with the results of differential subordination. In the next section, we derive the sharp constraints for the first three unknown coefficients, the sharp Fekete-Szegö problem, and the sharp estimate of the second-order Hankel determinant for the newly defined class of $q$-starlike functions. In addition, the consequences of these results are given in the form of corollaries. In the last section, we establish the sufficient criteria for functions belonging to the class $\mathcal{S}_{\tan }^{\ast }\left(q\right).$

Theorem 3.1. Let $f\left(\varepsilon \right) \in \mathcal{A}$ and $h\left(\varepsilon \right) \in \mathcal{H}\left(\mathfrak{D}\right)$. Suppose that

If the following condition holds:

Then

Proof. Suppose that

Also, for $\omega (\varepsilon) \in \mathcal{H}\left(\mathfrak{D}\right)$, we consider

It is sufficient to demonstrate that $\left \vert \omega \left(\varepsilon \right) \right \vert < 1$ in order to yield the required result. Using (3.3) and (3.4), we have

and hence,

If $\omega \left(\varepsilon \right)$ achieves, at some point $\varepsilon = \varepsilon _{0},$ its maximum value $\left \vert \omega \left(\varepsilon _{0}\right) \right \vert = 1$, i.e., $\omega \left(\varepsilon _{0}\right) = e^{i\theta}, \ \theta \in \left[ -\pi, \pi \right]$, then, by Lemma 2.1, for $m\geq 1,$

Thus,

A direct computation gives that

Since $\varphi \left(-\theta \right) = \varphi \left(\theta \right)$, for $\theta \in \left[ 0, \pi \right]$, then

Therefore,

Putting values of (3.6) into (3.5), we get

Let

Then

which shows the increasing behavior of $\phi \left(m\right)$, so the maximum of $\phi \left(m\right)$ will be obtained at $m = 1$. It follows that

From (3.1),

which contradicts (3.2), thus $\left \vert \omega \left(\varepsilon \right) \right \vert < 1$ and we achieve the intended outcome. □

By taking $h\left(\varepsilon \right) = \frac{\varepsilon D_{q}f\left(\varepsilon \right) }{f\left(\varepsilon \right) }$, we deduce the following result.

Corollary 3.1. Let $f\left(\varepsilon \right) \in \mathcal{A}$ and $h\left(\varepsilon \right) = \frac{\varepsilon D_{q}f\left(\varepsilon \right) }{f\left(\varepsilon \right) }$. Suppose that

If the following condition holds:

Then

Theorem 3.2. Let $h\left(\varepsilon \right) \in \mathcal{H}\left(\mathfrak{D}\right)$ with $h\left(0\right) = 1$. Suppose that

If the following subordination criteria hold:

Then the following subordination holds:

Proof. Define

Let $\omega (\varepsilon) \in \mathcal{H}\left(\mathfrak{D}\right)$, and consider

We need to show that $\left \vert \omega (\varepsilon)\right \vert \leq 1$. Using logarithmic differentiation on (3.13), we obtain from (3.12) that

and so

If at some $\varepsilon = \varepsilon _{0},$ $\omega \left(\varepsilon \right)$ attains its maximum value for example $\left \vert \omega \left(\varepsilon _{0}\right) \right \vert = 1,$ then, by Lemma 2.1,

Now, a direct, simple calculation gives us

Since $\varphi _{1}\left(-\theta \right) = \varphi _{1}\left(\theta \right)$, consider $\theta \in \left[ 0, \pi \right]$. Then

Therefore,

Now set (3.6) and (3.15) in Eq (3.14), and we get

Clearly, one can observe that $\phi \left(m\right)$ is increasing in nature so its maximum value is obtained at $m = 1$, thus

By (3.10), we have

which contradicts (3.11), therefore $\left \vert \omega \left(\varepsilon \right) \right \vert < 1$ and so the desired result is obtained. □

By taking $h\left(\varepsilon \right) = \frac{\varepsilon D_{q}f\left(\varepsilon \right) }{f\left(\varepsilon \right) }$, we deduce the following corollary.

