Bi-Concave Functions Connected with the Combination of the Binomial Series and the Confluent Hypergeometric Function

Abstract

In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this purpose, we make use of a combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we derive the estimates on the initial Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in these analytic and bi-concave function classes, which are introduced in this paper. We also derive a number of corollaries and consequences of our main results in this paper.

1. Introduction and Preliminaries

The Geometric Function Theory of Complex Analysis is the study of the relationship of analytic properties of a given function and the geometric properties of its image domain. This subject has a remarkably rich literature due mainly to its potential for applications in the mathematical and physical sciences, and also in other areas. In particular, the estimate problems for the Taylor-Maclaurin coefficients of analytic and univalent functions have been investigated extensively and widely. As an example, we cite the celebrated de Branges theorem, which asserts the truths of the Milin conjecture of 1971, the Robertson conjecture of 1936 and the Bieberbach conjecture of 1916 (see, for details, [1]).

Investigations of the estimate problems for the Taylor-Maclaurin coefficients of the analytic functions, which are bi-univalent (that is, both the function and its inverse are univalent), were initiated by Brannan et al. (see [2,3]) (see also the related work by Netanyahu [4]). More recently, after the publication of the pioneering work on the subject by Srivastava et al. [5], analogous coefficient problems have been studied for various interesting subclasses of the class of analytic and bi-univalent functions. These subclasses include the classes of bi-starlike functions, bi-convex functions, bi-close-to-convex functions, and so on (see, for example, [6,7,8,9,10,11]).

The coefficient estimate problems, which we consider in this article, involve some new subclasses of analytic and bi-concave functions. Some earlier studies of the coefficient estimate problems for other subclasses of analytic and bi-concave functions include (for example) the works in [12,13,14,15,16].

We begin now by supposing that $\mathcal{A}$ represents the class of functions of the form given by

$$\mathcal{F}\left(z\right)=z+\sum _{t=2}^{\infty }{a}_t{z}^t(z\in \mathsf{\Lambda }),$$

(1)

which are analytic in the open unit disk

$$\mathsf{\Lambda }=\{z:z\in \mathbb{C}\mathrm{and}\left|z\right|<1\}.$$

As usual, we denote by $\mathcal{S}$ the subclass of the analytic function class $\mathcal{A}$, which consists of functions that are also univalent in $\mathsf{\Lambda }$.

We also let the function $\mathcal{H}\in \mathcal{A}$ be given by

$$\mathcal{H}\left(z\right)=z+\sum _{t=2}^{\infty }{c}_t{z}^t(z\in \mathsf{\Lambda }).$$

(2)

The Hadamard product (or convolution) of $\mathcal{F}$ and $\mathcal{H}$ is given by

$$(\mathcal{F}\ast \mathcal{H})\left(z\right):=z+\sum _{t=2}^{\infty }{a}_t{c}_t{z}^t=:(\mathcal{H}\ast \mathcal{F})\left(z\right)(z\in \mathsf{\Lambda }).$$

The Koebe one-quarter theorem (see, for details, [17]) states that the image of $\mathsf{\Lambda }$ under every univalent function $\mathcal{F}\in \mathcal{S}$ contains the disk of radius ${\displaystyle \frac{1}{4}}$. Therefore, every function $\mathcal{F}\in \mathcal{S}$ has an inverse ${\mathcal{F}}^{-1}$ that satisfies the following property:

$$\mathcal{F}\left({\mathcal{F}}^{-1}\left(w\right)\right)=w\left(\left|w\right|<r_0\left(\mathcal{F}\right);r_0\left(\mathcal{F}\right)\geqq \frac{1}{4}\right),$$

where

$$\begin{array}{cc}\hfill \mathcal{G}\left(w\right)={\mathcal{F}}^{-1}\left(w\right)& =w-a_2{w}^2+\left(2a_2^2-a_3\right)w^3\hfill \\ \hfill & -\left(5a_2^3-5a_2{a}_3+a_4\right)w^4+\cdots (w\in \mathsf{\Lambda }),\hfill \end{array}$$

(3)

the function $\mathcal{G}$ being the analytic extension of ${\mathcal{F}}^{-1}$ to $\mathsf{\Lambda }$.

