Estimates of coefficients for bi-univalent Ma-Minda-type functions associated with $\mathfrak{q}$-Srivastava-Attiya operator

1.   Introduction

The class of functions $f$ that are analytic in the open unit disk $\mathbb D = \{ z : z \in \mathbb{C} \; \; and\; \; | z | < 1\}$ and are normalized by the criteria $f (0) = f ^{'}(0) -1 = 0$ is indicated by the symbol $\mathcal{H}$. Equivalently, if $f \in \mathcal{H}$, the Taylor-Maclaurin series representation takes the form:

In addition, let us designate $\mathcal{S}$ as the fundamental subclass of $\mathcal{H},$ whose functions are univalent in $\mathbb D$. Koebe one-quarter theorem ^[1], which is widely recognized, guarantees that the image of $\mathbb D$ under each function $f \in \mathcal{S}$ contains a disk with a radius of $\dfrac{1}{4}$. Therefore, it can be concluded that every univalent function $f$ possesses an inverse $f ^{-1}$ that satisfies the equation

and

where

If $f$ and $f ^{-1}$ are both univalent in $\mathbb D$, then a function $f \in \mathcal{H}$ is considered bi-univalent in $\mathbb D$. The class of bi-univalent functions defined in the unit disk $\mathbb D$ is indicated by $\tau$. Because it univalently maps the unit disk $\mathbb D$ onto the entire complex plane, minus a slit along the line from $-\dfrac{1}{4}$ to $-\infty$, the well-known Koebe function is not an element of $\tau$. Therefore, the unit disk $\mathbb D$ is absent from the image domain. The well-known Bieberbach conjecture, which asserts that the following coefficient inequality holds for each $f \in \mathcal{S}$ produced by the Taylor-Maclaurin series expansion (1.1), was established in 1985 by Louis de Branges ^[2],

The class of analytic bi-univalent functions was first introduced and studied by Lewin who proved $|a_2| < 1.51$. Brannan and Clunie subsequently enhanced Lewin's outcome to $|a_2| \leq \sqrt2$. Indeed, it is verified for the class of bi-close to convex functions in the article ^[3]. Researchers Brannan and Taha ^[4] and Taha ^[5] looked at different types of bi-univalent functions and found that they are similar to well-known types of univalent functions represented by strongly starlike, starlike and convex functions. We present bi-starlike functions and bi-convex functions, and established non-sharp estimates for the first two Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$.

Next, we recall the definition of subordination between analytic functions. For two functions $f, \mathcal G\in \mathcal{H}$, we say that the function $f$ is subordinate to $\mathcal G$, if there exists a Schwarz function $w$, which is analytic in $\mathbb{D}$ with the following property:

such that

This subordination is symbolically written as follows:

It is well known that if the function $\mathcal G$ is univalent in $\mathbb D$, then the following equivalence holds (see ^[6]):

The $\mathfrak{q}$-difference operator which was introduced by Jackson is

The following limit relationship is clear:

For a function $f \in \mathbb D$ defined by (1.1), we deduce the following result:

where $[l]_\mathfrak{q}$ is given by

As $\mathfrak{q}\to 1^{-}$, we have $[l]_\mathfrak{q}\rightarrow l$ and $[0]_\mathfrak{q} = 0$.

The Srivastava-Attiya operator, which has been extensively investigated, was defined by Srivastava and Attiya ^[7] by using the Hurwitz-Lerch zeta function $\Phi(z, \epsilon, \alpha)$, which is systematically discussed in recent survey papers ^[8,9]. To obtain comprehensive information regarding the interconnections between the function $\Phi(z, \epsilon, \alpha)$ and many significant functions within the realm of analytic number theory, readers may consult Chapter I in reference ^[10]. The following $\mathfrak{q}$-analogue of the Hurwitz-Lerch zeta function $\Phi(z, \epsilon, \alpha)$ was explored by Shah and Noor ^[11] (see also ^[12]):

where $\alpha\in\mathbb{C} \backslash \{\mathbb{Z}^{-}_{0}\}, \epsilon \in\mathbb{C}\; \text{when}\; | z | < 1, \, \Re(\epsilon) > 1,$ and $| z | = 1.$

$\mathbb{Z}^{-}_{0}$ being the set of negative integers. The normalized form of the series (1.4) is defined by

By using (1.1) and (1.5), Shah and Noor ^[11] defined the $\mathfrak{q}$-Srivastava-Attiya operator $\mathcal{J}^{\epsilon}_{\mathfrak{q}, \alpha}:\mathcal{H}\rightarrow \mathcal{H}$ as follows:

