On a New Class of Concave Bi-Univalent Functions Associated with Bounded Boundary Rotation

Abstract

In this research article, we introduce a new subclass of concave bi-univalent functions associated with bounded boundary rotation defined on an open unit disk. For this new class, we make an attempt to find the first two initial coefficient bounds. In addition, we investigate the very famous Fekete–Szegö inequality for functions belonging to this new subclass of concave bi-univalent functions related to bounded boundary rotation. For some particular choices of parameters, we derive the earlier estimates on the coefficient bounds, which are stated at the end.

1. Introduction

Indicate with $\mathcal{A}$ the family of holomorphic functions which are defined in the open unit disk

$${\mathbb{U}}_d=\left\{\xi :\left|\xi \right|<1\right\}$$

of the form

$$g\left(\xi \right)=\xi +\sum _{m=2}^{\infty }{b}_m{\xi }^m.$$

(1)

and normalized by the conditions $g\left(0\right)=g^{\prime }\left(0\right)-1=0.$ Let $\mathcal{S}\subset \mathcal{A}$ denote the class of univalent functions in the open unit disk ${\mathbb{U}}_d.$ For every $g\in \mathcal{S},$ the Koebe one-quarter theorem confirms that the image of every univalent function g on the open unit disk ${\mathbb{U}}_d$ contains a unit ball with its center at the origin and a radius of $1/4.$ Therefore, every $g\in \mathcal{S}$ has inverse (denote by $g^{-1}$), which satisfies

$$g^{-1}\left(g\left(\xi \right)\right)=\xi ,\xi \in {\mathbb{U}}_d$$

and

$$\omega =g\left(g^{-1}\left(\omega \right)\right),\left(r\ge \left|\omega \right|;{\displaystyle \frac{1}{4}}\le r\right).$$

For each $g\in \mathcal{S},$ we define the inverse of $g:$

$$g^{-1}\left(\omega \right)=\phi \left(\omega \right)=\omega +\sum _{m=2}^{\infty }{\phi }_m{\omega }^m,$$

(2)

where

$$\begin{array}{ccc}\hfill {\phi }_2& =& -b_2,\hfill \\ \hfill {\phi }_3& =& (2b_2^2-b_3)\hfill \\ \hfill {\phi }_4& =& -(5b_2^3-5b_2{b}_3+b_4).\hfill \end{array}$$

If both g and $g^{-1}$ are univalent in ${\mathbb{U}}_d$, then we say the function g is bi-univalent in ${\mathbb{U}}_d$. The class of all bi-univalent functions is denoted by $\sigma .$ The functions defined by

$$-log\sqrt{{\displaystyle \frac{1-\xi }{1+\xi }}},-log(1-\xi )and{\displaystyle \frac{\xi }{1-\xi }}$$

are in the class $\sigma$ with the corresponding inverse functions

$${\displaystyle \frac{1-e^{-2\omega }}{1+e^{-2\omega }}},1-e^{-\omega }and{\displaystyle \frac{\omega }{1+\omega }}.$$

But the function ${\displaystyle \frac{2\xi -{\xi }^2}{2}}\mathrm{and}{\displaystyle \frac{\xi }{1-{\xi }^2}}$ does not belongs to class $\sigma .$

Brannan and Taha [1] studied the class of bi-univalent functions. Furthermore, a few specific subclasses of the bi-univalent function class $\sigma$ have been established by Brannan and Taha [1]. The first mathematician to explore bi-univalent class functions was Lewin [2], who demonstrated that $\left|b_2\right|\le 1.51$ in 1967. Later, Srivastava et al. [3] gave an overview of the class $\sigma$ with some fascinating examples and presented the first two initial coefficient estimates of $|b_2|$ and $|b_3|$ for bi-starlike functions, which were discovered in [3] (also see [4]). These subclasses are comparable to the widely used subclasses bi-starlike, bi-convex, and bi-close-to-convex. Comparably, Li et al. [5], Orhan et al. [6] and Sharma et al. [7] presented specific subclasses of bi-univalent functions linked to bounded boundary rotation. These subclasses are frequently employed as bi-starlike, bi-convex, bi-quasi convex, and bi-close-to-convex with bounded boundary rotation (for additional information, see [6,8,9] and references cited therein). In this study, we present a novel subclass of $\sigma$ related to bounded boundary rotation, which we refer to as a concave bi-univalent function.

1.1. Concave Function

A domain $\mathbf{D}$ is said to be a concave domain if the closed set $\mathbb{C}\setminus \mathbf{D}$ is convex and unbounded. Hence, $\mathbf{D}$ is simply connected. A function $g:\mathbf{D}\to \mathbb{C}\cup \{\infty \}$ is said to be concave, if $g\left(\mathbf{D}\right)$ is a concave domain. The analytic description of the above class is given by

$${\mathcal{C}}_0=\left\{g\in \mathcal{S}:\Re \left(1+{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right)<0,\xi \in {\mathbb{U}}_d\right\}.$$

In 2006, Cruz and Pommerenke [10] and many other authors (see [11,12,13,14]) studied concave univalent functions.