As we mentioned above, a function $\mathcal{F}\in \mathcal{A}$ is said to be bi-univalent in $\mathsf{\Lambda }$ if both $\mathcal{F}$ and its inverse ${\mathcal{F}}^{-1}$ are univalent in $\mathsf{\Lambda }$. Let $\mathsf{\Sigma }$ denote the class of analytic and bi-univalent functions in $\mathsf{\Lambda }$ given by (1). Remarkably, in their pioneering work in the year 2010, Srivastava et al. [5] actually revived the study of analytic and bi-univalent functions in latter years. Moreover, the current literature is flooded by many sequels to their paper [5]. We recall from [5] that the following functions:

$${\mathcal{F}}_1\left(z\right)={\displaystyle \frac{z}{1-z}},{\mathcal{F}}_2\left(z\right)={\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{1+z}{1-z}}\right)\mathrm{and}{\mathcal{F}}_3\left(z\right)=-log(1-z),$$

together with their corresponding inverse functions:

$${\mathcal{F}}_1^{-1}\left(w\right)={\displaystyle \frac{w}{1+w}},{\mathcal{F}}_2^{-1}\left(w\right)={\displaystyle \frac{e^{2w}-1}{e^{2w}+1}}\mathrm{and}{\mathcal{F}}_3^{-1}\left(w\right)={\displaystyle \frac{e^w-1}{e^w}},$$

are elements of $\mathsf{\Sigma }$ (see [5]). For a brief history and interesting examples of functions in the class $\mathsf{\Sigma }$, see the work of Brannan et al. [2]. Brannan and Taha [3], and also the more recent work of Srivastava et al. [5]) who introduced certain subclasses of the bi-univalent function class $\mathsf{\Sigma }$ similar to the familiar subclasses ${\mathcal{S}}^{*}\left(\gamma \right)$ and $\mathcal{K}\left(\gamma \right)$ of starlike and convex functions of order $\gamma$$\left(0\leqq \gamma <1\right)$ in $\mathsf{\Lambda }$, respectively.

A function $\mathcal{F}:\mathsf{\Lambda }\to \mathbb{C}$ is said to belong to the family ${\mathcal{C}}_0\left(\gamma \right)$ if $\mathcal{F}$ satisfies the following conditions:

(i)

$\mathcal{F}$ is analytic in $\mathsf{\Lambda }$ with the standard normalization given by

$$\mathcal{F}\left(0\right)={\mathcal{F}}^{\prime }\left(0\right)-1=0.$$

(ii)

$\mathcal{F}$ maps $\mathsf{\Lambda }$ conformally onto a set whose complement with respect to $\mathbb{C}$ is convex.

(iii)

The opening angle of $\mathcal{F}\left(\mathsf{\Lambda }\right)$ at ∞ is less than or equal to $\pi \gamma \left(\gamma \in (1,2]\right).$

The class of concave univalent functions in $\mathsf{\Lambda }$ is usually represented by the notation ${\mathcal{C}}_0\left(\gamma \right)$. For a detailed description of concave functions, see [14,18]. The following inequality:

$$\Re \left(1+\frac{z{\mathcal{F}}^{″}\left(z\right)}{{\mathcal{F}}^{\prime }\left(z\right)}\right)<0(z\in \mathsf{\Lambda })$$

was used by Bhowmik et al. [19] in order to prove that an analytic function $\mathcal{F}$ maps $\mathsf{\Lambda }$ onto an angled concave domain $\pi \gamma$ if and only if

$$\Re \left(P_{\mathcal{F}}\left(z\right)\right)>0(z\in \mathsf{\Lambda }),$$

where

$$P_{\mathcal{F}}\left(z\right)=\frac{2}{\gamma -1}\left(\frac{\left(\gamma +1\right)\left(1+z\right)}{2\left(1-z\right)}-1-\frac{z{\mathcal{F}}^{″}\left(z\right)}{{\mathcal{F}}^{\prime }\left(z\right)}\right).$$

The existing literature contains a number of investigations on various subclasses of the class ${\mathcal{C}}_0\left(\gamma \right)$ of concave univalent functions in $\mathsf{\Lambda }$ (see [12,16]).

Next, we denote by ${\left(\lambda \right)}_t$ the Pochhammer symbol or the shifted factorial, since

$${\left(1\right)}_t=t!(t\in {\mathbb{N}}_0).$$

It is defined for $\lambda ,\nu \in \mathbb{C}$, and in terms of the familiar Gamma function, by

$${\left(\lambda \right)}_t:=\frac{\mathsf{\Gamma }(\lambda +t)}{\mathsf{\Gamma }\left(\lambda \right)}=\left\{\begin{array}{cc}1\hfill & (\nu =0;\lambda \in \mathbb{C}\setminus \{0\left\}\right)\hfill \\ \\ \lambda (\lambda +1)\cdots (\lambda +t-1)\hfill & (t\in \mathbb{N};\lambda \in \mathbb{C}).\hfill \end{array}\right.$$

Here, and in what follows, it is understood conventionally that ${\left(0\right)}_0:=1$ and assumed tacitly that the $\mathsf{\Gamma }$-quotient exists.

