Applications of a certain $q$-integral operator to the subclasses of analytic and bi-univalent functions

1.   Introduction and definitions

By $\mathcal{H}\left(\mathbb{U}\right)$ we denote the analytic function class in the open unit disk

where $\mathbb{C}$ represents the set of complex numbers.

The class $\mathcal{A}$ of normalized analytic functions consists of functions $f\in\mathcal{H}\left(\mathbb{U}\right)$, which have the following Taylor-Maclaurin series expansion:

and satisfy the normalization condition given by

Further, a noteworthy subclass of $\mathcal{A}$, which contains all univalent functions in the open unit disk $\mathbb{U}$, is denoted by $\mathcal{S}$.

All functions $f\in \mathcal{S}$ that satisfy the following condition:

are placed in the class $\mathcal{S}^{\ast}$ of starlike functions in $\mathbb{U}$.

For regular functions $f$ and $g$ in the unit disk $\mathbb{U}$, we say that the function $f$ is subordinate to the function $g$, and write

if there exists a Schwarz function $w$ of the class $\mathcal{B}$, where

such that

Specifically, when the given function $g$ is regular in $\mathbb{U}$, then the following equivalence holds true:

We next introduce the class $\mathcal{P}$ which consists of functions $p$, which are analytic in $\mathbb{U}$ and normalized by

such that

In the theory of analytic functions, the vital role of the function class $\mathcal{P}$ is obvious from the fact that there are many subclasses of analytic functions which are related to this class of functions denoted by $\mathcal{P}$.

In connection with functions in the class $\mathcal{S}$, on the account of the Koebe one-quarter theorem (see ^[9]), it is clear that, under every function $f\in \mathcal{S}$, the image of $\mathbb{U}$ contains a disk of radius $\frac{1}{4}$. Consequently, every univalent function $f\in \mathcal{S}$ has an inverse $f^{-1}$ given by

and

where

A function $f\in \mathcal{S}$ such that both $f$ and its inverse function $g = f^{-1}$ are univalent in $\mathbb{U}$ is known as bi-univalent in $\mathbb{U}$. The class of bi-univalent functions in $\mathbb{U}$ is symbolized by $\Sigma$. In their pioneering work, Srivastava et al. ^[46] basically resuscitated the study of the analytic and bi-univalent function class $\Sigma$ in recent years. In fact, as sequels to their investigation in ^[46], a number of different subclasses of $\Sigma$ have since then been presented and studied by many authors (see, for example, ^{[2,5,6,7,8,11,25,26,35,38,40,41,42,47,51,52,53,55,56,57]}). However, except for a few of the cited works using the Faber polynomial expansion method for finding upper bounds for the general Taylor-Maclaurin coefficients, most of these investigations are devoted to the study of non-sharp estimates on the initial coefficients $\left \vert a_{2}\right \vert$ and $\left \vert a_{3}\right \vert$ of the Taylor-Maclaurin series expansion.

Some important elementary concept details and definitions of the $q$-calculus which play vital role in our presentation will be recalled next.

Definition 1. Let $q\in \left(0, 1\right)$ and define the $q$-number $\left[ \lambda \right] _{q}$ by

Definition 2. Let $q\in \left(0, 1\right)$ and define the $q$-factorial $\left[n\right] _{q}!$ by

Definition 3. The generalized $q$-Pochhammer symbol is defined, for $t\in \mathbb{R}$ and $n\in \mathbb{N},$ by

Also, for $t > 0$, let the $q$-gamma function be defined as follows:

where

Definition 4. (see ^[13] and ^[14]) For a function $f$ in the class $\mathcal{A}$, the $q$-derivative (or $q$-difference) operator $D_{q}$ is defined, in a given subset of $\mathbb{C}$, by

We note from Definition 4 that the $q$-derivative operator $D_{q}$ converges to the ordinary derivative operator as follows:

for a differentiable function $f$ in a given subset of $\mathbb{C}$. Further, taking (1.1) and (1.6) into account, it is easy to observe that