A function $g:{\mathbb{U}}_d\to \mathbb{C}$ is said to be a concave univalent function with an opening angle $\vartheta \pi ,$ $1<\vartheta \le 2$ at infinity if g satisfies the given conditions:

• $g\in \mathcal{A}$ and satisfies the condition $g\left(1\right)=\infty .$
• g maps ${\mathbb{U}}_d$ conformally onto a concave set, i.e., g maps ${\mathbb{U}}_d$ conformally, a set whose complement with $\mathbb{C}$ is convex.
• The opening angle of the image of f (i.e., $g\left({\mathbb{U}}_d\right)$) at ∞ is equal to or less than $\vartheta \pi .$

Let us denote the class of all concave univalent functions with an opening angle of $\vartheta \pi ,$ as ${\mathcal{C}}_0^{\vartheta }.$ In 2009, Bhowmik [15] pointed out that an analytic function g is concave univalent with an opening angle of $\vartheta \pi ,$ if and only if

$$\Re \left({\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]\right)>0,\xi \in {\mathbb{U}}_d.$$

Sakar and Güney [16] extended the class ${\mathcal{C}}_0^{\vartheta }$ by introducing an order $\rho$, $0\le \rho <1$, called the class of all concave univalent functions of order $\rho$, $0\le \rho <1$ with an opening angle of $\vartheta \pi ,$ as follows:

For $\alpha \in [1,2)$, $0\le \rho <1$, $f\in {\mathcal{C}}_0^{\vartheta }\left(\rho \right)$, if and only if ${\displaystyle \Re \left(P_g\left(\xi \right)\right)>\rho ,\xi \in {\mathbb{U}}_d}$ where

$$P_g\left(\xi \right)={\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right],\xi \in {\mathbb{U}}_d,g\left(0\right)=g^{\prime }\left(0\right)-1=0.$$

1.2. Function with Bounded Variation

In 1975, Padmanabhan and Parvatham [17] introduced the class ${\mathcal{P}}_{\kappa }\left(\rho \right).$ For $2\le \kappa$ and $1>\rho \ge 0,$ let us indicate ${\mathcal{P}}_{\kappa }\left(\rho \right)$ as the class of functions q, which are holomorphic and normalized by $q\left(0\right)=1$ and $q^{\prime }\left(0\right)>1,$ satisfy the condition

$$\underset{0}{\overset{2\pi }{\int }}\left|{\displaystyle \frac{\Re \left(q\right(\xi \left)\right)-\rho }{1-\rho }}\right|dt\le \kappa \pi ,$$

where $\xi =r{e}^{it}\in {\mathbb{U}}_d$. For $\rho =0,$ the class ${\mathcal{P}}_{\kappa }\left(\rho \right)$ converts into the class ${\mathcal{P}}_{\kappa }$, which is defined by Pinchuk [18]. The class ${\mathcal{P}}_{\kappa }$ denotes the class of functions $q\left(0\right)=1$ that are analytic in ${\mathbb{U}}_d$ and with a representation of

$$q\left(\xi \right)={\displaystyle \frac{1}{2}}\underset{0}{\overset{2\pi }{\int }}\left|{\displaystyle \frac{e^{it}+\xi }{e^{it}-\xi }}\right|d\Lambda \left(t\right),$$

where $\Lambda$ is a function of bounded variation (which is real-valued) and satisfies

$${\displaystyle \underset{0}{\overset{2\pi }{\int }}d\Lambda \left(t\right)=2\mathrm{and}\underset{0}{\overset{2\pi }{\int }}\left|d\Lambda \left(t\right)\right|\le \kappa ,2\le \kappa .}$$

If we choose $\rho =0$ and $\kappa =2,$ then the class ${\mathcal{P}}_{\kappa }\left(\rho \right)$ reduces to the class $\mathcal{P}$, which is known as the class of Carathéodory functions. An interesting observation between the class ${\mathcal{P}}_{\kappa }$ and Carathéodory functions is that the function $q\left(\xi \right)\in {\mathcal{P}}_{\kappa }$ if there exist two Carathéodory functions, $q_1\left(\xi \right)$ and $q_2\left(\xi \right)$, such that

$$p\left(z\right)=\left({\displaystyle \frac{\kappa }{4}}+{\displaystyle \frac{1}{2}}\right)q_1\left(\xi \right)-\left({\displaystyle \frac{\kappa }{4}}-{\displaystyle \frac{1}{2}}\right)q_2\left(\xi \right).$$

