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
\begin{equation}\label{funAnalytic}
f(z)=z+\sum_{n=2}^{\infty}a_nz^n \quad (z\in\mathcal{U}),
\end{equation}
(1)
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
\begin{equation}\label{funCaratheodory}
p(z) = 1 + p_1z + p_2z^2 + \cdots \quad (z\in\mathcal{U}),
\end{equation}
(2)
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
\begin{equation}\label{funPW}
p(z)=\frac{1+w(z)}{1-w(z)} \Longleftrightarrow w(z)=\frac{p(z)-1}{p(z)+1}.
\end{equation}
(3)
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
\begin{equation}\label{funMaMinda}
\phi(z)=1+\beta_1z + \beta_2z^2+\cdots \quad (z\in\mathcal{U},\ \beta_1>0).
\end{equation}
(4)
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
\begin{equation}\label{funF(w)}
F(\omega)=f^{-1}(\omega) = \omega – a_2\omega^2 + (2a_2^2 – a_3)\omega^3 – (5a^3_2
– 5a_2a_3 + a_4)\omega^4 +\cdots.
\end{equation}
(5)
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
\begin{equation}\label{qDerivative}
\mathcal{D}_qf(z)=\frac{f(z)-f(qz)}{(1-q)z} \qquad (z\neq 0, \ 0< q< 1),
\end{equation}
(6)
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
\begin{equation}\label{funOperator}
\left.
\begin{array}{l}
\mathcal{D}_qf(z)=1+\sum\limits_{n=2}^{\infty}[n]_q a_n z^{n-1}\\
\mathcal{D}_qf(z)z=\sum\limits_{n=2}^{\infty}[n]_q[n-1]_q a_n z^{n-2}
\end{array}
\right\}
\end{equation}
(7)
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
\begin{equation}\label{myclassz}
1 + \frac{1}{\tau}[\mathcal{D}_qf(z) + \lambda z\mathcal{D}_qf(z)z – 1]\prec \phi(z)\qquad (z\in\mathcal{U}),
\end{equation}
(8)
and
\begin{equation}\label{myclassw}
1 + \frac{1}{\tau}[\mathcal{D}_q F(\omega) + \lambda \omega\mathcal{D}_q^2F(\omega) – 1]\prec\phi(\omega) \quad (\omega\in \mathcal{U}),
\end{equation}
(9)
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
\begin{align}
|a_2| &\leq \frac{\beta_1^{3/2}|\tau|} {\sqrt{\left|\beta_1^2\tau [3]_q(1+[2]_q\lambda)+[2]_q^2(1+ [1]_q\lambda)^2(\beta_1-\beta_2)\right|}}\,,\label{Resulta2}\\
\end{align}
(10)
\begin{align}
|a_3| &\leq \frac{ \beta_1^2|\tau|^2}{[2]^2_q(1+[1]_q\lambda)^2} + \frac{ \beta_1|\tau|}{[3]_q(1+[2]_q\lambda)}\label{Resulta3}\,,
\end{align}
(11)
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:
\begin{equation}\label{myclassz2}
1 + \frac{1}{\tau}[\mathcal{D}_q f(z) + \lambda z\mathcal{D}_qf(z)z – 1] = \phi (u(z)) \quad (z\in\mathcal{U})\,,
\end{equation}
(12)
and
\begin{equation}\label{myclassw2}
1 + \frac{1}{\tau}[\mathcal{D}_q F(\omega) + \lambda \omega\mathcal{D}_q^2 F(\omega) – 1] = \phi (v(\omega)) \quad (\omega\in \mathcal{U}).
