Bi-Univalent Problems Involving Generalized Multiplier Transform with Respect to Symmetric and Conjugate Points

Abstract

In the present article, using the subordination principle, the authors employed certain generalized multiplier transform to define two new subclasses of analytic functions with respect to symmetric and conjugate points. In particular, bi-univalent conditions for function $f\left(z\right)$ belonging to these new subclasses and their relevant connections to the famous Fekete-Szegö inequality $|a_3-v{a}_2^2|$ were investigated using a succinct mathematical approach.

1. Introduction and Definitions

Suppose that we denote by $\mathbb{A}$, the class of all analytic and univalent functions

$$f\left(z\right)=z+\sum _{k=2}^{\infty }{a}_k{z}^k$$

in the open unit disk $U=\left\{z:\left|z\right|<1\right\}$. Usually, $f\left(z\right)$ is normalized by $f\left(0\right)=f^{\prime }\left(0\right)-1=0$.

We recall that every function $f\in S$ has an inverse $f^{-1}\left(z\right)$, which is defined by

$$f^{-1}\left(f\left(z\right)\right)=z,z\in U$$

and

$$f\left(f^{-1}\left(w\right)\right)=w,\left[\right|w|<r_o\left(f\right):r_o\left(f\right)\ge \frac{1}{4}].$$

We can write that

$$g\left(w\right)=f^{-1}\left(w\right)=w-a_2{w}^2+(2a_2^2-a_3)w^3-(5a_2^3-5a_2{a}_3+a_4)w^4+\dots$$

$$=w+\sum _{k=2}^{\infty }{b}_k{w}^k$$

where,

$$b_2=-a_2,b_3=2a_2^2-a_3,b_4=-(5a_2^3-5a_2{a}_3+a_4),\dots .$$

A function $f\in A$ is said to be bi-univalent in U if both f and its inverse $g\in f^{-1}$ are univalent. Recently, the pioneering work of Srivastava et al. [1] has truly reignited interest in the study of analytic bi-univalent functions and a vast flood of follow-up work has resulted in the literature on the study of various subclasses of analytic univalent functions (see also [2,3,4]). In particular, Lewin [5] introduced the class $\sum$ and proved the bound for the second coefficients of every $f\in \sum$ satisfies the inequality $|a_2|\le 1.51$. After that, Brannan and Chunie [6] improved Lewin’s result to $|a_2|\le 1.41$ while Netanyahu [7] further showed that $|a_2|\le 1.33$. For more details on bi-univalent functions, interested reader can see the likes of Srivastava et al. [1,2,3,4,5,6,7,8,9,10,11], Frasin and Aouf [12], Hamidi and Jahangiri [13], Bulut [14], Caglar et al. [15] and Deniz [16] among others. Although investigations on the class $\sum$ are common in literatures and still on going, it is pertinent to say that the earlier investigations on the class $\sum$ seems to lack full breadth when addressing the problems involving coefficient estimates for the class $\sum$ with respect to symmetric and conjugate points. Consequently, in the present work, we shall consider the bi-univalent condition involving multiplier transform as related to symmetric and conjugate points. Some examples of bi-univalent functions and their inverses are given below:

• $log\frac{1}{1-z}$ and its corresponding inverse is $\frac{e^w-1}{e^w}$
• $\frac{z}{1-z}$ and its corresponding inverse $\frac{w}{w+1}$
• $\frac{1}{2}log\frac{1+z}{1-z}$ and its corresponding inverse $\frac{e^{2w}-1}{e^{2w}+1}$.

see [17].

Therefore, it is not out of point to say that the class of bi-univalent function is non-empty.