Definition 1.1. (see ^[11,12]) The $\mathfrak{q}$-Srivastava-Attiya operator: $\mathcal{J}^{\epsilon}_{\mathfrak{q}, \alpha}:\mathcal{H}\rightarrow \mathcal{H}$ is defined in terms of the Hadamard product (or convolution) by

where

The mathematical applications of $\mathfrak{q}$-calculus, fractional $\mathfrak{q}$-calculus, and fractional $\mathfrak{q}$-derivative operators in geometric function theory of complex analysis were investigated by Srivastava ^[13] in his recently published survey-cum-expository review article. Srivastava ^[13] also exposed the not-yet-widely-understood fact that the so-called $(p, \mathfrak{q})$-variation of classical $\mathfrak{q}$-calculus is a relatively trivial and inconsequential variation of classical $\mathfrak{q}$-calculus, the additional parameter $p$ being redundant or superfluous (see, for details, ^[13], p. 340).

In this paper, we utilize the fundamental or quantum (or $\mathfrak{q}-$) extension $\phi_{\mathfrak{q}}(\epsilon, \alpha; z).$ When $\mathfrak{q} \rightarrow 1^{-}$, it produces the well-known Hurwitz-Lerch zeta function $\Phi(z, \epsilon, \alpha)$. Local or non-local symmetries are observed in certain properties of many members of the Hurwitz-Lerch zeta function. family, as previously mentioned. Additional support for our investigation into the practical uses of quantum extensions (or $\mathfrak{q}-$) in this research can be located in the chapter titled "Symmetric Quantum Calculus" in reference ^[14].

Remark 1.1. The operator $\mathcal{J}^{\epsilon}_{\mathfrak{q}, \alpha}$ is a generalization of several known operators studied in earlier investigations, which are recalled below:

1) The operator $\mathcal{J}^{\epsilon}_{\mathfrak{q}, \alpha}$ coincides with the Srivastava-Attiya operator in ^[7], and for $\mathfrak{q}\rightarrow 1^{-},$ the function $\phi_{\mathfrak{q}}(\epsilon, \alpha; z)$ reduces to the Hurwitz-Lerch zeta function (see ^[8,9]). The Srivastava-Attiya operator has several uses, which can be found in ^[15,16,17] and the references listed in each of these previous publications.

2) The operator $\mathcal{J}^{\epsilon}_{\mathfrak{q}, \alpha}$ reduces to the $\mathfrak{q}$-Bernardi operator for $\epsilon = 1$ as stated in ^[18].

3) The operator $\mathcal{J}^{\epsilon}_{\mathfrak{q}, \alpha}$ reduces to the $\mathfrak{q}$-Libera operator for $\epsilon = \alpha = 1,$ as stated in ^[18].

4) The operator $\mathcal{J}^{\epsilon}_{\mathfrak{q}, \alpha}$ reduces to the Bernardi operator for $\mathfrak{q} \rightarrow 1^{-}$ and $\epsilon = 1$ (see ^[19]).

5) The operator $\mathcal{J}^{\epsilon}_{\mathfrak{q}, \alpha}$ reduces to the Alexander operator for $\mathfrak{q}\rightarrow 1^{-}, \epsilon = 1,$ and $\alpha = 0$ (see ^[20]).

We define the subclasses $S^{\ast, \epsilon}_{\mathfrak{q}, \alpha}(\delta)$ and $K^{\epsilon}_{\mathfrak{q}, \alpha}(\delta)$ of the class $\mathcal{H}$ for $0\leq \delta < 1$ using the $\mathfrak{q}$-Srivastava-Attiya operator.

Definition 1.2. A function $f (z)$ of the form (1.1) is in the class $S^{\ast, \epsilon}_{\mathfrak{q}, \alpha}(\delta)$ if it satisfies the following condition:

When $\epsilon = 0$, we obtain the result.

Corollary 1.1. A function $f (z)$ of the form (1.1) is in the class $S^{\ast, 0}_{\mathfrak{q}, \alpha}(\delta)$ if it satisfies the following condition ^[21]:

Definition 1.3. A function $f (z)$ of the form (1.1) is in the class $K^{\epsilon}_{\mathfrak{q}, \alpha}(\delta)$ if it satisfies the following condition:

Observe that $\mathcal{J}^{\epsilon}_{\mathfrak{q}, \alpha} f \in K^{\epsilon}_{\mathfrak{q}, \alpha}(\delta)$ if and only if $z (\mathcal{J}^{\epsilon}_{\mathfrak{q}, \alpha} f (z))^{'}\in S^{\ast, \epsilon}_{\mathfrak{q}, \alpha}(\delta).$ When $\epsilon = 0$, we obtain the result.