In terms of the above-defined Pochhammer symbol ${\left(\lambda \right)}_t$, the generalized hypergeometric function ${}_{p}F_q(p,q\in {\mathbb{N}}_0)$, with p numerator parameters

$${\alpha }_j\in \mathbb{C}(j=1,\cdots ,p)$$

and q denominator parameters

$${\beta }_j\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}(j=1,\cdots ,q),$$

is given by

$$\begin{array}{cc}\hfill {}_{p}F_q\left[\begin{array}{c}\hfill {\alpha }_1,\cdots ,{\alpha }_p;\\ \\ \hfill {\beta }_1,\cdots ,{\beta }_q;\end{array}z\right]& :=\sum _{t=0}^{\infty }\frac{{\left({\alpha }_1\right)}_t\cdots {\left({\alpha }_p\right)}_t}{{\left({\beta }_1\right)}_t\cdots {\left({\beta }_q\right)}_t}\frac{z^t}{t!}\hfill \\ \hfill & =:{}_{p}F_q\left({\alpha }_1,\cdots ,{\alpha }_p;{\beta }_1,\cdots ,{\beta }_q;z\right),\hfill \end{array}$$

(4)

in which the infinite series

(i)

converges absolutely for $\left|z\right|<\infty$ if $p\leqq q,$

(ii)

converges absolutely for $\left|z\right|<1$ if $p=q+1,$ and

(iii)

diverges for all z$(z\ne 0)$ if $p>q+1.$

For $p-1=q=1$, the Equation (4) defines the Gauss hypergeometric function ${}_{2}F_1,$ given by

$${}_{2}F_1\left({\alpha }_1,{\alpha }_2;{\beta }_1;z\right):=\sum _{t=0}^{\infty }\frac{{\left({\alpha }_1\right)}_t{\left({\alpha }_2\right)}_t}{{\left({\beta }_1\right)}_t}\frac{z^t}{t!}(z\in \mathsf{\Lambda }),$$

so that, by the principle of confluence, we have the confulent hypergeometric function ${}_{1}F_1$ defined by (see, for details, [20,21])

$$\begin{array}{cc}\hfill F(\alpha ;\beta ;z)& :={}_{1}F_1({\alpha }_1;{\beta }_1;z)=\underset{{\alpha }_1\to \infty }{lim}\left\{{}_{2}F_1\left({\alpha }_1,{\alpha }_2;{\beta }_1;\frac{z}{{\alpha }_1}\right)\right\}\hfill \\ \hfill & =\sum _{t=0}^{\infty }\frac{{\left(\alpha \right)}_t}{{\left(\beta \right)}_t}\frac{z^t}{t!}\left(z,\alpha \in \mathbb{C};\beta \in \mathbb{C}\setminus \left\{0,-1,-2,\cdots \right\}\right).\hfill \end{array}$$

Recently, Porwal and Kumar [22] (see also [23,24]) introduced the confluent hypergeometric distribution (CHD) whose probability mass function is given by

$$P\left(t\right)=\frac{{\left(\alpha \right)}_t}{{\left(\beta \right)}_{t}t!F(\alpha ;\beta ;m)}{m}^t\left(\alpha ,\beta ,m>0;t=0,1,2,\cdots \right).$$

On the other hand, El-Deeb [23] introduced the following series $\mathcal{I}(\alpha ;\beta ;m;z)$ whose coefficients are probabilities of the above-defined confluent hypergeometric distribution:

$$\mathcal{I}(\alpha ;\beta ;m;z)=z+\sum _{t=2}^{\infty }\frac{{\left(\alpha \right)}_{t-1}{m}^{t-1}}{{\left(\beta \right)}_{t-1}\left(t-1\right)!F(\alpha ;\beta ;m)}{z}^t\left(\alpha ,\beta ,m>0\right).$$

(5)