\end{equation}
(13)
By substituting (7) into LHS of (12) we respectively get
\begin{equation}\label{Seriesf}
1 + \frac{1}{\tau}[\mathcal{D}_qf(z) + \lambda z\mathcal{D}_qf(z)z – 1]= 1 + \frac{[2]_q(1+[1]_q\lambda)a_2}{\tau}z + \frac{[3]_q(1+[2]_q\lambda)a_3}{\tau}z^2+\cdots\,,
\end{equation}
(14)
and following the same process for \(F(\omega)\) in (5) gives
\begin{equation}\label{Seriesw}
1 + \frac{1}{\tau}[\mathcal{D}_q F(\omega) + \lambda \omega\mathcal{D}_q^2 F(\omega) – 1]
= 1 – \frac{[2]_q(1+[1]_q\lambda)a_2}{\tau}\omega + \frac{[3]_q(1+[2]_q\lambda)(2a_2^2 – a_3)}{\tau}\omega^2 + \cdots.
\end{equation}
(15)
Now to expand
\begin{equation}\label{Step3}
\phi(u(z)),
\end{equation}
(16)
and
\begin{equation}\label{Step4}
\phi(v(\omega)),
\end{equation}
(17)
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),
\begin{equation}\label{Eqn:p1}
\delta_1(z)=\frac{1+u(z)}{1-u(z)}\Longrightarrow u(z)=\frac{\delta_1(z)-1}{\delta_1(z)+1}=\frac{1}{2}\left[b_1z + \left(b_2-\frac{b^2_1}{2}\right)z^2+\left(\frac{b_1^3}{2^2}-b_1b_2 + b_3\right)z^3+\cdots\right]\,,
\end{equation}
(18)
and following the same process
\begin{equation}\label{Seriesv}
\delta_2(\omega)=\frac{1+v(\omega)}{1-v(\omega)}\Longrightarrow v(\omega)=\frac{\delta_2(\omega)-1}{\delta_2(\omega)+1}=\frac{1}{2}\left[c_1\omega + \left(c_2-\frac{c^2_1}{2}\right)\omega^2+\left(\frac{c_1^3}{2^2}-c_1c_2+c_3\right)\omega^3+\cdots\right].
\end{equation}
(19)
Substituting (18) into (16) as expressed by (4) we get
\begin{align}\label{Seriesphiu}
\phi(u(z))=1& + \frac{1}{2}\beta_1b_1z + \frac{1}{2}\left[\beta_1\left(b_2-\frac{b_1^2}{2}\right)+\frac{1}{2}\beta_2b_1^2\right]z^2\notag
\\&+\frac{1}{2}\left[\beta_1\left(\frac{b_1^3}{2^2} – b_1b_2 +b_3\right)+\beta_2b_1\left(b_2-\frac{b_1^2}{2}\right)+\frac{1}{4}\beta_3b_1^3\right]z^3+\cdots\,,
\end{align}
(20)
and substituting (19) into (17) as expressed by (4) we get
\begin{align}\label{Seriesphiv}
\phi(v(\omega))=
1 &+ \frac{1}{2}\beta_1c_1\omega + \frac{1}{2}\left[\beta_1\left(c_2-\frac{c_1^2}{2}\right)+\frac{1}{2}\beta_2c_1^2\right]\omega^2
\notag\\&+\frac{1}{2}\left[\beta_1\left(\frac{c_1^3}{2^2} – c_1c_2 + c_3\right)+\beta_2c_1\left(c_2-\frac{c_1^2}{2}\right)+\frac{1}{4}\beta_3c_1^3\right]\omega^3+\cdots.
\end{align}
(21)
Now comparing the coefficients in (14) and (20) we get
\begin{equation}\label{a1f}
\frac{[2]_q(1+[1]_q\lambda)a_2}{\tau}=\frac{\beta_1b_1}{2}\,,
\end{equation}
(22)
\begin{equation}\label{a2f}
\frac{[3]_q(1+[2]_q\lambda)a_3}{\tau}=\frac{1}{2}\left[\beta_1\left(b_2-\frac{b_1^2}{2}\right)+\frac{1}{2}\beta_2b_1^2\right]\,,
\end{equation}
(23)
and comparing the coefficients in (15) and (21) gives
\begin{equation}\label{a1g}
-\frac{[2]_q(1+\lambda[1]_q)a_2}{\tau}=\frac{\beta_1c_1}{2}\,,
\end{equation}
(24)
\begin{equation}\label{a2g}
\frac{[3]_q(1+[2]_q\lambda)(2a^2_2 – a_3)}{\tau}=\frac{1}{2}\left[\beta_1\left(c_2-\frac{c_1^2}{2}\right)+\frac{1}{2}\beta_2c_1^2\right].
