Some properties for certain class of bi-univalent functions defined by $q$-Cătaş operator with bounded boundary rotation

1.   Introduction

Let $\mathcal{A}$ denote the class of analytic functions of the form:

Let $S$ be the subclass of $\mathcal{A}$ consisting of univalent functions in $\mathbb{U}$ and let $K,$ $S_{\lambda }, \ S^{\ast }$ and $C$ be the usual subclasses of $S$ consisting of functions which are, respectively, close-to-convex, $\lambda$-spiral-like, starlike ($w.r.t.$ the origin) and convex in $\mathbb{U}$.

Definition 1. (^[4], with ${p = 1}$.) Let $\mathcal{P}_{m}^{\mu }(\rho)\, 0\leq \rho < 1, \ m\geq 2$ and $\left\vert \mu \right\vert < \frac{\pi }{2},$ be the class of analytic functions $p(z) = 1+\sum\limits_{k = 1}^{\infty }c_{k}z^{k}$ $\mathbb{\ }$and satisfy the conditions:

and

We note that:

(i) $\mathcal{P}_{m}^{\mu }(0) = \mathcal{P}_{m}^{\mu }\ (\ m\geq 2$ and $\left\vert \mu \right\vert < \frac{\pi }{2})$, the class of functions introduced by Robertson ^[34].

(ii) $\mathcal{P}_{m}^{0}(\rho) = \mathcal{P}_{m}(\rho)\ (0\leq \rho < 1, \ m\geq 2),$ the class of functions introduced by Padmanabhan and Parvatham ^[31].

(iii) $\mathcal{P}_{m}^{0}(0) = \mathcal{P}_{m}(m\geq 2),$ the class of functions having their real parts bounded in the mean on $\mathbb{U}$, introduced by Robertson ^[34] and further studied by Pinchuk ^[32]. (iv) $\mathcal{P}_{2}^{0}(\rho) = \mathcal{P}\left(\rho \right) \ (0\leq \rho < 1),$ the class of functions with positive real part of order $\rho$, $0\leq \rho < 1.$ (v) $\mathcal{P}_{2}^{0}(0) = \mathcal{P}$, the class of functions having positive real part for $z\in \mathbb{U}$.

Definition 2. (^[12,36,51]). The class of Bazilevi č functions in the open unit disc $\mathbb{U}$ was introduced by Bazilevi č ^[12], he defined Bazilevič functions by the following relation

where $g(z)\in S^{\ast }, p(z)\in \mathcal{P}_{m}^{\mu }(\rho)$ and $\beta, \tau > 0.$ Singh ^[36] studied the class $B_{1}(\beta)$ of Bazilevič functions by putting $g(z) = z$ and $\tau = 0$ in above equation. If $f(z)\in B_{1}(\beta)$ then $f(z)$ is a Bazilevič function of type $\beta.$

By the Koebe one-quarter theorem ^[15], we know that the image of $\mathbb{U\ }$under every univalent function $f\in \mathcal{A}$ contains the disc with the center in the origin and radius $1/4$. Therefore, every univalent function $f$ has an inverse $f^{-1}$ satisfies

It is easy to see that the inverse function has the form

A function $f\in \mathcal{A}$ is said to be bi-univalent in $\mathbb{U}$ if both $f$ and its inverse map $g = f^{-1}$are univalent in $\mathbb{U}$. Let $\sum$ denote the class of bi-univalent functions in $\mathbb{U}$ in the form (1.1). For interesting examples about the class $\sum$ (see ^{[8,11,16,18,25,46]}).

The pioneering work of Srivastava et al. ^[45] actually revived the study of bi-univalent functions in recent years. In a substantially large number of work subsequent to the work of Srivastava et al. ^[45], several distinct subclasses of the bi-univalent function class were presented and examined similarly by many authors. For example, the function classes $H_{\sum }(\tau, \mu, \lambda, \delta; \alpha)$ and $H_{\sum }(\tau, \mu, \lambda, \gamma; \beta)$ were defined and the estimates on the Taylor-Maclaurin coefficients $\left\vert a_{2}\right\vert$ and $\left\vert a_{3}\right\vert$ were obtained by Srivastava et al. ^[43]. The upper bounds for the second Hankel determinant for certain subclasses of analytic and bi-univalent functions were obtained by Caglar et al. ^[13]. Several new subclasses of the class of $m$-fold symmetric bi-univalent functions were introduced and the initial estimates of the Taylor-Maclaurin series as well as some Fekete-Szeg ö functional problems for each of their defined function classes were obtained by Tang et al. ^[50] and Srivastava et al. ^[42]. Several other well-known mathematicians gave their findings on this subject (e.g. ^[40,41,44]).