El-Deeb [23] also defined the following linear operator $Q^{\alpha ;\beta ;m}\mathcal{F}:\mathcal{A}\to \mathcal{A}$:

$$\begin{array}{cc}\hfill Q^{\alpha ;\beta ;m}\mathcal{F}\left(z\right)& =\mathcal{I}(\alpha ;\beta ;m;z)\ast \mathcal{F}\left(z\right)\hfill \\ \hfill & =z+\sum _{t=2}^{\infty }\frac{{\left(\alpha \right)}_{t-1}{m}^{t-1}}{{\left(\beta \right)}_{t-1}\left(t-1\right)!F(\alpha ;\beta ;m)}{a}_t{z}^t\left(b,c,m>0\right).\hfill \end{array}$$

Now, by making use of the binomial series:

$${\left(1-z\right)}^r=\sum _{j=0}^r\left(\genfrac{}{}{0pt}{}{r}{j}\right){(-z)}^j(r\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}),$$

we introduce the following linear differential operator for $\mathcal{F}\in \mathcal{A}$ given by (1):

$${\mathcal{D}}_r^{\delta ,0,\alpha ,\beta ,m}\mathcal{F}\left(z\right)=Q^{\alpha ;\beta ;m}\mathcal{F}\left(z\right),$$

(6)

$$\begin{array}{cc}\hfill {\mathcal{D}}_r^{\delta ,1,\alpha ,\beta ,m}\mathcal{F}\left(z\right)& ={\mathcal{D}}_r^{\delta ,\alpha ,\beta ;m}\mathcal{F}\left(z\right)={\left(1-\delta \right)}^r{Q}^{\alpha ;\beta ;m}\mathcal{F}\left(z\right)\hfill \\ \hfill & +\left[1-{\left(1-\delta \right)}^r\right]z{\left(Q^{\alpha ;\beta ;m}\mathcal{F}\right)}^{\prime }\left(z\right)\hfill \\ \hfill & =z+\sum _{t=2}^{\infty }\left[1+\left(t-1\right)c^r\left(\delta \right)\right]\left(\frac{{\left(\alpha \right)}_{t-1}{m}^{t-1}}{{\left(\beta \right)}_{t-1}\left(t-1\right)!F(\alpha ;\beta ;m)}\right)a_t{z}^t\hfill \end{array}$$

(7)

and

$$\begin{array}{cc}\hfill {\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}\mathcal{F}\left(z\right)& ={\mathcal{D}}_r^{\delta ,\alpha ,\beta ,m}\left({\mathcal{D}}_r^{\delta ,n-1,b,c,m}\mathcal{F}\left(z\right)\right)\hfill \\ \hfill & ={\left(1-\delta \right)}^r{\mathcal{D}}_r^{\delta ,n-1,b,c,m}\mathcal{F}\left(z\right)+\left[1-{\left(1-\delta \right)}^r\right]z{\left({\mathcal{D}}_r^{\delta ,n-1,b,c,m}\mathcal{F}\left(z\right)\right)}^{\prime }\hfill \\ \hfill & =z+\sum _{t=2}^{\infty }{\left[1+\left(t-1\right)c^r\left(\delta \right)\right]}^n\left(\frac{{\left(\alpha \right)}_{t-1}{m}^{t-1}}{{\left(\beta \right)}_{t-1}\left(t-1\right)!F(\alpha ;\beta ;m)}\right)a_t{z}^t\hfill \\ \hfill & =z+\sum _{t=2}^{\infty }{\psi }_t{a}_t{z}^t\hfill \end{array}$$

(8)

$$\left(\delta >0;\alpha ,\beta ,m>0;r,n\in {\mathbb{N}}_0\right),$$

where

$${\psi }_t={\left[1+\left(t-1\right)c^r\left(\delta \right)\right]}^n\left(\frac{{\left(\alpha \right)}_{t-1}{m}^{t-1}}{{\left(\beta \right)}_{t-1}\left(t-1\right)!F(\alpha ;\beta ;m)}\right)$$

(9)

and

$$c^r\left(\delta \right)=-\sum _{j=1}^r\left(\genfrac{}{}{0pt}{}{r}{j}\right){(-\delta )}^j\left(r\in \mathbb{N}\right).$$

(10)

We find from (8) that

$$c^r\left(\delta \right)z{\left({\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}\mathcal{F}\left(z\right)\right)}^{\prime }={\mathcal{D}}_r^{\delta ,n+1,\alpha ,\beta ,m}\mathcal{F}\left(z\right)-\left[1-c^r\left(\delta \right)\right]{\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}\mathcal{F}\left(z\right).$$

(11)

Remark 1.