Recently, the study of the $q$-calculus has fascinated the intensive devotion of researchers. The great concentration is because of its advantages in many areas of mathematics and physics. The significance of the $q$-derivative operator $D_{q}$ is quite obvious by its applications in the study of several subclasses of analytic functions. Initially, in the year 1990, Ismail et al. ^[12] gave the idea of $q$-starlike functions. Nevertheless, a firm foothold of the usage of the $q$-calculus in the context of Geometric Function Theory was effectively established, and the use of the generalized basic (or $q$-) hypergeometric functions in Geometric Function Theory was made by Srivastava (see, for details, [29, pp. 347 et seq.]). After that, notable studies have been made by numerous mathematicians which offer a momentous part in the advancement of Geometric Function Theory. For instance, Srivastava et al. ^[44] examined the family of $q$-starlike functions associated with conic region, and in ^[22] the estimate on the third Hankel determinant was settled. Recently, a set of articles were published by Srivastava et al. (see, for example, ^{[20,43,49,50]}) in which they studied various families of $q$-starlike functions related with the Janowski functions from different aspects. For some more recent investigations about $q$-calculus, we may refer the reader to ^{[1,3,4,15,16,27,33,36,48]}.

We remark in passing that, in the aforementioned recently-published survey-cum-expository review article ^[33], the so-called $(p, q)$-calculus was exposed to be a rather trivial and inconsequential variation of the classical $q$-calculus, the additional parameter $p$ being redundant or superfluous (see, for details, [33, p. 340]). Srivastava ^[33] also pointed out how the Hurwitz-Lerch Zeta function as well as its multi-parameter extension, which is popularly known as the $\lambda$-generalized Hurwitz-Lerch Zeta function (see, for details, ^[30]), have motivated the studies of several other families of extensively- and widely-investigated linear convolution operators which emerge essentially from the Srivastava-Attiya operator ^[37] (see also ^[31] and ^[32]).

Definition 5. (see ^[12]) A function $f$ in the function class $\mathcal{A}$ is said to belong to the function class $\mathcal{S}_{q}^{\ast }$ if

and

Then on the account of last inequality, it is obvious that, in limit case when $q\rightarrow 1-$

the above closed disk is merely the right-half plane and the class $\mathcal{S}_{q}^{\ast}$ of $q$-starlike functions turns into the familiar class $\mathcal{S}^{\ast}$ of starlike functions in $\mathbb{U}$. Analogously, in view of the principle of subordination, one may express the relations in $(1.8)$ and $(1.9)$ as follows: (see ^[54]):

In recent years, many integral and derivative operators were defined and studied from different viewpoints and different perspectives (see, for example, ^[10,19,23]). Motivated by the ongoing researches, Srivastava et al. ^[45] introduced the $q$-version of the Noor integral operator as follows.

Definition 6. (see ^[45]) Let a function $f\in \mathcal{A}$. Then the $q$-integral operator is given by

and

where

and

Specifically, we notice that

Clearly, in limit case when $q\rightarrow 1-$, th above $q$-integral operator simply becomes to the Noor integral operator (see ^[24]). It is straightforward to verify the following identity:

If $q\rightarrow 1-$, the equality (1.11) implies that

which is the known recurrence formula for the Noor integral operator (see ^[24]).

Motivated by the works, which we have mentioned above, we now define subfamilies of the normalized univalent function class $\mathcal{S}$ by means of the operator $\mathcal{I}_{q}^{\lambda }$ and the principle of subordination between analytic functions as follows.

Definition 7. Let a function $f \in \mathcal{S}$. Then $f$ belongs to the function class $\mathcal{H}_{q}(\lambda, \widehat{p})$ if it satisfies the following conditions:

and

where the function $\widehat{p}\left(z\right)$ is analytic and has positive real part in $\mathbb{U}$. Moreover, $\widehat{p}\left(0\right) = 1$, $\widehat{p}^{\prime} \left(0\right) > 0$, and $\widehat{p}\left(\mathbb{U}\right)$ is symmetric with respect to the real axis. Consequently, it has a series expansion of the form given by

noticing that $g(w) = f^{-1}(w)$.