As we know that the fractional $q$-calculus and the fractional of $q$ -derivative operators in Geometric Function Theory were investigated by sturdy of researchers (see ^{[2,3,7,9,10,17,19,20,21,22,23,24,25,37,38,39,47,48,49]}). The $q$-calculus is an important tool which is used to study various applications in mathematics, physics and chemistry and some basic sciences subjects. In the study of Geometric Function Theory, the versatile applications of the $q$-derivative operator $D_{q}$ make it remarkably significant. Inspired by the above-mentioned works, in recent years, important researches have played a significant part in the development of Geometric Function Theory of complex analysis. Several convolutional and fractional calculus $q$-operators were defined by many researchers, which were surveyed in the above-cited work by Srivastava ^[37]. For a function $f(z)\in \mathcal{A}$ given by (1.1) and $0 < q < 1.$ Jackson's $q$-derivative (or $q$-difference) $D_{q}$ of a function defined on a subset of the complex space $\mathbb{C}$ is defined as follows:

where $D_{q}f(0) = f^{\prime }(0),$ $D_{q}^{2}f(z) = D_{q}\left(D_{q}f(z)\right)$ and $\left[k\right] _{q} = \frac{1-q^{k}}{1-q},$ as $q\rightarrow 1\bf{-}$, then $\left[k\right] _{q}\rightarrow k$, hence we have

For a function $f(z)\in \mathcal{A},$ Aouf and Madian ^[10] defined the $q$ -analogue Cătaş operator as follows (see also ^[7,9] with $p = 1$]):

where

And introduced recurrent relation as follows:

We note that $I_{q}^{n}(\lambda, l)f(z)$ generalized many operators such as C ătaş operator, Multiplier operator and Sălăgean operator etc., for more details see ^{[1,5,6,7,14,28,35]}.

Definition 3. Let $f\in \sum, \ \alpha \in \mathbb{C} ^{\ast } = \mathbb{C} \backslash \{0\}, \ l, \lambda \ \beta \geq 0, \ 0\leq \rho < 1, \ m\geq 2, \ \left\vert \mu \right\vert < \frac{\pi }{2}, n\in \mathbb{N} _{0}$ and $0 < q < 1,$ then $I_{q}^{n}(\lambda, l)f(z)\in \sum$ is said to be in $\mathcal{H}_{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}\ f(z),$ if it satisfies the following conditions:

and

By choosing different values for $\alpha, \mu, \rho, m, \beta, n, q, \lambda$ and $l,$ in the a above definition we have:

(1) $\lim_{q\rightarrow 1-}\mathcal{H}_{1, 0, \rho, m, \beta }^{0, q, \lambda, l}\ f(z) = K_{\sum }(\rho, m, \beta)f(z)\ (f\in \sum, \ \beta \geq 0, \ 0\leq \rho < 1, \ m\geq 2)$ (see ^[29], with $\gamma = 1$ and $\delta = 0$).

(2) $\lim_{q\rightarrow 1-}\mathcal{H}_{\alpha, 0, \rho, 2, \beta }^{0, q, \lambda, l}\ f(z) = \mathcal{F}_{\rho, \beta }f(z)\ (f\in \sum, \ \beta \geq 0, \ 0\leq \rho < 1)$ (see ^[33]).

(3) $\mathcal{H}_{1, 0, \rho, 2, 1}^{0, q, \lambda, l}\ f(z) = \mathcal{F} _{\rho, q}(f)(z)( f\in \sum, \ 0\leq \rho < 1)$ (see ^[46]).