Corollary 3.2. Let $f\in \mathcal{A}$. Suppose that

If the following subordination holds:

then we have

Theorem 3.3. Let $h\left(\varepsilon \right) \in \mathcal{H}\left(\mathfrak{D}\right)$ with $h\left(0\right) = 1$. Suppose that

If the following subordination criteria are fulfilled:

then we have

Proof. We define a function:

If there exists $\omega (\varepsilon) \in \mathcal{H}\left(\mathfrak{D}\right)$ such that

then, we obtain that

and hence,

If $\omega \left(\varepsilon \right)$ attains its maximum value at some point $\varepsilon = \varepsilon _{0}$, which is $\left \vert \omega \left(\varepsilon _{0}\right) \right \vert = 1: \omega \left(\varepsilon _{0}\right) = e^{i\theta }$, for some $\theta \in \left[ -\pi, \pi\right]$, then, by Lemma 2.1,

Thus, we have

Applying (3.6) and (3.15) in Eq (3.19),

Thus, the function $\phi \left(m\right)$ is increasing; hence, it has its maximum value at $m = 1$. Now, we have

By (3.18),

It is a contradiction to

Therefore, $\left \vert \omega \left(\varepsilon \right) \right \vert < 1$, and the required result is obtained. □

By taking $h\left(\varepsilon \right) = \frac{\varepsilon \left(D_{q}f\right) \left(\varepsilon \right) }{\left(f\left(\varepsilon \right) \right) ^{2}}$, we deduce the following corollay.

Corollary 3.3. Let $f\in \mathcal{A}$. Suppose that

If the following holds:

then, we have

Theorem 3.4. Let $h\left(\varepsilon \right) \in \mathcal{H}\left(\mathfrak{D}\right)$ with $h\left(0\right) = 1$. Suppose that

If the following subordination holds:

then,

Proof. Suppose that

Now if

then we can easily obtain

and so,

If the function $\omega$ achieves its maximum value, $\left \vert \omega \left(\varepsilon _{0}\right)\right \vert = 1$, at some point $\varepsilon = \varepsilon _{0}$, applying Lemma 2.1 gives us,

Substituting (3.6) and (3.15) into Eq (3.24), we have

It demonstrates that the function $\phi \left(m\right)$ is increasing. So, its maximum value is obtained at $m = 1$. Thus,

and hence

By (3.23), a contradiction occurs. We must have $\left \vert \omega\left(\varepsilon \right) \right \vert < 1$, so we obtain the needed outcome. □

By taking $h\left(\varepsilon \right) = \frac{\varepsilon D_{q}f\left(\varepsilon \right) }{\left(f\left(\varepsilon \right) \right) ^{3}}$, we deduce the following result.

Corollary 3.4. Let $f\in \mathcal{A}$. Suppose that

If the following condition holds:

then,

Theorem 3.5. Let $h\left(\varepsilon \right) \in \mathcal{H}\left(\mathfrak{D}\right)$ with $h\left(0\right) = 1$. Suppose that

If the following subordination is provided:

then,

Proof. Suppose

Now consider

We can easily obtain that

and hence,

If the function $\omega$ accomplishes its maximum value, $\left \vert \omega \left(\varepsilon_{0}\right) \right \vert = 1$, at some point $\varepsilon = \varepsilon _{0}$, then, utilizing Lemma 2.1, we have

Now set (3.6) and (3.15) in (3.29), and we get

It demonstrates that $\phi \left(m\right)$ increases, achieving the maximum value at $m = 1$. Thus,

and hence

which contradicts the assumption (3.28), so $\left \vert \omega\left(\varepsilon \right) \right \vert < 1$. We complete the proof. □

By taking $h\left(\varepsilon \right) = \frac{\varepsilon D_{q}f\left(\varepsilon \right) }{\left(f\left(\varepsilon \right) \right) ^{n}}$, we deduce the following corollary.