In order to drive the main results in this paper, the following known lemma is needed.

Lemma 1. (see ^[9]) Let the function $p\in \mathcal{P}$ and let it have the form $(1.4)$. Then

and the bound is sharp.

2.   A set of main results

We begin by estimating the upper bound for the Taylor-Maclaurin coefficients of functions in the function class $\mathcal{H}_{q} (\lambda, \widehat{p})$.

Theorem 1. If the function $f\in \mathcal{H}_{q}(\lambda, \widehat{p})$ has the power series given by $(1.1),$ then

and

Proof. Since $f\in\mathcal{H}_{q}(\lambda, \widehat{p})$ and $f^{-1} = g$, by means of Definition 7 and by using the principle of subordination, there exit functions $s(z), \, r(z)\in \mathcal{B}$ such that

We define the following two functions:

and

Then it is clear that $p_{j}\in \mathcal{P}$ for $j = 1, 2$. Equivalently, the last relations in terms of $r(z)$ and $s(z)$ can be restated as follows:

and

Therefore, by means of (2.4), (2.5) and (2.3), if we take (1.14) into account, we have

and

Now, upon expanding the right-hand sides of both equations in (2.3), we find that

and

Substituting from (2.6), (2.7), (2.8) and (2.9) into (2.3) and then by equating the corresponding coefficients of $z$, $z^{2}$, $w$ and $w^{2}$, we get

and

From (2.10) and (2.12), it immediately follows that

and

Addition of (2.11) and (2.14) yields

Also, by using (2.14) in the last equation, we get

which, in view of Lemma 1, yields the required bound on $|a_{2}|$ as asserted in (2.1).

Further, in order to find the estimate on $|a_{3}|$, we subtract (2.14) from (2.11). Further computations by using (2.13) lead us to

Finally, by using (2.14) in conjunction with Lemma 1 on the coefficients of $r_{2}$ and $s_{2}$, we are led to the assertion given in (2.2). This completes our proof of Theorem 1.

In the next result, we solve the Fekete-Szegö problem for the function class $\mathcal{H}_{q}(\lambda, \widehat{p})$ by making use of the coefficients $a_2$, $a_3$ and a complex parameter $\nu$.

Theorem 2. Let the function $f$ belong to the class $\mathcal{H}_{q}(\lambda, \widehat{p})$ and let $\nu\in \mathbb{C}$. Then

where

Proof. On the account of (2.15) and (2.16), we have

which can be written in the following equivalent form:

Some suitable computations would yield

where $\Theta(\nu)$ is defined in (2.18). Since all $\widehat{p}_j\; \; (j = 1, 2)$ are real and $\widehat{p}_1 > 0$, we obtain

The proof of Theorem 2 is thus completed.

Remark 1. It follows from Theorem 2 when $\nu = 1$ that, if $f\in\mathcal{H}_{q}(\lambda, \widehat{p}),$ then

If we first set $\lambda = 1$ and then apply limit as $q\rightarrow1-$, then we have following consequence of Theorem 2.

Corollary 1. (see ^[57]) Let a function $f$ belong to the class given by

and $\nu\in \mathbb{C}$. Then

where

3.   Applications of the main results

For the class of $q$-starlike functions of order $\alpha$ with $0 < \alpha \leqq 1$, the function $\widehat{p}$ is given by

Then Definition 7 of the bi-univalent function class $\mathcal{H}_{q}(\lambda, \widehat{p})$ yields a presumably new class $\mathcal{H}_{q}^{1}(\lambda, \alpha)$, which is given below.

Definition 8. A function $f\in A$ is said to be in the class $\mathcal{H}_{q}^{1}(\lambda, \alpha)$ if it satisfies the following conditions:

and

where $g(w)-f^{-1}(w)$.

Hence, upon setting

in Definition 8, we are led to the following corollaries of Theorem 1 stated below.