Let ${\mathcal{V}}_k$ represent the class of analytic functions g in ${\mathbb{U}}_d$ with $g\left(0\right)=0$, $g^{\prime }\left(0\right)=1$, satisfying

$$1+{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\in {\mathcal{P}}_k.$$

In 1931, Paatero [19] proved that a function $g\in {\mathcal{V}}_{\kappa }$ given in the form (1) maps ${\mathbb{U}}_d$ conformally onto an image domain $f\left({\mathbb{U}}_d\right)$ at most $\kappa \pi .$ Paatero [19] has shown that $g\in {\mathcal{V}}_{\kappa }$ if and only if

$$g^{\prime }\left(\xi \right)=exp\left\{\underset{0}{\overset{2\pi }{\int }}log\left[{\displaystyle \frac{e^{it}}{e^{it}-\xi }}\right]d\Lambda \left(t\right)\right\},$$

where $\Lambda$ is a function of bounded variation (which is real-valued) and satisfies

$${\displaystyle \underset{0}{\overset{2\pi }{\int }}d\Lambda \left(t\right)=2\mathrm{and}\underset{0}{\overset{2\pi }{\int }}\left|d\Lambda \left(t\right)\right|\le \kappa ,\kappa \ge 2.}$$

Brannan [20] showed that the function $g\in {\mathcal{V}}_{\kappa }$ if there exist two starlike functions, $f_1\left(\xi \right)$ and $f_2\left(\xi \right)$, such that

$$f^{\prime }\left(\xi \right)={\displaystyle \frac{{\left(\frac{f_2\left(\xi \right)}{\xi }\right)}^{\frac{\kappa }{4}+\frac{1}{2}}}{{\left(\frac{f_1\left(\xi \right)}{\xi }\right)}^{\frac{\kappa }{4}-\frac{1}{2}}}}.$$

Paatero [19] gave the following distortion bounds: for the function, $g\in {\mathcal{V}}_{\kappa }$ for $\left|\xi \right|=r<1$

$${\displaystyle \frac{{(1-r)}^{\frac{\kappa }{2}-1}}{{(1+r)}^{\frac{\kappa }{2}+1}}\le \left|g^{\prime }\left(\xi \right)\right|\le \frac{{(1+r)}^{\frac{\kappa }{2}+1}}{{(1-r)}^{\frac{\kappa }{2}-1}}.}$$

In 1971, Pinchuk [18] introduced and studied the class ${\mathcal{R}}_{\kappa }$, that is, $g\in {\mathcal{R}}_{\kappa }$ if and only if

$$g\left(\xi \right)=\xi exp\left\{\underset{0}{\overset{2\pi }{\int }}log\left[{\displaystyle \frac{e^{it}}{e^{it}-\xi }}\right]d\Lambda \left(t\right)\right\},$$

where $\Lambda$ is a function of bounded variation (which is real-valued) and satisfies

$${\displaystyle \underset{0}{\overset{2\pi }{\int }}d\Lambda \left(t\right)=2\mathrm{and}\underset{0}{\overset{2\pi }{\int }}\left|d\Lambda \left(t\right)\right|\le \kappa ,\kappa \ge 2.}$$

Thus, ${\mathcal{R}}_{\kappa }$ is the class of functions of bounded radius rotation bounded by $\kappa \pi .$

Lemma 1

([21]). Let an analytic function $q\left(\xi \right)=1+{\sum }_{m=1}^{\infty }{q}_m{\xi }^m$, $\xi \in {\mathbb{U}}_d$ belong to the class ${\mathcal{P}}_{\kappa }\left(\rho \right)$. Then,

$$\left|q_m\right|\le \kappa (1-\rho ),m\ge 1.$$

(3)

In this study, we present a new subclass of concave bi-univalent functions related to bounded boundary rotation as given in Definition 1. We obtain the first two initial bounds, $|a_2|$ and $|a_3|$, and also find the well-known Fekete–Szegö inequality $|a_3-\aleph a_2^2|$ for functions in this new class. For specific parameter selections, the previous findings are pointed out as corollaries, which have not been discussed so far.

2. Examples for the Class Concave Functions with Bounded Boundary Rotation

2.1. Concave Functions with Bounded Boundary Rotation

Suppose $1<\vartheta \le 2$ and $4\ge \kappa \ge 2.$ A function g given in the form (1) is said to be a concave function with bounded boundary rotation if the function g satisfies the following condition:

$${\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]\in {\mathcal{P}}_{\kappa }.$$

The class of all concave functions with bounded boundary rotation is denoted by ${\mathcal{CO}}^{\vartheta }\left(\kappa \right).$

2.2. Integral Representation of ${\mathcal{CO}}^{\vartheta }\left(\kappa \right)$

Theorem 1.

Suppose $1<\vartheta \le 2$ and $4\ge \kappa \ge 2.$ If a function $g\in {\mathcal{CO}}^{\vartheta }\left(\kappa \right),$ then

$$g^{\prime }\left(\xi \right)={(1-\xi )}^{-(\vartheta +1)}exp\left\{{\displaystyle \frac{1}{2}}{\int }_0^{2\pi }(\vartheta -1)log(1-\xi e^{-it})\right\}d\Lambda \left(t\right),$$

(4)

where Λ is a function of bounded variation (which is real-valued) and satisfies

$${\displaystyle \underset{0}{\overset{2\pi }{\int }}d\Lambda \left(t\right)=2and\underset{0}{\overset{2\pi }{\int }}\left|d\Lambda \left(t\right)\right|\le \kappa ,2\le \kappa .}$$

Proof. 