For function $f\left(z\right)$ of the form (1), we consider a multiplier differential operator $L_{\rho ,\sigma ,\psi }^{n,\theta }f\left(z\right)$ such that

$$L_{\rho ,\sigma ,\psi }^{1,\theta }f\left(z\right)=\frac{\rho f\left(z\right)+\sigma e^{i\theta }z{f}^{\prime }\left(z\right)+\psi z{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }}{\rho +\sigma e^{i\theta }+\psi }$$

$$L_{\rho ,\sigma ,\psi }^{2,\theta }f\left(z\right)=L_{\rho ,\sigma ,\psi }f\left(z\right)\left(L_{\rho ,\sigma ,\psi }^{1,\theta }f\left(z\right)\right)$$

$$\begin{gathered} L_{\rho ,\sigma ,\psi }^{3,\theta }f\left(z\right)=L_{\rho ,\sigma ,\psi }f\left(z\right)\left(L_{\rho ,\sigma ,\psi }^{2,\theta }f\left(z\right)\right) \\ ⋮ \end{gathered}$$

$$L_{\rho ,\sigma ,\psi }^{n,\theta }f\left(z\right)=L_{\rho ,\sigma ,\psi }f\left(z\right)\left(L_{\rho ,\sigma ,\psi }^{n-1,\theta }f\left(z\right)\right)$$

(1)

where $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$, $\left|\theta \right|<\frac{\pi }{2}$ and $\rho$ is real such that $\rho +\sigma +\psi >0$. It follows from (6) that

$$L_{\rho ,\sigma ,\psi }^{n,\theta }f\left(z\right)=z+\sum _{k=2}^{\infty }{\left(\frac{\rho +\sigma e^{i\theta }k+\psi k^2}{\rho +\sigma e^{i\theta }+\psi }\right)}^n{a}_k{z}^k$$

(2)

Suppose that the function $f\left(z\right)$ has the form (1), it is easily verified that

$$L_{\rho ,0,0}^{n,0}f\left(z\right)=f\left(z\right)\in \mathbb{A}.$$

Clearly, the operator $L_{\rho ,\sigma ,\psi }^{n,\theta }f\left(z\right)$ generalizes many existing operators of this kind which were introduced the following authors.

(i)

$L_{\rho ,\sigma ,\psi }^{n,0}{f}_1\left(z\right)=I_{\rho ,\sigma ,\psi }^{n}f\left(z\right)$ examined by Makinde et al. [18].

(ii)

$L_{\rho ,0,0}^{n,0}{f}_1\left(z\right)=I_{\rho ,\sigma }^{n}f\left(z\right)$ investigated by Swamy [19].

(iii)

$L_{\rho ,1,\psi }^{n,0}{f}_1\left(z\right)=I_{\rho }^{n}f\left(z\right),\rho >-1$ studied by Cho and Srivastava [20] and Cho and Kim [21].

(iv)

$L_{1,\sigma ,0}^{n,0}f\left(z\right)=N_{\sigma }^{n}f\left(z\right)$ studied by Swamy [19]. See also Hamzat and El-Ashwah [22].

Now in view (7), the following definitions are given:

Definition 1.

Let h be an analytic function with positive real part in the unit disk U such that $h\left(z\right)=1+D_{1}z+D_2{z}^2+D_3{z}^3+...$ with $D_n>0,n\in \mathbb{N}$. Also, let f be of the form (1), then $f\in {\mathcal{H}}_s^{n,\theta }(\rho ,\sigma ,\psi )$ if it satisfies the condition that

$$\frac{2z{\left(L_{\rho ,\sigma ,\psi }^{n,\theta }f\left(z\right)\right)}^{\prime }}{L_{\rho ,\sigma ,\psi }^{n,\theta }f\left(z\right)-L_{\rho ,\sigma ,\psi }^{n,\theta }f(-z)}\prec h\left(z\right)(z\in U)$$

(3)

and

$$\frac{2\omega {\left(L_{\rho ,\sigma ,\psi }^{n,\theta }g\left(\omega \right)\right)}^{\prime }}{L_{\rho ,\sigma ,\psi }^{n,\theta }g\left(z\right)-L_{\rho ,\sigma ,\psi }^{n,\theta }g(-\omega )}\prec h\left(\omega \right)(\omega \in U)$$

(4)

for $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$, $\left|\theta \right|<\frac{\pi }{2}$ and ρ is real such that $\rho +\sigma +\psi >0$.