\end{equation}
(25)
Now adding (22) and (24) and simplifying we get
\begin{equation}\label{Eqn:a1}
b_1=-c_1 \text{    and    } b^2_1 = c^2_1.
\end{equation}
(26)
Also from (22) and (24) we get
\begin{equation}\label{a^2_2}
8[2]^2_q(1+[1]_q\lambda)^2a^2_2 = \tau^2 \beta^2_1(b^2_1 + c^2_1)\,,
\end{equation}
(27)
and adding (23) and (25) and using (26) we get
\begin{equation}\label{Eqn:b^2_1}
4[3]_q(1+[2]_q\lambda)a_2^2 =\tau \beta_1(b_2+c_2)-\tau b_1^2(\beta_1-\beta_2).
\end{equation}
(28)
From (27) and using (26) we get
\begin{equation}
b_1^2=\frac{4[2]^2_q(1+ [1]_q\lambda)^2a^2_2}{\tau^2 \beta^2_1}.
\end{equation}
(29)
So that by substituting for \(b_1^2\) in (28) we get
\begin{equation}\label{Eqn:a22}
a_2^2 = \frac{\tau^2 \beta_1^3 (b_2+c_2)}{4\{\tau \beta_1^2 [3]_q(1+[2]_q\lambda)+[2]_q^2(1+ [1]_q\lambda)^2(\beta_1-\beta_2)\}}\,,
\end{equation}
(30)
and applying Lemma 2 gives (10).
Again by subtracting (23) from (25), using (26) and simplifying we get
\begin{equation}\label{a3witha22}
a_3=a_2^2+\frac{\tau \beta_1(b_2-c_2)}{4[3]_q(1+[2]_q\lambda)}.
\end{equation}
(31)
Thus, from (27), using (26) and simplifying we get
\begin{equation}
a_3 = \frac{\tau^2\beta^2_1 b^2_1}{4[2]^2_q(1+[1]_q\lambda)^2} + \frac{\tau \beta_1(b_2-c_2)}{4[3]_q(1+[2]_q\lambda)}\label{a3}\,,
\end{equation}
(32)
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
\begin{equation}\label{h(rho)}
h(\varrho) = \frac{\tau \beta_1^2(1-\varrho)}{\{\tau \beta_1^2[3]_q(1+[2]_q\lambda) + [2]_q^2(1+[1]_q\lambda)^2(\beta_1-\beta_2)\}}\,.
\end{equation}
(33)
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
\begin{equation}\label{Step5}
\mbox{\(|a_3 – \rho a_2^2|\)}\leq \left\{
\begin{array}{rl}
\frac{|\tau|\beta_1}{[3]_q(1+[2]_q\lambda)} & \mbox{for \(|1-\rho|\in [0,\xi)\);}\\
\frac{\beta^2_1|\tau|^2}{[2]^2_q(1+[1]_q\lambda)^2}|1-\rho| & \mbox{for \(|1-\rho| \in [\xi,\infty)\),}
\end{array}\right.
\end{equation}
(34)
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),
\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 (b_2 – c_2)}{4[3]_q(1+[2]_q\lambda)}\label{a3-mua22}.
\end{equation}
(35)
From Lemma 3 and (26)
\begin{equation}\label{b2-c2}
b_2 – c_2 = \frac{1}{2}(4-b_1^2)(x-y)\,,
\end{equation}
(36)
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 \(b\in[0,2]\);
\begin{equation}
G'(b) = \frac{\beta_1^2 |\tau|^2}{2[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\,,
\end{equation}
(37)
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.