Since $g\in {\mathcal{CO}}^{\vartheta }\left(\kappa \right),$ there exists an analytic function $r\left(\xi \right)$ belonging to the class ${\mathcal{P}}_{\kappa }$ such that

$${\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]=r\left(\xi \right).$$

(5)

Hence, from (5), we obtain

$${\displaystyle \frac{\vartheta +1}{1-\xi }}-{\displaystyle \frac{g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}={\displaystyle \frac{\vartheta -1}{2}}{\displaystyle \frac{r\left(\xi \right)-1}{\xi }}.$$

(6)

Since $r\left(\xi \right)\in {\mathcal{P}}_{\kappa },$

$$\begin{array}{cc}\hfill {\displaystyle \frac{r\left(\xi \right)-1}{\xi }}=& {\displaystyle \frac{1}{\xi }}\left[{\displaystyle \frac{1}{2}}\underset{0}{\overset{2\pi }{\int }}{\displaystyle \frac{1+e^{-it}\xi }{1-e^{-it}\xi }}d\Lambda \left(t\right)-1\right],\hfill \\ \hfill =& {\displaystyle \frac{1}{2\xi }}\left[{\int }_0^{2\pi }{\displaystyle \frac{1+e^{-it}\xi }{1-e^{-it}\xi }}d\Lambda \left(t\right)-{\int }_0^{2\pi }d\Lambda \left(t\right)\right],\hfill \\ \hfill =& {\int }_0^{2\pi }{\displaystyle \frac{e^{-it}}{1-e^{-it}\xi }}d\Lambda \left(t\right).\hfill \end{array}$$

Therefore,

$${\int }_0^{\xi }{\displaystyle \frac{r\left(\zeta \right)-1}{\zeta }}d\zeta =-{\int }_0^{2\pi }log(1-e^{-it}\xi )d\Lambda \left(t\right).$$

(7)

From (6) and (7), we obtain (4). This completes the proof of Theorem 1. □

2.3. Relation Between Class ${\mathcal{CO}}^{\vartheta }\left(\kappa \right)$ and ${\mathcal{V}}_{\kappa }$

Theorem 2.

Suppose $1<\vartheta \le 2$ and $4\ge \kappa \ge 2.$ If the function $g\in {\mathcal{CO}}^{\vartheta }\left(\kappa \right),$ then there exists a function $\psi \in {\mathcal{V}}_{\kappa }$ such that

$$g^{\prime }\left(\xi \right)={\displaystyle \frac{{\left[{\psi }^{\prime }\left(\xi \right)\right]}^{-\frac{\vartheta +1}{2}}}{{(1-\xi )}^{\vartheta +1}}}.$$

(8)

Proof. 

Since $g\in {\mathcal{CO}}^{\vartheta }\left(\kappa \right),$ there exists an analytic function $r\left(\xi \right)$ belonging to the class ${\mathcal{P}}_{\kappa }$ such that

$${\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]=r\left(\xi \right).$$

(9)

Since $r\in {\mathcal{P}}_{\kappa },$ there exists a function $\psi \in {\mathcal{V}}_{\kappa }$ such that

$$r\left(\xi \right)=1+{\displaystyle \frac{\xi {\psi }^{″}\left(\xi \right)}{{\psi }^{\prime }\left(\xi \right)}}.$$

(10)

Hence, from (9) and (10), we obtain

$${\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]=1+{\displaystyle \frac{\xi {\psi }^{″}\left(\xi \right)}{{\psi }^{\prime }\left(\xi \right)}}.$$

(11)

Equation (11) can be written as

$${\displaystyle \frac{\vartheta +1}{1-\xi }}-{\displaystyle \frac{g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}={\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{{\psi }^{″}\left(\xi \right)}{{\psi }^{\prime }\left(\xi \right)}}.$$

(12)

Integrating (12), we obtain (8). This completes the proof of Theorem 2. □

2.4. Examples

Let us define some examples of functions belonging to the class ${\mathcal{CO}}^{\vartheta }\left(\kappa \right).$

Example 1.