Each of the following special cases is worthy of note:

(i)

Putting $n=0,$ we obtain

$${\mathcal{D}}_r^{\delta ,0,\alpha ,\beta ,m}=Q^{\alpha ;\beta ;m},$$

where $Q^{b;c;m}$ is given by

$$Q^{\alpha ;\beta ;m}\mathcal{F}\left(z\right)=z+\sum _{t=2}^{\infty }\left(\frac{{\left(\alpha \right)}_{t-1}{m}^{t-1}}{{\left(\beta \right)}_{t-1}\left(t-1\right)!F(\alpha ;\beta ;m)}\right)a_t{z}^t;$$

(12)

(ii)

Putting $r=0,$ we obtain

$${\mathcal{D}}_0^{\delta ,n,\alpha ,\beta ,m}=:{\mathcal{I}}^{n,\alpha ,\beta ,m},$$

where ${\mathcal{I}}^{n,\alpha ,\beta ,m}$ is given by

$${\mathcal{I}}^{n,\alpha ,\beta ,m}\mathcal{F}\left(z\right)=z+\sum _{t=2}^{\infty }{\left(1+\left(t-1\right)\right]}^n\left(\frac{{\left(\alpha \right)}_{t-1}{m}^{t-1}}{{\left(\beta \right)}_{t-1}\left(t-1\right)!F(\alpha ;\beta ;m)}\right)a_t{z}^t.$$

(13)

Definition 1.

We define the functions $h,p:\mathsf{\Lambda }\to \mathbb{C},$ so that

$$min\left\{\Re \left(h\left(z\right)\right),\Re \left(p\left(z\right)\right)\right\}>0$$

and

$$h\left(0\right)=p\left(0\right)=1,$$

that is,

$$h\left(z\right)=1+\sum _{n=1}^{\infty }{h}_n{z}^n\mathit{and}p\left(z\right)=1+\sum _{n=1}^{\infty }{p}_n{z}^n(z\in \mathsf{\Lambda }).$$

(14)

Under the above-mentioned conditions, the functions h and p are said to belong to the class $\mathcal{P}$.

The subclass ${\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right)$ of bi-concave functions in $\mathsf{\Lambda }$ is now introduced by Definition 2 below.

Definition 2.

A function $\mathcal{F}$ of the form (1) is said to be in the class ${\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right)$ of bi-concave functions $\mathcal{F}\in \mathsf{\Sigma }$ if the following conditions are satisfied:

$$\mathcal{F}\in \mathsf{\Sigma }\mathit{and}\frac{2}{\gamma -1}\left(\frac{\left(\gamma +1\right)\left(1+z\right)}{2\left(1-z\right)}-1-\frac{z{\left({\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}\mathcal{F}\left(z\right)\right)}^{″}}{{\left({\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}\mathcal{F}\left(z\right)\right)}^{\prime }}\right)\in h(\mathsf{\Lambda })$$

(15)

and

$$\mathcal{F}\in \mathsf{\Sigma }\mathit{and}\frac{2}{\gamma -1}\left(\frac{\left(\gamma +1\right)\left(1+w\right)}{2\left(1-w\right)}-1-\frac{w{\left({\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}\mathcal{G}\left(w\right)\right)}^{″}}{{\left({\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}\mathcal{G}\left(w\right)\right)}^{\prime }}\right)\in p(\mathsf{\Lambda }),$$

(16)

where $\delta >0,$ $\alpha ,\beta ,m>0,$ $r\in \mathbb{N},$ $n\in {\mathbb{N}}_0$ and $\gamma \in (1,2]$. Moreover, the function $\mathcal{G}$ in the condition (16) is the analytic extension of ${\mathcal{F}}^{-1}$ to $\mathsf{\Lambda },$ which is given by the Equation (3), and the functions $hp\in \mathcal{P}$ are defined and used as in Definition 1.

Remark 2.