Corollary 1.2. A function $f (z)$ of the form (1.1) is in the class $K^{0}_{\mathfrak{q}, \alpha}(\delta)$ if it satisfies the following condition ^[22]:

Definition 1.4. A function $f (z)$ of the form (1.1) is in the class $H^{\epsilon}_{\mathfrak{q}, \alpha}(\delta)$ if it satisfies the following condition:

When $\epsilon = 0$, we obtain the result.

Corollary 1.3. A function $f (z)$ of the form (1.1) is in the class $H^{0}_{\mathfrak{q}, \alpha}(\delta)$ if it satisfies the following condition ^[23]:

In this study, we derive estimates for the initial coefficients $a_2$ and $a_3$ of three novel subclasses of the class $\tau$ of bi-univalent functions.

Let $\psi$ be an analytic function with a positive real part in $\mathbb D$ such that $\psi(0) = 1, \; \psi(0) > 0$, and $\psi(\mathbb D)$ is symmetric with respect to the real axis. Such a function has a series expansion of the form

With this brief introduction, we define the following classes of bi-univalent functions and find the coefficient estimates with the help of the $\mathfrak{q}$-Srivastava-Attiya operator.

Definition 1.5. A function $f \in \tau$ is said to be in the class $\mathcal{T}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\psi)$ if the following subordinations hold

Definition 1.6. A function $f (z) \in \tau$ is said to be in the class $\mathcal{KH}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\delta, \psi), \delta\geq0,$ if the following subordinations hold:

and

where $\mathcal G(\mu) = f ^{-1}(\mu).$

Definition 1.7. A function $f \in \tau$ is said to be in the class $\mathcal{A}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\psi), \; \delta > 0,$ if the following subordinations hold:

and

where $\mathcal G(\mu) = f ^{-1}(\mu).$

In order to derive our main results, we have to recall the following lemma here.

Lemma 1.1. ^[24] If the function $p \in \mathcal{P}$ is given by the series

where $\mathcal{P}$ is the family of all functions $p(z)$ analytic in $\mathbb D$ and satisfying $\Re \{p(z)\} > 0.$ Then the following sharp estimate holds:

Also, $\gamma\in \mathbb{R}$ for all, we obtain

2.   Main results

Theorem 2.1. Let $f (z)\in \mathcal{T}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\psi)$ and be given by (1.1). Then

Proof. Let $f \in \mathcal{T}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\psi)$ and $\mathcal G = f ^{ -1}$. Then there are analytic functions $r, \mathcal{F} : \mathbb D \rightarrow \mathbb D,$ with $r(0) = \mathcal{F}(0) = 0,$ satisfying

The functions $p_1$ and $p_2$ are defined as follows:

or, equivalently,

and

It is clear that $p_{1}$ and $p_{2}$ are analytic in $\mathbb D$ and $p_{1}(0) = p_{2}(0) = 1$. Also $p_{1}$ and $p_{2}$ have a positive real part in $\mathbb D$ and hence $|b_{i}| \leq 2$ and $|c_{i}| \leq 2\; \; (i \in \mathbb{N}\backslash \{1\}).$

Substituting (2.3) and (2.4) into (2.2) and using (1), we can obtain

and

Since $f \in \tau$ has the Maclaurin series given by (1.1), a computation shows that its inverse $\mathcal G = f ^{-1}$ has the expansion $\mathcal G(\mu) = f ^{-1}(\mu) = \mu-a_{2}\mu^{2} + (2a^{2}_{2}-a_{3})\mu^{3} +....$.

Since

and

it follows from (2.5), (2.6), and (2.2) that

From (2.7) and (2.9), we obtain

and

Adding Eqs (2.8) and (2.10) and using (2.12), we now obtain

Applying Lemma 1.1 for the coefficients $b_{2}$ and $c_{2}$, we immediately have

As stated in (2.7), this provides us with the bound on $|a_{2}|$. Next, in order to find the bound on $|a_{3}|$, by subtracting (2.10) from (2.8) and also from (2.11), we get $c^{2}_{1} = b^{2}_{1}$, hence

Using (2.12) and applying Lemma 1.1 once again for the coefficients $b_{2}$ and $c_{2}$, we have

This completes the proof of Theorem 2.1. □

When $\epsilon = 0$, we obtain the result presented by Ali et al. ^[25].

