On a generalized class of bi-univalent functions defined by subordination and \(q\)-derivative operator

1. Introduction

Let \(\mathcal{U}=\{z:z\in\mathbb{C}, |z|< 1\}\) be a unit disk and let \(\mathcal{A}\) denote the class of analytic functions of the form

normalized by the conditions \(f(0)=f'(0)-1=0\). Let \(\mathcal{S}\subset\mathcal{A}\) be the class of analytic and univalent functions in \(\mathcal{U}\).

Let \(\mathcal{W}\) denote the class of functions

\begin{equation*}\label{funSchwarz} w(z)=w_1z + w_2z^2 + w_3z^3+\cdots\quad (z\in\mathcal{U}), \end{equation*} such that \(w(0)=0\) and \(|w(z)|< 1\). The class \(\mathcal{W}\) is known as the class of Schwarz functions.

By [1], let \(j(z)\), \(J(z)\in\mathcal{A}\), then \(j(z)\prec J(z)\), \(z\in\mathcal{U}\), if \(\exists w(z)\) analytic in \(\mathcal{U}\), such that \(w(0)=0\), \(|w(z)|< 1\) and \(j(z)=J(w(z))\). If the function \(J(z)\) is univalent in \(\mathcal{U}\), then \(j(z)\prec J(z)\implies j(0) = J(0)\) and \(j(\mathcal{U})\subset J(\mathcal{U})\).

Let \(\mathcal{P}\) denote the class of functions

which are analytic in \(\mathcal{U}\) such that \(\mathcal{R}e(p(z))>0\) and \(p(0)=1\). It is known that functions in classes \(\mathcal{P}\) and \(\mathcal{W}\) are related such that In [2], Ma and Minda defined a function \(\phi\in\mathcal{P}\ (z\in\mathcal{U})\) such that \(\phi(0)=1\), \(\phi'(0)>0\) and \(\phi(\mathcal{U})\) is starlike with respect to 1 and symmetric with respect to the real axis. Such function \(\phi\) can be expressed as Fekete and Szegö [3] investigated the coefficient functional \begin{equation*}\label{funFSFunctional} g_\rho(f)=|a_3 - \rho a_2^2|, \end{equation*} which arose from the disproof of Littlewood-Parley conjecture (see [1]) that says modulus of coefficients of odd univalent functions are less than 1. This functional has been investigated by many researchers, see for instance [4,5].

Historically, Lewin [6] introduced a subclass of \(\mathcal{A}\) called the class of bi-univalent functions and established that \(|a_2|\leq 1.51\) for all bi-univalent functions. Also, the Koebe 1/4 theorem (see [1]) states that the range of every function \(f\in\mathcal{S}\) contains the disk \(D=\{\omega:|\omega|< 0.25\}\subseteq f(\mathcal{U})\). This implies that \(\forall f\in\mathcal{S}\) has an inverse function \(f^{-1}\) such that

\begin{align*} f^{-1}(f(z)) &= z \quad (z\in\mathcal{U}), \end{align*} and \begin{align*} f(f^{-1}(\omega)) &= \omega\quad (\omega:|\omega| < r_0(f);\; r_0(f)\geq 0.25), \end{align*} where \(f^{-1}(\omega)\) is expressed as Thus, a function \(f\in\mathcal{A}\) is said to be bi-univalent in \(\mathcal{U}\) if both \(f(z)\) and \(F(\omega)\) are univalent in \(\mathcal{U}\). Let \(\mathcal{B}\) denote the class of analytic and bi-univalent functions in \(\mathcal{U}\).

Some functions \(f\in\mathcal{B}\) includes \(f(z)=z\), \(f(z)=z/(1-z)\), \(f(z)=-\log(1-z)\) and \(f(z)=\frac{1}{2}\log[(1+z)/(1-z)]\). Observe that some familiar functions \(f\in\mathcal{S}\) such as the Koebe function \(K(z)=z/(1-z)^2\), its rotation function \(K_\sigma(z)=z/(1-e^{i\sigma}z)^2\), \(f(z)=z - z^2/2\) and \(f(z)=z/(1-z^2)\) are nonmembers of \(\mathcal{B}\). See [4,5,7,8,9,10,11] for more details.

Jackson [12] (see also [8,13,14]) introduced the concept of \(q\)-derivative operator. For functions \(f\in\mathcal{A}\), the \(q\)-derivative of \(f\) can be defined by

where \(\mathcal{D}_q f(0)=f'(0)\) and \(\mathcal{D}_qf(z)z=\mathcal{D}_q(\mathcal{D}_qf(z))\). From (1) and (6) we get where \([n]_q=\frac{1-q^n}{1-q}\), \([n-1]_q=\frac{1-q^{n-1}}{1-q}\), \(\lim\limits_{q\uparrow 1}[n]_q = n\) and \(\lim\limits_{q\uparrow 1}[n-1]_q = n-1\).