Our present investigation of the coefficient estimate problems makes use of Definition 2 involving the functions p and h of the class $\mathcal{P}$ for which the following important coefficient inequalities hold true for functions $p,h\in \mathcal{P}$ given by (14):

$$|p_n|\leqq 2\mathit{and}{h}_n|\leqq 2(n\in \mathbb{N}:=\{1,2,3,\cdots \}),$$

wherein the equality is valid for the extremal functions:

$$h\left(z\right)=\frac{1+z}{1-z}=1+2z+2z^2+\cdots \mathit{and}p\left(z\right)=\frac{1-z}{1+z}=1-2z+2z^2-\cdots (z\in \mathsf{\Lambda }).$$

Several interesting examples of the functions p and $h,$ which are motivated by the above extremal functions, and also such examples as those given below:

$${\left(\frac{1+z}{1-z}\right)}^{\alpha }\mathit{and}{\left(\frac{1-z}{1+z}\right)}^{\alpha }(0<\alpha \leqq 1)$$

and

$$\frac{1+(1-2\sigma )z}{1-z}\mathit{and}\frac{1-(1-2\sigma )z}{1+z}(0<\sigma \leqq 1),$$

will be considered in the corollaries and consequences of our main results in Theorem 1 below (see also Remark 4).

It may be possible to suitably replace the functions $p,h\in \mathcal{P}$ in Definition 2 by members of other function classes defined in Λ. However, the investigation of the analogous coefficient problems for the correspondingly-defined bi-concave function classes will presumably be much more involved than what we have presented in this article.

Remark 3.

Each of the following special cases is worthy of note:

(i)

Putting $n=0,$ we obtain

$${\mathcal{D}}_r^{\delta ,0,\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right)=:{\mathcal{G}}^{\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right),$$

where ${\mathcal{G}}^{\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right)$ represents the functions $\mathcal{F}\in \mathsf{\Sigma }$ that satisfy the conditions in (15) and (16) with ${\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}$ replaced by $Q^{\alpha ;\beta ;m},$ which is given by (12);

(ii)

Putting $r=0,$ we obtain

$${\mathcal{D}}_0^{\delta ,n,\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right)=:{\mathcal{R}}^{n,\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right),$$

where ${\mathcal{R}}^{n,\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right)$ represents the functions $\mathcal{F}\in \mathsf{\Sigma }$ that satisfy the conditions in (15) and (16) with ${\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}$ replaced by ${\mathcal{I}}^{n,\alpha ,\beta ,m},$ which is given by (13).

2. Coefficient Bounds for the Bi-Concave Function Class ${\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right)$

Throughout this section, we assume that $\delta >0,$ $\alpha ,\beta ,m>0,$ $t\in \mathbb{N}$ and $n\in {\mathbb{N}}_0.$

Theorem 1.

If the function $\mathcal{F}$ given by (1) belongs to the class ${\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right),$ and if $\gamma \in (1,2],$ then

$$\left|a_2\right|\leqq min\{\sqrt{\frac{{\left(\gamma +1\right)}^2}{4{\psi }_2^2}+\frac{{\left(\gamma -1\right)}^2\left({\left|h^{\prime }\left(0\right)\right|}^2+{\left|p^{\prime }\left(0\right)\right|}^2\right)}{32{\psi }_2^2}+\frac{\left({\gamma }^2-1\right)\left(\left|h^{\prime }\left(0\right)\right|+\left|p^{\prime }\left(0\right)\right|\right)}{8{\psi }_2^2}},$$

$$\sqrt{\frac{\left(\gamma -1\right)\left(\left|h^{″}\left(0\right)\right|+\left|p^{″}\left(0\right)\right|\right)}{16\left(2{\psi }_2^2-3{\psi }_3\right)}+\frac{\gamma +1}{2\left(2{\psi }_2^2-3{\psi }_3\right)}}\}$$

(17)

and

$$\left|a_3\right|\leqq min\{\frac{8{\left(\gamma +1\right)}^2+{\left(\gamma -1\right)}^2\left({\left|h^{\prime }\left(0\right)\right|}^2+{\left|p^{\prime }\left(0\right)\right|}^2\right)}{32{\psi }_2^2}+\frac{\left({\gamma }^2-1\right)\left(\left|h^{\prime }\left(0\right)\right|+\left|p^{\prime }\left(0\right)\right|\right)}{8{\psi }_2^2},$$

$$\begin{array}{cc}\hfill & +\frac{\left(\gamma -1\right)\left(\left|h^{″}\left(0\right)\right|+\left|p^{″}\left(0\right)\right|\right)}{48{\psi }_3}\frac{\left(\gamma -1\right)\left(3{\psi }_3-{\psi }_2^2\right)\left|h^{″}\left(0\right)\right|+\left(\gamma -1\right){\psi }_2^2\left|p^{″}\left(0\right)\right|}{24{\psi }_3\left(2{\psi }_2^2-3{\psi }_3\right)}\hfill \\ \hfill & +\frac{\gamma +1}{2\left(2{\psi }_2^2-3{\psi }_3\right)}\},\hfill \end{array}$$

(18)

where ${\psi }_t$$(t\in \{2,3\left\}\right)$ is given by (9).