The function $g_1:{\mathbb{U}}_d\to \mathbb{C}$ defined by

$$g_1\left(\xi \right)={\int }_0^{\xi }{\displaystyle \frac{e^{\left(\frac{1-\vartheta }{2}\right)t}}{{(t-1)}^{\vartheta +1}}}dt$$

is in the class ${\mathcal{CO}}^{\vartheta }\left(\kappa \right).$ Since

$$g_1^{\prime }\left(\xi \right)={\displaystyle \frac{e^{\left(\frac{1-\vartheta }{2}\right)\xi }}{{(\xi -1)}^{\vartheta +1}}}$$

we have

$$\Re \left({\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]\right)=1+\Re \left(\xi \right).$$

Therefore,

$${\int }_0^{2\pi }\left|\Re \left({\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]\right)\right|={\int }_0^{2\pi }(1+cos\theta )d\theta =2\pi \le \kappa \pi .$$

Hence, $g_1\in {\mathcal{CO}}^{\vartheta }\left(\kappa \right).$

Example 2.

The function $g_2:{\mathbb{U}}_d\to \mathbb{C}$ defined by

$$g_2\left(\xi \right)={\int }_0^{\xi }{\displaystyle \frac{e^{\left[\frac{\kappa (1-\vartheta )}{4}\right]t}}{{(t-1)}^{\vartheta +1}}}dt$$

is in the class ${\mathcal{CO}}^{\vartheta }\left(\kappa \right).$ Since

$$g_2^{\prime }\left(\xi \right)={\displaystyle \frac{e^{\left[\frac{\kappa (1-\vartheta )}{4}\right]\xi }}{{(\xi -1)}^{\vartheta +1}}},$$

we have

$$\Re \left({\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]\right)=1+{\displaystyle \frac{\kappa }{2}}\Re \left(\xi \right).$$

Therefore,

$${\int }_0^{2\pi }\left|\Re \left({\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]\right)\right|={\int }_0^{2\pi }(1+{\displaystyle \frac{\kappa }{2}}cos\theta )d\theta =2\pi \le \kappa \pi .$$

Hence, $g_2\in {\mathcal{CO}}^{\vartheta }\left(\kappa \right).$

Example 3.

The function $g_3:{\mathbb{U}}_d\to \mathbb{C}$ defined by

$$g_3\left(\xi \right)={\int }_0^{\xi }{\displaystyle \frac{e^{\left(\frac{1-\vartheta }{4}\right)t}}{{(t-1)}^{\vartheta +1}}}dt$$

and is in the class ${\mathcal{CO}}^{\vartheta }\left(\kappa \right).$ Since

$$g_3^{\prime }\left(\xi \right)={\displaystyle \frac{e^{\left(\frac{1-\vartheta }{4}\right)\xi }}{{(\xi -1)}^{\vartheta +1}}},$$

we have

$$\Re \left({\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]\right)=1+{\displaystyle \frac{1}{2}}\Re \left(\xi \right).$$

Therefore,

$${\int }_0^{2\pi }\left|\Re \left({\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]\right)\right|={\int }_0^{2\pi }(1+{\displaystyle \frac{1}{2}}cos\theta )d\theta =2\pi \le \kappa \pi .$$

Hence, $g_3\in {\mathcal{CO}}^{\vartheta }\left(\kappa \right).$

Example 4.

The function $g_4:{\mathbb{U}}_d\to \mathbb{C}$ defined by

$$g_4\left(\xi \right)={\int }_0^{\xi }{\displaystyle \frac{e^{\left[\frac{\kappa (1-\vartheta )}{8}\right]t}}{{(t-1)}^{\vartheta +1}}}dt$$

belongs to the class ${\mathcal{CO}}^{\vartheta }\left(\kappa \right).$ Since

$$g_4^{\prime }\left(\xi \right)={\displaystyle \frac{e^{\left[\frac{\kappa (1-\vartheta )}{8}\right]\xi }}{{(\xi -1)}^{\vartheta +1}}},$$

we have

$$\Re \left({\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]\right)=1+{\displaystyle \frac{\kappa }{4}}\Re \left(\xi \right).$$

Therefore,

$${\int }_0^{2\pi }\left|\Re \left({\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]\right)\right|={\int }_0^{2\pi }(1+{\displaystyle \frac{\kappa }{4}}cos\theta )d\theta =2\pi \le \kappa \pi .$$

Hence, $g_4\in {\mathcal{CO}}^{\vartheta }\left(\kappa \right).$

3. Concave Bi-Univalent Functions with Bounded Boundary Rotation

Definition 1.

Suppose $1<\vartheta \le 2,$ $\rho \in [0,1)$ and $4\ge \kappa \ge 2.$ A function $g\in \sigma$ given in the form (1) is said to be a bi-concave function with bounded boundary rotation of order ρ if the functions g and $g^{-1}=\phi$ satisfy the following conditions:

$${\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]\in {\mathcal{P}}_{\kappa }\left(\rho \right)$$

and

$${\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1-\omega }{1+\omega }}-1-{\displaystyle \frac{\omega {\phi }^{″}\left(\omega \right)}{{\phi }^{\prime }\left(\omega \right)}}\right]\in {\mathcal{P}}_{\kappa }\left(\rho \right).$$

The class of all concave bi-univalent functions with bounded boundary rotation of order ρ is denoted by ${\mathcal{CO}}_{\sigma }^{\vartheta }(\kappa ,\rho ).$

Remark 1.