Definition 2.

Let h be an analytic function with positive real part in the unit disk U such that $h\left(z\right)=1+D_{1}z+D_2{z}^2+D_3{z}^3+...$ with $D_n>0,n\in \mathbb{N}$. Also, let f be of the form (1), then $f\left(z\right)\in {\mathcal{H}}_c^{n,\theta }(\rho ,\sigma ,\psi )$ if it satisfies the condition that

$$\frac{2z{\left(L_{\rho ,\sigma ,\psi }^{n,\theta }f\left(z\right)\right)}^{\prime }}{L_{\rho ,\sigma ,\psi }^{n,\theta }f\left(z\right)+\overline{L_{\rho ,\sigma ,\psi }^{n,\theta }\overline{f\left(z\right)}}}\prec h\left(z\right)(z\in U)$$

(5)

and

$$\frac{2\omega {\left(L_{\rho ,\sigma ,\psi }^{n,\theta }g\left(\omega \right)\right)}^{\prime }}{L_{\rho ,\sigma ,\psi }^{n,\theta }g\left(\omega \right)+\overline{L_{\rho ,\sigma ,\psi }^{n,\theta }\overline{g\left(\omega \right)}}}\prec h\left(\omega \right)(\omega \in U)$$

(6)

for $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$, $\left|\theta \right|<\frac{\pi }{2}$ and ρ is real such that $\rho +\sigma +\psi >0$.

Definition 3.

Let $\gamma :U\to \mathbb{C}$ be a convex univalent function in U and satisfying the following conditions:

$$\gamma \left(0\right)=1and\Re \left\{\gamma \left(z\right)\right\}>0(z\in U).$$

Further, let $\gamma \left(s\right)$ be defined such that

$$\gamma \left(z\right)=1+\sum _{\kappa =1}^{\infty }{B}_k{z}^k.$$

(7)

The present study becomes necessary since the results obtained have various applications in different areas of science and engineering. In particular, the bounds would be relevant in calculating discrete Fourier Transform (DFT) or Inverse Discrete Fourier Transform (IDFT) and in AC circuit analysis, among others.

2. Results

Lemma 1 ([23]).

Let a function $p\in P$ be given by

$$p\left(z\right)=1+\sum _{k=1}^{\infty }{p}_k{z}^k,z\in U.$$

(8)

Then

$$|p_k|\le 2,k\in \mathbb{N},$$

(9)

where p is the family of functions analytic in U for which

$$p\left(0\right)=1,Re\left\{p\right(z\left)\right\}>0,z\in U$$

(10)

Lemma 2 ([24,25]).

Let the function r given by

$$r\left(z\right)=1+\sum _{k=1}^{\infty }{C}_k{z}^k,z\in U.$$

(11)

be convex in $U.$ Also, let the function h given by

$$l\left(z\right)=1+\sum _{k=1}^{\infty }{L}_k{z}^k,$$

(12)

be holomorphic in U. If

$$l\left(z\right)\prec r\left(z\right),z\in U$$

then

$$|L_k|\le |C_1|,k\in \mathbb{N}$$

Theorem 1.

Suppose that $f\left(z\right)\in {\mathcal{H}}_s^{n,\theta }(\rho ,\sigma ,\psi )$. Then, for $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$, $\left|\theta \right|<\frac{\pi }{2}$ and ρ is real such that $\rho +\sigma +\psi >0$

$$|a_2|\le \sqrt{\frac{|B_1|\left[2D_1+\left|B_1\right|\left(D_2-D_1\right)\right]}{8{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}},$$

$$|a_3|\le \frac{|B_1|\left[2D_1+\left|B_1\right|\left(D_2-D_1\right)\right]}{8{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{|B_1{|}^2{D}_1^2}{16{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}},$$

and

$$|a_3-v{a}_2^2|\le \frac{|B_1|\left[2D_1+\left|B_1\right|\left(D_2-D_1\right)\right]}{8{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{(1-v)|B_1{|}^2{D}_1^2}{16{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

for complex number v.