(4) $\lim_{q\rightarrow 1-}\mathcal{H}_{\alpha, 0, \rho, 2, \beta }^{0, q, \lambda, l}\ f(z) = \mathcal{MP}_{\sum }^{\beta, \alpha }(0, \rho)f(z)\ (f\in \sum, \ \beta, \alpha \geq 0, \ 0\leq \rho < 1)$ (see ^[30], with $\beta = 0$]).

(5) $\lim_{q\rightarrow 1-}\mathcal{H}_{\alpha, \mu, \rho, 2, \beta }^{0, q, \lambda, l}\ f(z) = \mathcal{L}_{\sum }(\beta, \rho)f(z)\ (\ f\in \sum, \ \beta \geq 0, \ 0\leq \rho < 1)$ (see ^[27]).

(6) $\lim_{q\rightarrow 1-}\mathcal{H}_{\alpha, \mu, \rho, m, \beta }^{0, q, \lambda, l}$ $f(z) = \mathcal{K}(\alpha, \mu, \rho, m, \beta)f(z)\ (\ f\in \sum, \ \alpha \in \mathbb{C} ^{\ast },$ $\beta \geq 0, \ 0\leq \rho < 1, \ m\geq 2, \ \left\vert \mu \right\vert < \frac{\pi }{2}, \ 0\leq \rho < 1)$ (see ^[11], with $h = \frac{z}{1-z }$ and ^[8], with $h = \frac{z}{1-z}, b = 1$ and $\delta = 0$).

As well as, we obtain new subclasses as follows:

(a)

(b)

(c)

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

Lemma 1. (^[4] with $p$ = 1.) If $p(z) = 1+\sum\limits_{n = 1}^{ \infty }c_{n}z^{n}\in \mathcal{P}_{m}^{\mu }(\rho),$ then

The result is sharp. Equality is attained for the odd coefficients and even coefficients, respectively, for the functions

We note that for $\mu = \rho = 0$ in Lemma 1, we obtain the result obtained by Goswami et al. [16, Lemma 1] for the class $\mathcal{P}_{m}.$

The object of this paper is to introduce a new subclass of the class $\sum$ by using the definition of Bazilevič functions, bi-univalent functions with bounded boundary rotation and $q$-analogue Cătaş operator. As well as I calculate the coefficient estimates for functions in the subclass $\mathcal{H}_{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}f(z)$. Also I get coefficients bounds for the subclasses of our main class.

2.   Coefficient estimates for functions in the subclass $\mathcal{H}_{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}f(z)$

Theorem 1. Let $f\in \sum, \ \alpha \in \mathbb{C} ^{\ast }\backslash \{-1, \frac{-1}{2}\}, \ l, \lambda, \beta \geq 0, \ 0\leq \rho < 1, \ m\geq 2, \ \left\vert \mu \right\vert < \frac{\pi }{2}, n\in \mathbb{N} _{0}$ and $0 < q < 1,$ then $I_{q}^{n}(\lambda, l)f(z)\in \mathcal{H}_{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}f(z)$ if satisfies

and

Proof. Let $I_{q}^{n}(\lambda, l)f(z)\in \mathcal{H} _{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}f(z)$, then from Definition 1, we have

and

where $p$ and $q$ have Taylor expansions as follows:

By comparing coefficients in (2.3) with (2.5) and coefficients in (2.4) with (2.6), we obtain

and

Since $p, q\in \mathcal{P}_{m}^{\mu }(\rho)$ and by applying Lemma 1, we have

and

From (2.8), (2.10) and using inequalities (2.11) and (2.12), we obtain

Also, from (2.7) and (2.11), we obtain

By subtracting (2.10) from (2.8), we have

Also, we have

After using (2.15), (2.16), (2.11) and (2.12) and some easily calculations, we obtain

which completes the proof of Theorem 1. The result is sharp in view of the fact that assertion (1.11) of Lemma 1 is sharp. By following the observation of Thomas ^[52], the equality is attained for the odd coefficients and even coefficients, respectively, for the functions

By using Ma and Minda Lemma (see ^[26]), we obtain the following corollary.