(1) Upon fixing $\rho =0$, we have a new class, ${\mathcal{CO}}_{\sigma }^{\vartheta }(\kappa ,\rho )\equiv {\mathcal{CO}}_{\sigma }^{\vartheta }\left(\kappa \right),$ the class of all bi-concave functions with bounded boundary rotation.

(2) Assuming $\kappa =2$, the class ${\mathcal{CO}}_{\sigma }^{\vartheta }(\kappa ,\rho )\equiv {\mathcal{CO}}_{\sigma }^{\vartheta }\left(\rho \right),$ which was called the class of all bi-concave functions of order ρ, which was defined by Sakar and Güney [16].

Theorem 3.

Suppose $1<\vartheta \le 2,$ $\rho \in [0,1)$ and $4\ge \kappa \ge 2.$ If the function $g\in {\mathcal{CO}}_{\sigma }^{\vartheta }(\kappa ,\rho ),$ then

$$\left|b_2\right|\le \sqrt{{\displaystyle \frac{1+\vartheta }{2}}+{\displaystyle \frac{\kappa (1-\rho )(\vartheta -1)}{4}}},$$

(13)

$$\left|b_3\right|\le {\displaystyle \frac{1+\vartheta }{2}}+{\displaystyle \frac{\kappa (1-\rho )(\vartheta -1)}{4}}$$

(14)

and for any $\aleph \in \mathbb{R}$ that is a real number,

$$|b_3-\aleph b_2^2|\le \left\{\begin{array}{cc}\left[{\displaystyle \frac{1+\vartheta }{2}+\frac{\kappa (1-\rho )(\vartheta -1)}{4}}\right](1-\aleph )\hfill & {\displaystyle for\aleph <\frac{2}{3},}\hfill \\ {\displaystyle \frac{1+\vartheta }{2}(1-\aleph )+\frac{\kappa (1-\rho )(\vartheta -1)}{12}}\hfill & {\displaystyle for\frac{2}{3}\le \aleph <1,}\hfill \\ {\displaystyle \frac{1+\vartheta }{2}(\aleph -1)+\frac{\kappa (1-\rho )(\vartheta -1)}{12}}\hfill & {\displaystyle for1\le \aleph <\frac{4}{3},}\hfill \\ \left[{\displaystyle \frac{1+\vartheta }{2}+\frac{\kappa (1-\rho )(\vartheta -1)}{4}}\right](\aleph -1)\hfill & {\displaystyle for\aleph \ge \frac{4}{3}.}\hfill \end{array}\right.$$

(15)

Proof. 

Since $g\in {\mathcal{CO}}_{\sigma }^{\vartheta }(\kappa ,\rho ),$ from Definition 1, there exist two analytic functions $r\left(\xi \right)$, and $u\left(\omega \right)$ belongs to the class ${\mathcal{P}}_{\kappa }\left(\rho \right)$ such that

$${\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]=r\left(\xi \right)$$

(16)

and

$${\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1-\omega }{1+\omega }}-1-{\displaystyle \frac{\omega {\phi }^{″}\left(\omega \right)}{{\phi }^{\prime }\left(\omega \right)}}\right]=u\left(\omega \right),$$

(17)

where

$$r\left(\xi \right)=1+\sum _{m=1}^{\infty }{r}_m{\xi }^m=1+r_1\xi +r_2{\xi }^2+\cdots .$$

(18)

and

$$u\left(\omega \right)=1+\sum _{m=1}^{\infty }{u}_m{\omega }^m=1+u_1\omega +u_2{\omega }^2+\cdots .$$

(19)

Since

$$\begin{array}{c}{\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1+\xi }{1-\xi }}-1-{\displaystyle \frac{\xi g^{″}\left(\xi \right)}{g^{\prime }\left(\xi \right)}}\right]=\hfill \\ \hfill 1+\left[{\displaystyle \frac{2(\vartheta +1)}{\vartheta -1}}-{\displaystyle \frac{4b_2}{\vartheta -1}}\right]\xi +\left[{\displaystyle \frac{2(\vartheta +1)}{\vartheta -1}}-{\displaystyle \frac{2(6b_3-4b_2^2)}{\vartheta -1}}\right]{\xi }^2+\cdots \end{array}$$

(20)

and

$$\begin{array}{c}{\displaystyle \frac{2}{\vartheta -1}}\left[{\displaystyle \frac{\vartheta +1}{2}}{\displaystyle \frac{1-\omega }{1+\omega }}-1-{\displaystyle \frac{\omega {\phi }^{″}\left(\omega \right)}{{\phi }^{\prime }\left(\omega \right)}}\right]=\hfill \\ \hfill 1-\left[{\displaystyle \frac{2(\vartheta +1)}{\vartheta -1}}-{\displaystyle \frac{4b_2}{\vartheta -1}}\right]\omega +\left[{\displaystyle \frac{2(\vartheta +1)}{\vartheta -1}}-{\displaystyle \frac{2(8b_2^2-6b_3)}{\vartheta -1}}\right]{\omega }^2+\cdots ,\end{array}$$