Proof. 

If $\mathcal{F}\in {\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right),$ then it follows from (15) and (16) that

$$\frac{2}{\gamma -1}\left(\frac{\left(\gamma +1\right)\left(1+z\right)}{2\left(1-z\right)}-1-\frac{z{\left({\mathcal{D}}_t^{\delta ,n,b,c,m}\mathcal{F}\left(z\right)\right)}^{″}}{{\left({\mathcal{D}}_t^{\delta ,n,b,c,m}\mathcal{F}\left(z\right)\right)}^{\prime }}\right)=h\left(z\right)$$

(19)

and

$$\frac{2}{\gamma -1}\left(\frac{\left(\gamma +1\right)\left(1+w\right)}{2\left(1-w\right)}-1-\frac{w{\left({\mathcal{D}}_t^{\delta ,n,b,c,m}\mathcal{G}\left(w\right)\right)}^{″}}{{\left({\mathcal{D}}_t^{\delta ,n,b,c,m}\mathcal{G}\left(w\right)\right)}^{\prime }}\right)=p\left(w\right),$$

(20)

where the functions h and p satisfy the conditions in Definition 1 and the function $\mathcal{G}$ in the assertion (20) is the analytic extension of ${\mathcal{F}}^{-1}$ to $\mathsf{\Lambda },$ which is given by the Equation (3). Furthermore, the functions $h\left(z\right)$ and $p\left(w\right)$ have the following Taylor-Maclaurin series expansions:

$$h\left(z\right)=1+h_{1}z+h_2{z}^2+\cdots$$

(21)

and

$$p\left(w\right)=1+p_{1}w+p_2{w}^2+\cdots ,$$

(22)

respectively.

In view of the above Equations (21) and (22), we compare the coefficients of z and w in (19) and (20). We thus find that

$$\frac{2\left[\left(\gamma +1\right)-2{\psi }_2{a}_2\right]}{\gamma -1}=h_1,$$

(23)

$$\frac{2\left[\left(\gamma +1\right)+4{\psi }_2^2{a}_2^2-6{\psi }_3{a}_3\right]}{\gamma -1}=h_2,$$

(24)

$$-\frac{2\left[\left(\gamma +1\right)-2{\psi }_2{a}_2\right]}{\gamma -1}=p_1$$

(25)

and

$$\frac{2\left[\left(\gamma +1\right)+4{\psi }_2^2{a}_2^2-6{\psi }_3\left(2a_2^2-a_3\right)\right]}{\gamma -1}=p_2.$$

(26)

Now, by using (23) and (25), we obtain

$$h_1=-p_1.$$

(27)

From (23), we can write

$$a_2=\frac{\gamma +1}{2{\psi }_2}-\frac{\gamma -1}{4{\psi }_2}{h}_1.$$

(28)

Upon squaring (23) and (25), if we add the resulting equations, we get

$$a_2^2=\frac{{\left(\gamma +1\right)}^2}{4{\psi }_2^2}+\frac{{\left(\gamma -1\right)}^2\left(h_1^2+p_1^2\right)}{32{\psi }_2^2}-\frac{\left({\gamma }^2-1\right)}{8{\psi }_2^2}\left(h_1-p_1\right)$$

(29)

Adding (24) and (26), we have

$$a_2^2=\frac{\left(\gamma -1\right)\left(h_2+p_2\right)}{8\left(2{\psi }_2^2-3{\psi }_3\right)}-\frac{\gamma +1}{2\left(2{\psi }_2^2-3{\psi }_3\right)}.$$

(30)

By taking the moduli in (29) and (30), we conclude that

$$\left|a_2\right|\leqq \sqrt{\frac{{\left(\gamma +1\right)}^2}{4{\psi }_2^2}+\frac{{\left(\gamma -1\right)}^2\left({\left|h^{\prime }\left(0\right)\right|}^2+{\left|p^{\prime }\left(0\right)\right|}^2\right)}{32{\psi }_2^2}+\frac{\left({\gamma }^2-1\right)\left(\left|h^{\prime }\left(0\right)\right|+\left|p^{\prime }\left(0\right)\right|\right)}{8{\psi }_2^2}}$$

and

$$\left|a_2\right|\leqq \sqrt{\frac{\left(\gamma -1\right)\left(\left|h^{″}\left(0\right)\right|+\left|p^{″}\left(0\right)\right|\right)}{16\left(2{\psi }_2^2-3{\psi }_3\right)}+\frac{\gamma +1}{2\left(2{\psi }_2^2-3{\psi }_3\right)}},$$

which gives the bound for $\left|a_2\right|$ as we asserted in Theorem 1.