For instance, if \(\alpha\) is a constant, then for the function \(f(z)=\alpha z^n\),

\[\mathcal{D}_qf(z)=\mathcal{D}_q(\alpha z^n)=\frac{1-q^n}{1-q}\alpha z^{n-1}=[n]_q\alpha z^{n-1}\,,\] and note that \[\lim\limits_{q\uparrow 1}\mathcal{D}_qf(z)=\lim\limits_{q\uparrow 1}[n]_q\alpha z^{n-1}=n\alpha z^{n-1}=:f'(z)\,,\] where \(f'(z)\) is the classical derivative.

In this study, the \(q\)-derivative operator and the subordination principle are used to define and generalize a subclass of bi-univalent functions. Afterwards, some coefficient bounds and some Fekete-Szegö estimates were investigated. Some of our results generalised that of Srivastava and Bansal in [10] and some new results are added.

Definition 1. Let \(0< q< 1\), \(\tau \in\mathbb{C}\setminus\{0\}\), \(0\leq \lambda\leq 1\) and \(\phi\) is defined in (4). A function \(f\in\mathcal{B}\) is said to be in the class \(\mathcal{B}_q(\tau,\lambda,\phi)\) if the subordination conditions

and where \(F(\omega)=f^{-1}(\omega)\) are satisfied.

Remark 1. Let \(q\uparrow 1\) in (8) and (9), then \(\mathcal{B}_q(\tau,\lambda,\phi)\) becomes the class \(\mathcal{B}(\tau, \lambda, \phi)\) investigated by Srivastava and Bansal [10].

2. Preliminary Lemmas

To establish our results, we shall need the following lemmas. Let \(p(z)\) be as defined in (2).

Lemma 2 ([1]). If \(p(z)\in\mathcal{P}\), then \(|p_n|\leq 2\ (n\in\mathbb{N}). \) The result is sharp for the well-known Möbius function.

Lemma 3 ([15,16]). If \(p(z)\in\mathcal{P}\), then \(2p_2 = p^2_1 + (4-p^2_1)x \) for some \(x\) and \(|x|\leq 1\).

3. Main Results

Unless otherwise mentioned in what follows, we assume throughout this work that \(0< q< 1\), \(\tau\in\mathbb{C}\setminus\{0\}\), \(0\leq \lambda \leq 1\), \(\phi\) is as defined in (4) and \(f\in\mathcal{B}\), hence our results are as follows:

Theorem 4. Let \(f\in\mathcal{B}_q(\tau,\lambda,\phi)\), then

where \(\beta_1>0\) and \(\beta_n\ (n\in\mathbb{N})\) are coefficients of \(\phi(z)\) in (4).

Proof. Let \(f(z)\in\mathcal{B}\) and \(F(\omega)=f^{-1}(\omega)\), then there exists the analytic functions \(u(z), v(\omega)\in\mathcal{W}\), \(z,\omega\in\mathcal{U}\) such that \(u(0)=0=v(0)\), \(|u(z)|< 1\), \(|v(\omega)|< 1\) so that they satisfy the subordination conditions:

and By substituting (7) into LHS of (12) we respectively get and following the same process for \(F(\omega)\) in (5) gives Now to expand and in series form, let \(\delta_1(z)=1+b_1z+b_2z^2+\dots\), \(\delta_2(\omega)=1+c_1\omega+c_2\omega^2+\dots\in\mathcal{P}\), then by (3), and following the same process Substituting (18) into (16) as expressed by (4) we get and substituting (19) into (17) as expressed by (4) we get Now comparing the coefficients in (14) and (20) we get and comparing the coefficients in (15) and (21) gives Now adding (22) and (24) and simplifying we get Also from (22) and (24) we get and adding (23) and (25) and using (26) we get From (27) and using (26) we get So that by substituting for \(b_1^2\) in (28) we get and applying Lemma 2 gives (10).

Again by subtracting (23) from (25), using (26) and simplifying we get

Thus, from (27), using (26) and simplifying we get and applying Lemma 2 gives (11).

Let \(q\uparrow 1\), then Theorem 4 becomes

Corollary 5. Let \(f(z)\in\mathcal{B}_q(\tau, \lambda, \phi)\), then as \(q\uparrow 1\), \begin{align*} |a_2| &\leq \frac{|\tau|\beta_1^{3/2}} {\sqrt{|\tau[3]_q \beta_1^2 +[2]_q^2 (\beta_1-\beta_2)|}}\,,\\ |a_3| &\leq \frac{|\tau|^2 \beta_1^2}{[2]^2_q} + \frac{ |\tau| \beta_1}{[3]_q}\,. \end{align*} which is the result of Srivastava and Bansal [10].