Proof. 

Since $f\left(z\right)\in {\mathcal{H}}_s^{n,\theta }(\rho ,\sigma ,\psi )$, there exists a Schwarz function $\mu \left(z\right)$ such that $\mu \left(0\right)=0$ and $\left|\mu \right(z\left)\right|<1$ in U, then from Definition 1

$$\frac{2z{\left(L_{\rho ,\sigma ,\psi }^{n,\theta }f\left(z\right)\right)}^{\prime }}{L_{\rho ,\sigma ,\psi }^{n,\theta }f\left(z\right)-L_{\rho ,\sigma ,\psi }^{n,\theta }f(-z)}=h\left({\mu }_1\left(z\right)\right)(z\in U)$$

(13)

and

$$\frac{2\omega {\left(L_{\rho ,\sigma ,\psi }^{n,\theta }g\left(\omega \right)\right)}^{\prime }}{L_{\rho ,\sigma ,\psi }^{n,\theta }g\left(z\right)-L_{\rho ,\sigma ,\psi }^{n,\theta }g(-\omega )}=h\left({\mu }_2\left(\omega \right)\right)(\omega \in U)$$

(14)

We define the functions ${\delta }_1\left(z\right)$ and $p_1\left(z\right)$ by

$${\delta }_1\left(z\right)=\frac{1+{\mu }_1\left(z\right)}{1-{\mu }_1\left(z\right)}=1+c_{1}z+c_2{z}^2+...$$

(15)

and

$$p_1\left(z\right)=\frac{1+{\mu }_2\left(\omega \right)}{1-{\mu }_2\left(\omega \right)}=1+q_1\omega +q_2{\omega }^2+....$$

(16)

Then ${\delta }_1\left(z\right)$ and $p_1\left(z\right)$ are analytic in U with positive real part with ${\delta }_1\left(0\right)=p_1\left(0\right)=1$. Now set

$$\delta \left(z\right)=\frac{2z{\left(L_{\rho ,\sigma ,\psi }^{n,\theta }f\left(z\right)\right)}^{\prime }}{L_{\rho ,\sigma ,\psi }^{n,\theta }f\left(z\right)-L_{\rho ,\sigma ,\psi }^{n,\theta }f(-z)}=h\left({\mu }_1\left(z\right)\right)=1+c_{1}z+c_2{z}^2+...$$

(17)

and

$$p\left(z\right)=\frac{2\omega {\left(L_{\rho ,\sigma ,\psi }^{n,\theta }g\left(\omega \right)\right)}^{\prime }}{L_{\rho ,\sigma ,\psi }^{n,\theta }g\left(z\right)-L_{\rho ,\sigma ,\psi }^{n,\theta }g(-\omega )}=h\left({\mu }_2\left(\omega \right)\right)=1+q_1\omega +q_2{\omega }^2+....$$

(18)

Solving for ${\mu }_1\left(z\right)$ and ${\mu }_2\left(z\right)$, we have

$${\mu }_1\left(z\right)=\frac{{\delta }_1\left(z\right)-1}{{\delta }_1\left(z\right)+1}=\frac{1}{2}\left[c_{1}z+\left(c_2-\frac{c_1^2}{2}\right)z^2+\left(c_3+\frac{c_1^3}{4}-c_1{c}_2\right)z^3+...\right]$$

and

$${\mu }_2\left(\omega \right)=\frac{p_1\left(z\right)-1}{p_1\left(\omega \right)+1}=\frac{1}{2}\left[q_1\omega +\left(q_2-\frac{q_1^2}{2}\right){\omega }^2+\left(q_3+\frac{q_1^3}{4}-q_1{q}_2\right){\omega }^3+...\right].$$