Corollary 1. Let $I_{q}^{n}(\lambda, l)f(z)\in H_{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}f(z), \ \alpha \in \mathbb{C} ^{\ast }\backslash \{-1, \frac{-1}{2}\}, \ l, \lambda, \beta \geq 0, n\in \mathbb{N} _{0}$ and $0 < q < 1,$ then (i) For any real number $\zeta,$ we have

(ii) For any complex number $\zeta$, we have

Putting $\rho = 0,$ in Theorem 1, we obtain the following corollary.

Corollary 2. Let $f\in \sum, \ \alpha \in \mathbb{C} ^{\ast }\backslash \{-1, \frac{-1}{2}\}, \ l, \lambda, \beta \geq 0, \ m\geq 2, \ \left\vert \mu \right\vert < \frac{\pi }{2}, n\in \mathbb{N} _{0}$ and $0 < q < 1,$ then $I_{q}^{n}(\lambda, l)f(z)\in \mathcal{P}_{\alpha, \mu, m, \beta }^{n, q, \lambda, l}f(z)$ if satisfies

and

Putting $\mu = 0$ and $m = 2$ in Theorem 1, we obtain the following corollary.

Corollary 3. Let $f\in \sum, \ \alpha \in \mathbb{C} ^{\ast }\backslash \{-1, \frac{-1}{2}\}, \ l, \lambda, \beta \geq 0, \ 0\leq \rho < 1, n\in \mathbb{N} _{0}$ and $0 < q < 1,$ then $I_{q}^{n}(\lambda, l)f(z)\in \mathcal{D}_{\alpha, \rho, \beta }^{n, q, \lambda, l}f(z)$ if satisfies

and

Putting $\alpha = 1$ in Theorem 1, we obtain the following corollary.

Corollary 4.Let $f\in \sum, \ l, \lambda, \beta \geq 0, \ 0\leq \rho < 1, \ m\geq 2, \ \left\vert \mu \right\vert < \frac{\pi }{2}, n\in \mathbb{N} _{0}$ and $0 < q < 1,$ then $I_{q}^{n}(\lambda, l)f(z)\in \mathcal{U}_{\mu, \rho, m, \beta }^{n, q, \lambda, l}f(z)$ if satisfies

and

Remarks. (i) Putting, $\beta = \alpha = 1, m = 2$ and$\ n = \mu = 0$ in Theorem 1, we obtain the result obtained by Srivastava et al. [46, Theorem 2].

(ii) Letting $q\rightarrow 1-$ and $n = 0$ in Theorem 1, we obtain the result obtained by Aouf and Madian [11, with $h = \frac{z}{1-z},$ Theorem 1] and [8, with $h = \frac{z}{1-z}, b = 1$ and $\delta = 0,$ Theorem 1]).

3.   Conclusions

Throughout the paper, I used the definition of Bazilevič function, bi-univalent functions with bounded boundary rotation and the definition of $q$-analogue Cătaş operator to introduce the new subclass $\mathcal{H} _{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}f(z)$. I estimated the coefficients bounds for the functions belong to the subclass $\mathcal{H}_{\alpha, \mu, \rho, m, \beta }^{n, q, \lambda, l}f(z)$. In addition, through this paper we presented coefficients bounds for the functions belong to the subclasses of our main class.

Srivastava [37, p. 340] discussed the connection between the classical $q$ -analysis, which we used in this article, and its so-called trivial and inconsequential $(p, q)$-variation involving an obviously superfluous parameter $p$. Specifically, the results in this article for the $q$ -analogues $(0 < q < 1)$, can easily translated into the corresponding $(p, q)$ -variants $(0 < q < p\leq 1)$ by following the observation by Srivastava [37, p. 340] who applied some obvious parametric and argument variations, the additional parameter $p$ being redundant. As clearly and significantly pointed out by Srivastava et al. ^[46,48], some group of authors have made use of the so-called trivial and inconsequential $(p, q)$-variation by introducing a seemingly redundant parameter $p$ in the already known results dealing with the classical $q$-analysis. For further details, see the survey-cum-expository review article by Srivastava [37, p. 340].

Availability of data and material

During the current study the data sets are derived arithmetically.

Acknowledgements

I would like to thank the referees for their valuable comments and helpful suggestions, which have led to an improvement of the presentation of this paper. Also, I would like to express my sincere thanks to the editor for handling the article.

Conflict of interest

The authors don't have competing for any interests.