Theorem 6.( Fekete-Szegö Estimate, \(\varrho\in\mathbb{R}\)). If \(f\in\mathcal{B}_q(\tau,\lambda,\phi)\) and \(\varrho\in\mathbb{R}\), then \[ \mbox{\(|a_3 - \varrho a_2^2|\)}\leq \left\{ \begin{array}{rl} \frac{|\tau|\beta_1}{[3]_q(1+[2]_q\lambda)} & \mbox{for \(0\leq |h(\varrho)|\leq\frac{1}{[3]_q(1+[2]_q\lambda)}\);}\\ |\tau|\beta_1|h(\rho)| & \mbox{for \(|h(\varrho)|\geq \frac{1}{[3]_q(1+[2]_q\lambda)}\),} \end{array}\right. \] where

Proof. From (30) and (31), \begin{align*} |a_3 - \varrho a_2^2| &= \left|\frac{\tau \beta_1 (b_2 - c_2)}{4[3]_q(1+[2]_q\lambda)} + (1 - \varrho)a^2_2\right|\\ &= \left|\frac{\tau \beta_1}{4}\left\{\frac{(b_2 - c_2)}{[3]_q(1+[2]_q\lambda)} + (b_2+c_2)h(\varrho)\right\}\right|\,, \end{align*} where \(h(\varrho) \) is given in (33), so that by applying triangle inequality, (4), Lemma 2 and simplifying complete the proof.

Theorem 7( Fekete-Szegö Estimate, \(\rho\in\mathbb{C}\)). If \(f\in\mathcal{B}_q(\tau, \lambda, \phi)\) and \(\rho\in\mathbb{C}\), then

where \begin{equation*} \xi = \frac{[2]^2_q(1+[1]_q\lambda)^2}{|\tau|\beta_1[3]_q(1+[2]_q\lambda)}\,. \end{equation*}

Proof. From (27) and (31) and using (26),

From Lemma 3 and (26) for some \(x, y,|x|\leq 1,|y|\leq 1\) and \(|b_1|\in[0,2]\). Thus using (36) in (35) simplifies to \begin{equation*} a_3 - \rho a^2_2 = (1-\rho)\frac{\beta_1^2 b_1^2 \tau^2}{4[2]^2_q(1+[1]_q\lambda)^2} + \frac{\beta_1\tau (4-b^2_1)}{8[3]_q(1+[2]_q\lambda)}(x-y). \end{equation*} For \(\delta(z)=1+b_1z + b_2z^2 + \cdots\in\mathcal{P}\), \(|b_1| \leq 2\) by Lemma 2. Letting \(b= b_1\), we may assume without any restriction that \(b\in [0,2]\). Now using triangle inequality, letting \(X=|x|\leq 1\) and \(Y = |y|\leq 1\), then we get \begin{align*} |a_3 - \rho a^2_2|&\leq |1-\rho|\frac{\beta_1^2 b^2|\tau|^2}{4[2]^2_q(1+[1]_q\lambda)^2} + \frac{\beta_1|\tau|(4-b^2)}{8[3]_q(1+[2]_q\lambda)}(X + Y) = H(X,Y). \end{align*} For \(X,Y\in [0,1]\); \begin{equation*} \max\{H(X,Y)\} = H(1,1) = \frac{\beta_1^2 |\tau|^2}{4[2]^2_q(1+[1]_q\lambda)^2}\left\{ |1-\rho| - \frac{[2]_q^2(1+[1]_q\lambda)^2}{\beta_1|\tau|[3]_q(1+[2]_q\lambda)}\right\}b^2 + \frac{\beta_1|\tau|}{[3]_q(1+[2]_q\lambda)} = G(b)\label{G(t)}. \end{equation*} For \(b\in[0,2]\); which implies that there is a critical point at \(G'(b)=0\), that is at \(b=0\). Hence for \[G'(b)< 0; \; |1-\rho|\in \left[0,\frac{[2]_q^2(1+[1]_q\lambda)^2}{\beta_1|\tau|[3]_q(1+[2]_q\lambda)}\right)\,,\] thus, \(G(b)\) is strictly a decreasing function of \(|1-\rho|\), therefore from (3), \[\max\{G(b):b\in[0,2]\}=G(0)=\frac{\beta_1|\tau|}{[3]_q(1+[2]_q\lambda)}.\] Also for \[G'(b)\geq 0; \; |1-\rho|\in \left[\frac{[2]_q^2(1+[1]_q\lambda)^2}{\beta_1|\tau|[3]_q(1+[2]_q\lambda)}, 0\right)\,,\] thus, \(G(b)\) is an increasing function of \(|1-\rho|\), therefore from (3), \[\max\{G(b):b\in[0,2]\}=G(2)=\frac{|1-\rho|\beta_1^2|\tau|^2}{[2]_q^2(1+[1]_q\lambda)^2}.\] So that by putting the results together leads to (34).

4. Conclusion

In this work, we were able to establish the first two coefficient bounds and also solve the Fekete-Szegö problem for the class \(\mathcal{B}_q(\tau,\lambda,\phi)\) of analytic and bi-univalent functions in \(\mathcal{U}\). The results in the first theorem generalized that of Srivastava and Bansal [10].

Acknowledgments

The authors thank the referees for their valuable suggestions to improve the paper.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.