(21)

from (18)–(21), we obtain

$${\displaystyle \frac{2(\vartheta +1)}{\vartheta -1}}-{\displaystyle \frac{4b_2}{\vartheta -1}}=r_1,$$

(22)

$${\displaystyle \frac{2(\vartheta +1)}{\vartheta -1}}-{\displaystyle \frac{2(6b_3-4b_2^2)}{\vartheta -1}}=r_2,$$

(23)

$$-{\displaystyle \frac{2(\vartheta +1)}{\vartheta -1}}+{\displaystyle \frac{4b_2}{\vartheta -1}}=u_1$$

(24)

and

$${\displaystyle \frac{2(\vartheta +1)}{\vartheta -1}}-{\displaystyle \frac{2(8b_2^2-6b_3)}{\vartheta -1}}=u_2.$$

(25)

Hence, from (22) and (24), we obtain

$${\displaystyle \frac{4b_2}{\vartheta -1}}={\displaystyle \frac{2(\vartheta +1)}{\vartheta -1}}-r_1=u_1+{\displaystyle \frac{2(\vartheta +1)}{\vartheta -1}}.$$

(26)

By using Lemma 1 in (26), we obtain

$$\left|b_2\right|\le {\displaystyle \frac{\vartheta +1}{2}}+{\displaystyle \frac{\kappa (1-\rho )(\vartheta -1)}{4}}.$$

(27)

From (23) and (25), we obtain

$${\displaystyle \frac{8b_2^2}{\vartheta -1}}={\displaystyle \frac{4(\vartheta +1)}{\vartheta -1}}-r_2-u_2.$$

(28)

Hence, by using Lemma 1 in (28), we obtain

$$\left|b_2^2\right|\le {\displaystyle \frac{\vartheta +1}{2}}+{\displaystyle \frac{\kappa (1-\rho )(\vartheta -1)}{4}}.$$

(29)

Equations (27) and (29) give the bound of $\left|b_2\right|$ given in (13). Now, subtracting (22) from (24), we obtain

$${\displaystyle \frac{24b_3}{\vartheta -1}}={\displaystyle \frac{24b_2^2}{\vartheta -1}}-u_2+r_2.$$

(30)

By using (28) in (30), we obtain

$${\displaystyle \frac{12b_3}{\vartheta -1}}={\displaystyle \frac{6(\vartheta +1)}{\vartheta -1}}-2r_2-u_2.$$

(31)

Hence, by using Lemma 1 in (31), we obtain

$$\left|b_3\right|\le {\displaystyle \frac{\vartheta +1}{2}}+{\displaystyle \frac{\kappa (1-\rho )(\vartheta -1)}{4}}.$$

(32)

Equation (32) gives the bound of $\left|b_3\right|$ given in (14). For any $\aleph \in \mathbb{R}$ and from (28) and (31), we obtain

$${\displaystyle \frac{b_3-\aleph b_2^2}{\vartheta -1}}={\displaystyle \frac{(\vartheta +1)(1-\aleph )}{2(\vartheta -1)}}-{\displaystyle \frac{r_2(4-3\aleph )}{24}}-{\displaystyle \frac{u_2(2-3\aleph )}{24}}.$$

(33)

Hence, by using Lemma 1 in (33), we obtain

$${\displaystyle \frac{|b_3-\aleph b_2^2|}{\vartheta -1}}\le {\displaystyle \frac{(\vartheta +1)|1-\aleph |}{2(\vartheta -1)}}+{\displaystyle \frac{\kappa (1-\rho )}{24}}\left[\right|4-3\aleph |+|2-3\aleph \left|\right].$$

(34)

For the different choice of ℵ, Equation (34) gives (15). This completes the proof of Theorem 3. □

By fixing $\rho =0$, we state the following result for $g\in {\mathcal{CO}}_{\sigma }^{\vartheta }\left(\kappa \right).$

Corollary 1.

Suppose $1<\vartheta \le 2$ and $4\ge \kappa \ge 2.$ If a function $g\in {\mathcal{CO}}_{\sigma }^{\vartheta }\left(\kappa \right),$ then

$$\left|b_2\right|\le \sqrt{{\displaystyle \frac{\vartheta +1}{2}}+{\displaystyle \frac{\kappa (\vartheta -1)}{4}}},$$

$$\left|b_3\right|\le {\displaystyle \frac{\vartheta +1}{2}}+{\displaystyle \frac{\kappa (\vartheta -1)}{4}}$$