Corollary 2.1. Let $f (z)\in \mathcal{T}_{\tau}(\psi)$ and be given by (1.1). Then

Theorem 2.2. Let $f (z)$ given by (1.1) be in the class $\mathcal{KH}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\delta, \psi)$. Then

and

Proof. Let $f (z)\in \mathcal{KH}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\delta, \psi).$ Then there are analytic functions $r, \mathcal{F} : \mathbb D \rightarrow \mathbb D,$ with $r(0) = \mathcal{F}(0) = 0,$ satisfying

and

where $\mathcal G(\mu) = f ^{-1}(\mu).$ Since $\mathcal{J}^{\epsilon}_{\mathfrak{q}, \alpha} f (z) = z +C^{\epsilon}_{\mathfrak{q}, \alpha}(2)a_{2} z ^{2}+C^{\epsilon}_{\mathfrak{q}, \alpha}(3)a_{3} z ^{3}+...$ and $\; \; \mathcal{J}^{\epsilon}_{\mathfrak{q}, \alpha}\mathcal G(\mu) = \mu-C^{\epsilon}_{\mathfrak{q}, \alpha}(2)a_{2}\mu^{2}+C^{\epsilon}_{\mathfrak{q}, \alpha}(3)(2a^{2}_{2}-a_{3})\mu^{3}+...,$ we have

and

Equating the coefficients (2.16), (2.17), (2.5), and (2.6) on both sides, we have

From (2.18) and (2.20), we obtain

and

Adding Eqs (2.19) and (2.21) and using (2.24), we now obtain

Applying Lemma 1.1 for the coefficients $b_{2}$ and $c_{2}$, we immediately get

Since $\nu_{1} > 0,$ the last inequality gives the desired estimate on $|a_{2}|$ given in (2.14). Next, in order to find the bound on $|a_{3}|$, by subtracting (2.21) from (2.19) and also from (2.23), we get $c^{2}_{1} = b^{2}_{1}$, hence

Applying Lemma 1.1 once again for the coefficients $b_{2}$ and $c_{2}$, we obtain

This estimate is exactly that found in (2.15).

When $\epsilon = 0$, we obtain the result presented by Ali et al. ^[25]. □

Corollary 2.2. Let $f (z)$ given by (1.1) be in the class $\mathcal{KH}_{\tau}(\delta, \psi).$ Then

and

When $\epsilon = 0$ and $\delta = 0$, the coefficient estimates for Ma-Minda bi-starlike functions are obtained ^[26].

Corollary 2.3. Let $f (z)$ given by (1.1) be in the class $\mathcal{KH}_{\tau}(0, \psi).$ Then

and

Theorem 2.3. Let $f$ given by (1.1) be in the class $\mathcal{A}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\psi)$. Then

and

Proof. Let $f (z)\in \mathcal{A}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\delta, \psi).$ Then there are analytic functions $r, \mathcal{F} : \mathbb D \rightarrow \mathbb D,$ with $r(0) = \mathcal{F}(0) = 0,$ satisfying

and

where $\mathcal G(\mu) = f ^{-1}(\mu).$ By (2.27) and (2.28), we have

and

Equating the coefficients (2.27), (2.28), (2.5), and (2.6) on both sides, we have

From (2.29) and (2.31), we obtain

and

Adding Eqs (2.30) and (2.32) and using (2.34), we now obtain

Applying Lemma 1.1 for the coefficients $b_{2}$ and $c_{2}$, we immediately get

which yields the desired estimate on $|a_{2}|$ as described in (2.25). Next, in order to find the bound on $|a_{3}|$, by subtracting (2.32) from (2.30) and also from (2.33), we get $c^{2}_{1} = b^{2}_{1}$, hence

Applying Lemma 1.1 once again for the coefficients $b_2$ and $c_2$, we obtain

This estimate is exactly that found in (2.26). □

When $\epsilon = 0$, we obtain the result presented by Ali et al. ^[25].

Corollary 2.4. Let $f (z)$ given by (1.1) be in the class $\mathcal{A}_{\tau}(\delta, \psi).$ Then

and

The coefficient estimates for Ma-Minda bi-starlike functions are obtained when $\epsilon = 0$ and $\delta = 0$. Conversely, for Ma-Minda bi-convex functions, the coefficient estimates are obtained when $\delta = 1$ ^[26].