So that

$$\delta \left(z\right)=h\left({\mu }_1\left(z\right)\right)=h\left(\frac{{\delta }_1\left(z\right)-1}{{\delta }_1\left(z\right)+1}\right)=1+d_{1}z+d_2{z}^2+d_3{z}^3+...$$

$$=1+\frac{1}{2}{D}_1{c}_{1}z+\left[\frac{1}{2}{D}_1\left(c_2-\frac{c_1^2}{2}\right)+\frac{1}{4}{D}_2{c}_1^2\right]z^2+...$$

(19)

and

$$p\left(z\right)=h\left({\mu }_2\left(\omega \right)\right)=h\left(\frac{p_1\left(\omega \right)-1}{p_1\left(\omega \right)+1}\right)=1+e_1\omega +e_2{\omega }^2+e_3{\omega }^3+...$$

$$=1+\frac{1}{2}{D}_1{q}_1\omega +\left[\frac{1}{2}{D}_1\left(q_2-\frac{q_1^2}{2}\right)+\frac{1}{4}{D}_2{q}_1^2\right]{\omega }^2+...,$$

(20)

where

$$d_1=\frac{1}{2}{D}_1{c}_1,d_2=\frac{1}{2}{D}_1\left(c_2-\frac{c_1^2}{2}\right)+\frac{1}{4}{D}_2{c}_1^2,...$$

and

$$e_1=\frac{1}{2}{D}_1{q}_1,e_2=\frac{1}{2}{D}_1\left(q_2-\frac{q_1^2}{2}\right)+\frac{1}{4}{d}_2{q}_1^2,....$$

In view of (13) and (19) and (14) and (20), we obtain

$$2{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n{a}_2=\frac{1}{2}{D}_1{c}_1$$

(21)

and

$$2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n{a}_3=\frac{1}{2}{D}_1{c}_2+\frac{1}{4}(D_2-D_1)c_1^2.$$

(22)

Similarly,

$$2{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n{b}_2=\frac{1}{2}{D}_1{q}_1$$

(23)

and

$$2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n{b}_3=\frac{1}{2}{D}_1{q}_2+\frac{1}{4}(D_2-D_1)q_1^2.$$

(24)

Since $b_2=-a_2$ and $b_3=2a_2^2-a_3$. Then it is observed from (21) and (23) that

$$c_1=-q_1.$$

(25)

If both sides of (21) and (23) are squared and then summed together, we obtain

$$32{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}{a}_2^2=D_1^2\left(c_1^2+q_1^2\right).$$

(26)

Furthermore, the addition of (22) and (24) yields

$$a_2^2=\frac{2D_1\left(c_2+q_2\right)+\left(D_2-D_1\right)\left(c_1^2+q_1^2\right)}{16{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}.$$

(27)

Recall that $\delta \left(z\right),p\left(z\right)\in P$ (class of Caratheodory functions) and $\delta \left(z\right),p\left(z\right)\subset \gamma \left(U\right)$. Using Lemma (9), we have

$$|c_k|=|{\displaystyle \frac{c^k\left(0\right)}{k!}}|\le |B_1|,k\in N$$

(28)

and

$$|q_k|=|{\displaystyle \frac{q^k\left(0\right)}{k!}}|\le |B_1|,k\in N.$$

(29)

Then, the application of (28) and (29) in (27) yields

$$|a_2{|}^2=\frac{|B_1|\left[2D_1+\left|B_1\right|\left(D_2-D_1\right)\right]}{8{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}$$

(30)

which is the desired bound on $a_2$ as seen in Theorem 1.

If we subtract Equation (24) from (22), then

$$a_3=\frac{2D_1\left(c_2-q_2\right)+\left(D_2-D_1\right)\left(c_1^2-q_1^2\right)}{16{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{D_1^2\left(c_1^2+q_1^2\right)}{32{\left(\frac{\rho +\sigma e^{i\theta }(p+1)+\psi {(p+1)}^2}{\rho +\sigma e^{i\theta }p+\psi p^2}\right)}^{2n}}.$$

(31)

Applying Lemma 2 alongside (28) and (29), we obtain the desired result as contained in Theorem 1.