and for any $\aleph \in \mathbb{R}$ that is a real number,

$$|b_3-\aleph b_2^2|\le \left\{\begin{array}{cc}\left[{\displaystyle \frac{\vartheta +1}{2(\vartheta -1)}+\frac{\kappa }{4}}\right](1-\aleph )\hfill & {\displaystyle for\aleph <\frac{2}{3},}\hfill \\ {\displaystyle \frac{\vartheta +1}{2(\vartheta -1)}(1-\aleph )+\frac{\kappa }{12}}\hfill & {\displaystyle for\frac{2}{3}\le \aleph <1,}\hfill \\ {\displaystyle \frac{\vartheta +1}{2(\vartheta -1)}(\aleph -1)+\frac{\kappa }{12}}\hfill & {\displaystyle for1\le \aleph <\frac{4}{3},}\hfill \\ \left[{\displaystyle \frac{\vartheta +1}{2(\vartheta -1)}+\frac{\kappa }{4}}\right](\aleph -1)\hfill & {\displaystyle for\aleph \ge \frac{4}{3}.}\hfill \end{array}\right.$$

By taking $\kappa =2$ we obtain the following result for $g\in {\mathcal{CO}}_{\sigma }^{\vartheta }\left(\rho \right).$

Corollary 2.

Suppose $1<\vartheta \le 2$ and $\rho \in [0,1).$ If the function $g\in {\mathcal{CO}}_{\sigma }^{\vartheta }\left(\rho \right),$ then

$$\left|b_2\right|\le \sqrt{{\displaystyle \frac{(\vartheta +1)+(1-\rho )(\vartheta -1)}{2}}},$$

$$\left|b_3\right|\le {\displaystyle \frac{(\vartheta +1)+(1-\rho )(\vartheta -1)}{2}}$$

and for any $\aleph \in \mathbb{R}$ that is a real number,

$$|b_3-\aleph b_2^2|\le \left\{\begin{array}{cc}\left[{\displaystyle \frac{(\vartheta +1)+(1-\rho )(\vartheta -1)}{2}}\right](1-\aleph )\hfill & {\displaystyle for\aleph <\frac{2}{3},}\hfill \\ {\displaystyle \frac{3(\vartheta +1)(1-\aleph )+(1-\rho )(\vartheta -1)}{6}}\hfill & {\displaystyle for\frac{2}{3}\le \aleph <1,}\hfill \\ {\displaystyle \frac{3(\vartheta +1)(\aleph -1)+(1-\rho )(\vartheta -1)}{6}}\hfill & {\displaystyle for1\le \aleph <\frac{4}{3},}\hfill \\ \left[{\displaystyle \frac{(\vartheta +1)+(1-\rho )(\vartheta -1)}{2}}\right](\aleph -1)\hfill & {\displaystyle for\aleph \ge \frac{4}{3}.}\hfill \end{array}\right.$$

For the particular choice of $\kappa =2$ and $\rho =0,$ the class ${\mathcal{CO}}_{\sigma }^{\vartheta }(\kappa ,\rho )$ converts into the class ${\mathcal{CO}}_{\sigma }^{\vartheta }.$ Hence, we obtain the following result for the functions belonging to the class ${\mathcal{CO}}_{\sigma }^{\vartheta }.$

Corollary 3.

Suppose $1<\vartheta \le 2.$ If the function $g\in {\mathcal{CO}}_{\sigma }^{\vartheta },$ then

$$\left|b_2\right|\le \sqrt{\vartheta },\left|b_3\right|\le \vartheta$$

and for any $\aleph \in \mathbb{R}$ that is a real number,

$$|b_3-\aleph b_2^2|\le \left\{\begin{array}{cc}\vartheta (1-\aleph )\hfill & {\displaystyle for\aleph <\frac{2}{3},}\hfill \\ {\displaystyle \frac{3(\vartheta +1)(1-\aleph )+(\vartheta -1)}{6}}\hfill & {\displaystyle for\frac{2}{3}\le \aleph <1,}\hfill \\ {\displaystyle \frac{3(\vartheta +1)(\aleph -1)+(\vartheta -1)}{6}}\hfill & {\displaystyle for1\le \aleph <\frac{4}{3},}\hfill \\ \vartheta (\aleph -1)\hfill & {\displaystyle for\aleph \ge \frac{4}{3}.}\hfill \end{array}\right.$$

4. Concluding Remarks and Observations

In this research article, we introduce and examine some properties of a new class of concave bi-univalent functions associated with bounded boundary rotation in open unit disk ${\mathbb{U}}_d.$ We investigates the initial Taylor–Maclaurin coefficients, $\left|b_2\right|$ and $\left|b_3\right|$, for functions belonging to concave bi-univalent functions associated with bounded boundary rotation. Note that by using different values of parameters, we obtain certain corollaries, which verifies earlier results on the bounds $|b_2|$ and $|b_3|$ which were obtained by Sakar and Güney [16]. In addition, for the first time, we investigate the very famous Fekete–Szegö inequality $|b_3-\aleph b_2^2|$ for g belonging to this new subclass of concave bi-univalent functions related to bounded boundary rotation.