Corollary 3.5. Let $f\in \mathcal{A}$. Suppose that

If the following is given:

then,

Remark 3.1. One can obtain new results for the class $\mathcal{S}_{\tan }^{\ast }$ by taking the limit $q\rightarrow 1^{-}$ in our results.

4.   Coefficient estimates for class $\mathcal{S}_{\tan }^{\ast }\left(q\right)$

Theorem 4.1. If $f\in \mathcal{S}_{\tan }^{\ast }\left(q\right)$ is of the form (1.1), then

All these estimates are extreme for a function defined below:

Proof. Suppose $f\in \mathcal{S}_{\tan }^{\ast }\left(q\right),$ and then there exists a Schwarz function $\omega \left(\varepsilon \right)$ such that

where $\omega \left(\varepsilon \right) = \frac{p\left(\varepsilon \right) -1}{p\left(\varepsilon \right) +1}.$ If $p\left(\varepsilon \right)$ follows the form of (1.2), then

Using this, one can easily find

Then

Comparing (4.2) and (4.3), we obtain

Applying (2.1) to (4.4), and we have

To find the bound of $\mathfrak{\xi }_{3}$, apply (2.2) to (4.5) with $0.5 < q < 1,$ and we have

From Lemma 2.5,

and

so the conditions $0\leq \alpha _{2}\leq 1$ and $\alpha _{2}\left(2\alpha _{2}-1\right) \leq \alpha _{1}\leq \alpha _{2}$ are satisfied for $0.41310 < q < 0.57708$. Applying Lemma 2.5, we get

□

Corollary 4.1. If $f\in \mathcal{S}_{\tan }^{\ast },$ then

The above three bounds are sharp for the function defined below:

Theorem 4.2. If $f\in \mathcal{S}_{\tan }^{\ast }\left(q\right),$ then

The result is sharp for the function defined in (4.1), for $n = 2.$

Proof. From (4.4) and (4.5), we have

Applying Lemma 2.3 to the preceding equation yields the desired outcome. □

Corollary 4.2. If $f\in \mathcal{S}_{\tan }^{\ast },$ then

The result is sharp for the function defined in (4.7), for $n = 2.$

Theorem 4.3. If $f\in \mathcal{S}_{\tan }^{\ast }\left(q\right),$ then

The outcome is sharp for the function defined in (4.1), for $n = 3.$

Proof. From (4.4)–(4.6), we have

Comparing with Lemma 2.5, we have

and the conditions $0\leq \alpha _{2}\leq 1$ and $\alpha _{2}\left(2\alpha_{2}-1\right) \leq \alpha _{1}\leq \alpha _{2}$ are satisfied for $0.14527 < q < 0.22265,$ so Lemma 2.5 is valid to apply. Hence

□

Corollary 4.3. If $f\in \mathcal{S}_{\tan }^{\ast },$ then

The result is sharp for the function defined in (4.7), for $n = 3.$

Theorem 4.4. Let $f\in \mathcal{S}_{\tan }^{\ast }\left(q\right)$ be given in the form (1.1), and then

Proof. From (4.4)–(4.6), we have

Using Lemma 2.4 for $c_{1} = c$, we have

For $c\in \left[ 0, 2\right]$, the simple calculation gives

where $x$ and $\delta$ satisfy $\left \vert x\right \vert \leq 1$ and $\left \vert \delta \right \vert \leq 1$.

Next, we will find the maximum value of $\left \vert \mathfrak{\xi }_{2} \mathfrak{\xi }_{4}-\mathfrak{\xi }_{3}^{2}\right \vert$ for $c\in \left[ 0, 2\right]$.