Ultimately,

$$a_3-v{a}_2^2=\frac{2D_1\left(c_2-q_2\right)+\left(D_2-D_1\right)\left(c_1^2-q_1^2\right)}{16{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{D_1^2\left(c_1^2+q_1^2\right)}{32{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}-v\frac{D_1^2\left(c_1^2+q_1^2\right)}{32{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}.$$

(32)

Applying Lemma 2, we obtain the desired result as seen in Theorem 1 and this ends the proof. □

In a situation whereby $h\left(z\right)=\frac{1+\alpha z}{1-\beta z}$$(-1\le \beta <\alpha \le 1)$ or equivalently, $D_1=\alpha -\beta$ and $D_2=-\beta (\alpha -\beta )$, then the following corollary is obtained.

Corollary 1.

Let $f\left(z\right)\in {\mathcal{H}}_s^{n,\theta }(\rho ,\sigma ,\psi )$. Then, for $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$, $\left|\theta \right|<\frac{\pi }{2}$ and ρ is real such that $\rho +\sigma +\psi >0$

$$|a_2|\le \sqrt{\frac{|B_1|\left[2(\alpha -\beta )-|B_1|(\alpha -\beta )(\beta +1)\right]}{8{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}},$$

$$|a_3|\le \frac{|B_1|\left[2(\alpha -\beta )-|B_1|(\alpha -\beta )(\beta +1)\right]}{8{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{|B_1{|}^2{(\alpha -\beta )}^2}{16{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}},$$

and

$$|a_3-v{a}_2^2|\le \frac{|B_1|\left[2(\alpha -\beta )-|B_1|(\alpha -\beta )(\beta +1)\right]}{8{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{(1-v)|B_1{|}^2(\alpha -\beta )}{16{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

for complex number v.

Setting $h\left(z\right)=\frac{1+(1-2i)z}{1-\beta z}$$(0\le i<1)$ or equivalently, $D_1=D_2=2(1-i)$ in Definition 1, then the following corollary is obtained.

Corollary 2.

Let $f\left(z\right)\in {\mathcal{H}}_s^{n,\theta }(\rho ,\sigma ,\psi )$. Then, for $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$, $\left|\theta \right|<\frac{\pi }{2}$ and ρ is real such that $\rho +\sigma +\psi >0$

$$|a_2|\le \sqrt{\frac{|B_1|(1-i)}{2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}}$$

$$|a_3|\le \frac{|B_1|(1-i)}{2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{|B_1{|}^2{(1-i)}^2}{4{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}$$

and

$$|a_3-v{a}_2^2|\le \frac{|B_1|(1-i)}{2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{(1-v)|B_1{|}^2{(1-i)}^2}{4{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

for complex number $v.$

Setting $h\left(z\right)=\frac{1+z}{1-z}$$(0\le i<1)$ or equivalently, $D_1=D_2=2$, then the following corollary is obtained.

Corollary 3.

Let $f\left(z\right)\in {\mathcal{H}}_s^{n,\theta }(\rho ,\sigma ,\psi )$. Then, for $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$, $\left|\theta \right|<\frac{\pi }{2}$ and ρ is real such that $\rho +\sigma +\psi >0$

$$|a_2|\le \sqrt{\frac{|B_1|}{2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}},$$

$$|a_3|\le \frac{|B_1|}{2{\left(\frac{\rho +3\sigma e^{i\theta }+9}{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{|B_1{|}^2}{4{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

and

$$|a_3-v{a}_2^2|\le \frac{|B_1|}{2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{(1-v)|B_1{|}^2}{4{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

for complex number v.

Letting $h\left(z\right)=\frac{1+z}{1-z}$$(0\le i<1)$ or equivalently, $D_1=D_2=2$, $p=1$, $\theta =0$ and $H_1=H_2=2$, then the following corollary is obtained.