Corollary 2. Let the function $f\in \mathcal{H}_{q}^{1}(\lambda, \alpha)$ have the form $(1.1)$. Then

and

Corollary 3. Let the function $f\in \mathcal{H}_{q}^{1}(\lambda, \alpha)$ have the form $(1.1)$. Then

In Corollary 2, we set $\lambda = 1$. Then we arrive at the following result.

Corollary 4. Let the function $f$ be in the class given by

and have the form $(1.1)$. Then

and

On the other hand, for $0 < \alpha \leqq 1,$ if we set

then Definition 7 of the bi-univalent function class $\mathcal{H}_{q}(\lambda, \widehat{p})$ gives a new class $\mathcal{H}_{q}^{3}(\lambda, \alpha)$, which is given below.

Definition 9. A function $f\in \mathcal{A}$ is said to belong to the class $\mathcal{H}_{q}^{3}(\lambda, \alpha)$ if the following inequalities hold true:

and

where the inverse function is given by $f^{-1}(w) = g(w)$.

Using the parameter setting given by

in Definition 9, it leads to the following consequences of Theorem 1.

Corollary 5. Let the function $f\in \mathcal{H}_{q}^{3}(\lambda, \alpha)$ be given by $(1.1)$. Then

and

Corollary 6. Let the function $f\in\mathcal{H}_{q}^{3}(\lambda, \alpha)$ be given by $(1.1)$. Then

Next, if we take

where

The function given in (3.1) is not univalent in $\mathbb{U}$. However, it is univalent in

It is noteworthy that

which ensures the division of the interval $[0, 1]$ by the above-mentioned number $|\mathcal{T}|$ such that it fulfills the golden section. Since the equation:

holds true for $\mathcal{T}$, in order to attain higher powers $\mathcal{T}^{n}$ as a linear function of the lower powers, this relation can be used. In fact, it can be decomposed all the way down to a linear combination of $\mathcal{T}$ and $1$. The resulting recurrence relations yield the Fibonacci numbers $\mathfrak{u}_{n}$:

For the function $\widehat{p}$ represented in (3.1), Definition 7 of the bi-univalent function class $\mathcal{H}_{q}(\lambda, \widehat{p})$ gives a new class $\mathcal{H}_{q}^{4}(\lambda, \widehat{p})$, which (by using the principle of subordination) gives the following definition.

Definition 10. A function $f\in \mathcal{A}$ is said to be in the class $\mathcal{H}_{q}^{4}(\lambda, \alpha)$ if it satisfies the following subordination conditions:

and

where $g(w) = f^{-1}(w)$.

Using similar arguments as in proof of Theorem 1, we can obtain the the upper bounds on the Taylor-Maclaurin coefficients $a_2$ and $a_3$ given in Corollary 7.

Corollary 7. Let the function $f\in \mathcal{H}_{q}^{4}(\lambda, \alpha)$ have the form $(1.1)$. Then

and

4.   Concluding remarks and observations

Here, in our present investigation, we have successfully examined the applications of a certain $q$-integral operator to define several new subclasses of analytic and bi-univalent functions in the open unit disk $\mathbb{U}$. For each of these newly-defined analytic and bi-univalent function classes, we have settled the problem of finding the upper bounds on the coefficients $\left \vert a_{2}\right \vert$ and $\left \vert a_{3}\right \vert$ in the Taylor-Maclaurin series expansion subject to a gap series condition. By means of corollaries of our main theorems, we have also highlighted some known consequences and some applications of our main results.

Studies of the coefficient problems (including the Fekete-Szegö problems and the Hankel determinant problems) continue to motivate researchers in Geometric Function Theory of Complex Analysis. With a view to providing incentive and motivation to the interested readers, we have chosen to include several recent works (see, for example, ^{[17,18,21,28,33,34,39]}), on various bi-univalent function classes as well as the ongoing usages of the $q$-calculus in the study of other analytic or meromorphic univalent and multivalent function classes.

Acknowledgements

The fourth author is supported by UKM Grant: GUP-2019-032.

Conflicts of interest

The authors declare that they have no competing interests.