Case 1. When $c = 0$, we have

Case 2. When $c = 2$, we get

Case 3. Assume $c\in \left(0, 2\right)$. Then, by the above equation and the triangular inequality,

By Lemma 2.6, we can write it as

where

and

Clearly, for all $q\in \left(0, 1\right)$ and $c\in \left(0, 2\right)$, we have

where

Now we consider $\left \vert Q\right \vert -2\left(1-\left \vert R\right \vert \right)$. Let

which implies

Differentiating $\varphi \left(c\right)$ twice, we have

This shows that $\varphi ^{\prime }\left(c\right)$ is an increasing function and

It follows that

Hence, $\varphi \left(c\right)$ is a decreasing function and

This implies $\left \vert Q\right \vert -2\left(1-\left \vert R\right \vert \right) > 0$, and then by Lemma 2.6,

where

By differentiating, we have

where

and

which implies

We conclude that the function $h\left(c\right)$ is a decreasing function and

From all of the above discussion, we conclude that

□

5.   Sufficiency criteria for class $\mathcal{S}_{\tan }^{\ast }\left(q\right)$

Theorem 5.1. A function $f\in \mathcal{S}_{\tan }^{\ast }\left(q\right)$ if and only if

where

Proof. If a function $f\in$ $\mathcal{S}_{\tan }^{\ast }\left(q\right)$, then $f$ is analytic in $\mathfrak{D},$ and hence $\frac{1}{\varepsilon } f(\varepsilon)\neq 0\ $for all $\varepsilon$ in $\mathfrak{D}^{\ast } = \mathfrak{D}-\{0\}.$ Thus, we have the Eq (5.1) for $G = H = 1.$

Now, by using (1.9) along with the principle of subordination, there exists a Schwarz function $\omega$ such that

Taking into consideration $\omega \left(\varepsilon \right) = e^{i\theta }, \ $for $0\leq \theta \leq 2\pi$, then the above expression becomes

Now, by using the relations

the Eq (5.4) becomes

After some simple calculations, we get

where $H$ and $G$ are given in $\left(1.3\right),$ and thus the the necessary condition holds.

Conversely, assume that the condition in (5.1) satisfies, then $\frac{1}{\varepsilon}f\left(\varepsilon \right) \neq 0$, for all $\varepsilon \in \mathfrak{D}.$ Let $K(\varepsilon) = \frac{\varepsilon \left(D_{q}f\right) \left(\varepsilon \right) }{f\left(\varepsilon \right) }$, which is regular in $\mathfrak{D}$ and $K(0) = 1.\ $

Also, assume that $f(\varepsilon) = 1+\frac{ \tan \left(\varepsilon \right) }{2}$, and from (5.3), $f(\partial \mathfrak{D})\cap K(\varepsilon) = \emptyset$.

Therefore, the connected component $\mathbb{C}-f(\partial \mathfrak{D})$ containing the domain $K(\varepsilon)$ is connected as well. Given the univalence of "$K$" and the supposition that $f(0) = K(0) = 1, \ $ it is evident that $K\prec f$, indicating that $f\in \mathcal{S} _{\tan }^{\ast }\left(q\right).$ □

6.   Conclusions

In conclusion, leveraging the framework of $q$-calculus, we have introduced a novel class of $q$-starlike functions associated with the eight-shaped image domain, offering new insights into their geometric behavior. Our research successfully established results in differential subordination and derived sharp inequalities for the first three unknown coefficients of the Taylor series. Additionally, we provided precise solutions to the Fekete-Szegö problem and second-order Hankel determinants for this newly defined class, with the broader implications demonstrated through a series of corollaries.

The scope of these findings extends beyond the specific class of functions examined. Our methodology offers a flexible foundation for analyzing other image domains and subclasses within geometric function theory. This opens avenues for future work that could explore more general families of functions, potentially leading to new results in related areas, such as higher-order Hankel determinants, multi-variable quantum calculus, and further applications of $q$-differential operators. The results enhance the current understanding of $q$-starlike functions and provide a robust platform for future investigations into the more profound applications of quantum calculus in geometric and analytic function theory.

Author contributions

Jianhua Gong, Muhammad Ghaffar Khan, Hala Alaqad and Bilal Khan: Writing-original draft, writing-review and editing. All authors contributed equally to the manuscript. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The first author is supported by the UAEU Program for Advanced Research (UPAR12S127) from United Arab Emirates University.

Conflict of interest

The authors declare that they have no competing interests.