Corollary 4.

Let $f\left(z\right)\in {\mathcal{H}}_s^{n,\theta }(\rho ,\sigma ,\psi )$. Then, for $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$, $\left|\theta \right|<\frac{\pi }{2}$ and ρ is real such that $\rho +\sigma +\psi >0$

$$|a_2|\le \sqrt{\frac{|B_1|}{4{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma +\psi }\right)}^n}},|a_3|\le \frac{|B_1|}{2{\left(\frac{\rho +3\sigma +9\psi }{\rho +\sigma +\psi }\right)}^n}+\frac{|B_1{|}^2}{4{\left(\frac{\rho +2\sigma +4\psi }{\rho +\sigma +\psi }\right)}^{2n}}$$

and

$$|a_3-v{a}_2^2|\le \frac{|B_1|}{2{\left(\frac{\rho +3\sigma +9\psi }{\rho +\sigma +\psi }\right)}^n}+\frac{(1-v)|B_1{|}^2}{4{\left(\frac{\rho +2\sigma +4\psi }{\rho +\sigma +\psi }\right)}^{2n}}$$

for complex number v.

Theorem 2.

Suppose that $f\left(z\right)\in {\mathcal{H}}_c^{n,\theta }(\rho ,\sigma ,\psi )$. Then, for $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$, $\left|\theta \right|<\frac{\pi }{2}$ and ρ is real such that $\rho +\sigma +\psi >0$

$$|a_2|\le \sqrt{\frac{|B_1|\left[2D_1+\left|B_1\right|\left(D_2-D_1\right)\right]}{4\left[2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n-{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}\right]}},$$

$$|a_3|\le \frac{|B_1|\left[2D_1+\left|B_1\right|\left(D_2-D_1\right)\right]}{8{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{|B_1{|}^2{D}_1^2}{4{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

and

$$|a_3-v{a}_2^2|\le \frac{|B_1|\left[2D_1+\left|B_1\right|\left(D_2-D_1\right)\right]}{8{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{(1-v)|B_1{|}^2{D}_1^2}{4{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

for complex number v.

Proof. 

With reference to Definition 2, the argument follows the same pattern as that of Theorem 1. □

Suppose that $h\left(z\right)=\frac{1+\alpha z}{1-\beta z}$$(-1\le \beta <\alpha \le 1)$ or equivalently, $D_1=\alpha -\beta$ and $D_2=-\beta (\alpha -\beta )$, then the following corollary is obtained.

Corollary 5.

Let $f\left(z\right)\in {\mathcal{H}}_c^{n,\theta }(\rho ,\sigma ,\psi )$. Then, for $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$, $\left|\theta \right|<\frac{\pi }{2}$ and ρ is real such that $\rho +\sigma +\psi >0$

$$|a_2|\le \sqrt{\frac{|B_1|\left[2(\alpha -\beta )-|B_1|(\alpha -\beta )(\beta +1)\right]}{4\left[2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n-{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}\right]}},$$

$$|a_3|\le \frac{|B_1|\left[2(\alpha -\beta )-|B_1|(\alpha -\beta )(\beta +1)\right]}{8{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{|B_1{|}^2{(\alpha -\beta )}^2}{4{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

and

$$|a_3-v{a}_2^2|\le \frac{|B_1|\left[2(\alpha -\beta )-|B_1|(\alpha -\beta )(\beta +1)\right]}{8{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{(1-v)|B_1{|}^2{(\alpha -\beta )}^2}{4{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

for complex number v.

Letting $h\left(z\right)=\frac{1+(1-2i)z}{1-\beta z}$$(0\le i<1)$ or equivalently, $D_1=D_2=2(1-i)$, then the following corollary is obtained.

Corollary 6.