Corollary 2.5. Let $f (z)$ given by (1.1) be in the class $\mathcal{CN}_{\tau}(\psi).$ Then

and

3.   Fekete-Szegö inequality for the functions classes

Fekete and Szegö published their findings in 1933 ^[27], setting a precise limit for the functional $a_{3}-\gamma a^{2}_{2}$. This limit, known as the conventional Fekete-Szegö inequality, was determined using real values of $\gamma \; (0\leq\gamma\leq1)$. Establishing accurate bounds for a function within a compact family of functions $(f \in \mathcal{H})$ for a real parameter $\gamma$ is a tough problem. The Fekete-Szegö coefficient bounds for different analytic subclasses have been established by other authors ^[28,29,30]. In this context, the Fekete-Szegö inequality for functions belonging to the classes $f (z)\in\mathcal{T}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\psi), \; \mathcal{KH}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\delta, \psi), \text{and}\; \mathcal{A}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\delta, \psi),$ is studied, using the findings of Zaprawa ^[31].

Theorem 3.1. Let the function $f (z)$ given by (1.1) be in the class $\mathcal{T}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\psi).$ Then for some $\gamma\in\mathbb{R},$

where

and

Proof. Using for those in the Eqs (2.9) and (2.10), we get

and

Now, from (3.2) and (3.3), we can easily see that

and

Therefore, in view of Lemma 1.1, we conclude that

Moreover, we have

Taking $\epsilon = 0$ in Theorem 3.1, we obtain the following corollary.

Corollary 3.1. Let $f (z)$ given by (1.1) be in the class $\mathcal{T}_{\tau}(\psi).$ Then

where

and

Theorem 3.2. Let the function $f (z)$ given by (1.1) be in the class $\mathcal{KH}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\delta, \psi).$ Then for some $\gamma\in\mathbb{R},$

where

and

Proof. Using the relations in Eqs (2.20) and (2.21), we get

Now, from (3.5) and (3.6), we can see that

Therefore, in view of Lemma 1.1, we conclude that

Moreover, we have

□

Taking $\epsilon = 0$ and $\delta = 0$ in Theorem 3.2, we obtain the following corollary.

Corollary 3.2. Let $f (z)$ given by (1.1) be in the class $\mathcal{KH}_{\tau}(0, \psi)$. Then

and

Theorem 3.3. Let the function $f (z)$ given by (1.1) be in the class $\mathcal{A}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\delta, \psi).$ Then for some $\gamma\in\mathbb{R},$

where

and

Proof. Using for those in Eqs (2.29) and (2.30), we get

Now, from (3.8) and (3.9), we can easily see that

Therefore, in view of Lemma 1.1, we conclude that

Moreover, we have

□

Taking $\epsilon = 0$ and $\delta = 0$ from Theorem 3.3, we obtain the following corollary.

Corollary 3.3. Let $f (z)$ given by (1.1) be in the class $\mathcal{A}_{\tau} (0, \psi).$ Then

where

and

In these Theorems 3.1–3.3, we use the technique of ^[32].

4.   Conclusions

In our present investigation, we have introduced and studied the coefficient problems associated with each of the following three new subclasses: $\mathcal{T}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\psi),$ $\mathcal{KH}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\delta, \psi)$, $\mathcal{A}^{\epsilon}_{\tau, \mathfrak{q}, \alpha}(\psi)$ of the class of bi-univalent Ma-Minda-type functions associated with the $\mathfrak{q}$-Srivastava-Attiya operator in the open unit disk $\mathbb{D}$. These bi-univalent Ma-Minda-type functions associated with $\mathfrak{q}$-Srivastava-Attiya operator classes are given by Definitions 1.2 to 1.7, respectively. For functions in each of these three bi-univalent Ma-Minda-type functions associated with $\mathfrak{q}$-Srivastava-Attiya operator classes, we have derived the estimates of the Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions belonging to these new subclasses, along with estimates for the Fekete-Szegö functional problem, for functions belonging to each of these bi-univalent function classes. Our results also further generalize the results of Theorems 2.1–2.3 of R. M. Ali et al. and some results of W. Ma and D. Minda. The results presented in this paper are a beneficial supplement for the research of geometric function theory of complex analysis.

Author contributions

Norah Saud Almutairi: Investigation, supervision, writing–original draft, writing–review, and editing; Adarey Saud Almutairi: Supervision; Awatef Shahen: Supervision; Hanan Darwish: Supervision, writing–review and editing. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The first author would like to thank her father Saud Dhaifallah Almutairi for supporting this work.

Funding

The author(s) express their sincere gratitude and appreciation to [Onaizah Colleges, Saudi Arabia] for providing APC funding for this research.

Conflicts of interest

The authors declare that they have no conflict of interest.