Let $f\left(z\right)\in {\mathcal{H}}_c^{n,\theta }(\rho ,\sigma ,\psi )$. Then, for $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$, $\left|\theta \right|<\frac{\pi }{2}$ and ρ is real such that $\rho +\sigma +\psi >0$

$$|a_2|\le \sqrt{\frac{|B_1|(1-i)}{2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n-{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}},\left|a_3\right|\le \frac{|B_1|(1-i)}{2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{|B_1{|}^2{(1-i)}^2}{{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

and

$$|a_3-v{a}_2^2|\le \frac{|B_1|(1-i)}{2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{(1-v)|B_1{|}^2{(1-i)}^2}{{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

for complex number v.

Setting $h\left(z\right)=\frac{1+z}{1-z}$$(0\le i<1)$ or equivalently, $D_1=D_2=2$, then the following corollary is obtained.

Corollary 7.

Let $f\left(z\right)\in {\mathcal{H}}_c^{n,\theta }(\rho ,\sigma ,\psi )$. Then, for $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$, $\left|\theta \right|<\frac{\pi }{2}$ and ρ is real such that $\rho +\sigma +\psi >0$

$$|a_2|\le \sqrt{\frac{|B_1|}{2{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n-{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}},\left|a_3\right|\le \frac{|B_1|}{\left(\frac{2}{p}\right){\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{|B_1{|}^2}{{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

and

$$|a_3-v{a}_2^2|\le \frac{|B_1|}{{\left(\frac{\rho +3\sigma e^{i\theta }+9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{(1-v)|B_1{|}^2}{{\left(\frac{\rho +2\sigma e^{i\theta }+4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

for complex number v.

Letting $h\left(z\right)=\frac{1+z}{1-z}$$(0\le i<1)$ or equivalently, $D_1=D_2=2$, $p=1$, $\theta =0$ and $H_1=H_2=2$, then the following corollary is obtained.

Corollary 8.

Let $f\left(z\right)\in {\mathcal{H}}_c^{n,\theta }(\rho ,\sigma ,\psi )$. Then, for $n\in \mathbb{N}$, $\psi ,\sigma \ge 0$ and ρ is real such that $\rho +\sigma +\psi >0$

$$|a_2|\le \sqrt{\frac{|B_1|}{2{\left(\frac{\rho +3\sigma +9\psi }{\rho +\sigma e^{i\theta }p+\psi p^2}\right)}^n-{\left(\frac{\rho +2\sigma +4\psi }{\rho +\sigma +\psi }\right)}^{2n}}},\left|a_3\right|\le \frac{|B_1|}{2{\left(\frac{\rho +3\sigma +9\psi }{\rho +\sigma +\psi }\right)}^n}+\frac{|B_1{|}^2}{{\left(\frac{\rho +2\sigma +4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

and

$$|a_3-v{a}_2^2|\le \frac{|B_1|}{2{\left(\frac{\rho +3\sigma +9\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^n}+\frac{(1-v)|B_1{|}^2}{{\left(\frac{\rho +2\sigma +4\psi }{\rho +\sigma e^{i\theta }+\psi }\right)}^{2n}}$$

for complex number v.

3. Conclusions

The present investigation is mainly concerned with the study of bi-univalent problems associated with the generalized multiplier transform with respect to symmetric and conjugate points. In the study, two new subclasses ${\mathcal{H}}_s^{n,\theta }(\rho ,\sigma ,\psi )$ and ${\mathcal{H}}_c^{n,\theta }(\rho ,\sigma ,\psi )$ of bi-univalent functions are introduced using subordination principle.

The main results are established in Theorems 1 and 2, whereby the coefficient bounds for each of the two classes defined are obtained. Several new and interesting consequences of these results follow as corollaries.

It is important to note that the results obtained in this paper can be used in the future to establish the Fekete-Szego relation as well as the Hankel determinants for the two new subclasses introduced in Definitions 1 and 2.

The present work becomes more relevant and unique with the involvement of the generalized multiplier transform associated with the symmetric and conjugate points since the complex conjugate facilitates the analysis and design of systems—including electrical systems—that are modelled using complex numbers.