In order to find the bound for $\left|a_3\right|$, by subtracting (26) from (24), we get

$$a_3=a_2^2-\frac{\left(\gamma -1\right)\left(h_2-p_2\right)}{24{\psi }_3}.$$

(31)

Also, upon substituting the value of $a_2^2$ from (29) and (30) into (31), we obtain

$$a_3=\frac{{\left(\gamma +1\right)}^2}{4{\psi }_2^2}+\frac{{\left(\gamma -1\right)}^2\left(h_1^2+p_1^2\right)}{32{\psi }_2^2}-\frac{\left({\gamma }^2-1\right)}{8{\psi }_2^2}\left(h_1-p_1\right)-\frac{\left(\gamma -1\right)\left(h_2-p_2\right)}{24{\psi }_3}$$

(32)

and

$$a_3=\frac{\left(\gamma -1\right)\left(h_2+p_2\right)}{8\left(2{\psi }_2^2-3{\psi }_3\right)}-\frac{\gamma +1}{2\left(2{\psi }_2^2-3{\psi }_3\right)}-\frac{\left(\gamma -1\right)\left(h_2-p_2\right)}{24{\psi }_3}.$$

(33)

Finally, by taking the moduli in (32) and (33), we obtain

$$\begin{array}{cc}\hfill \left|a_3\right|& \leqq \frac{8{\left(\gamma +1\right)}^2+{\left(\gamma -1\right)}^2\left({\left|h^{\prime }\left(0\right)\right|}^2+{\left|p^{\prime }\left(0\right)\right|}^2\right)}{32{\psi }_2^2}+\frac{\left({\gamma }^2-1\right)\left(\left|h^{\prime }\left(0\right)\right|+\left|p^{\prime }\left(0\right)\right|\right)}{8{\psi }_2^2}\hfill \\ \hfill & +\frac{\left(\gamma -1\right)\left(\left|h^{″}\left(0\right)\right|+\left|p^{″}\left(0\right)\right|\right)}{48{\psi }_3}\hfill \end{array}$$

(34)

and

$$\left|a_3\right|\leqq \frac{\left(\gamma -1\right)\left(3{\psi }_3-{\psi }_2^2\right)\left|h^{″}\left(0\right)\right|+\left(\gamma -1\right){\psi }_2^2\left|p^{″}\left(0\right)\right|}{24{\psi }_3\left(2{\psi }_2^2-3{\psi }_3\right)}+\frac{\gamma +1}{2\left(2{\psi }_2^2-3{\psi }_3\right)}.$$

(35)

which completes the proof of Theorem 1. □

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

Corollary 1.

If the function $\mathcal{F}$ given by (1) belongs to the class ${\mathcal{R}}^{n,\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right),$ and if $\gamma \in (1,2],$ then

$$\begin{array}{cc}\hfill \left|a_2\right|& \leqq min\left\{\right(\frac{{\left[\beta \left(\gamma +1\right)F(\alpha ;\beta ;m)\right]}^2}{2^{2(n+1)}{\alpha }^2{m}^2}+\frac{{\left[\beta \left(\gamma -1\right)F(\alpha ;\beta ;m)\right]}^2\left({\left|h^{\prime }\left(0\right)\right|}^2+{\left|p^{\prime }\left(0\right)\right|}^2\right)}{2^{2(n+1)+3}{\alpha }^2{m}^2}\hfill \\ \hfill & +\frac{\left({\gamma }^2-1\right){\left[\beta F(\alpha ;\beta ;m)\right]}^2\left(\left|h^{\prime }\left(0\right)\right|+\left|p^{\prime }\left(0\right)\right|\right)}{2^{2(n+1)+1}{\alpha }^2{m}^2}{)}^{\frac{1}{2}},\hfill \\ \hfill & \sqrt{\frac{8\left(\gamma +1\right)+\left(\gamma -1\right)\left(\left|h^{″}\left(0\right)\right|+\left|p^{″}\left(0\right)\right|\right)}{16\left[2^{n+1}{\left(\frac{\alpha m}{\beta F(\alpha ;\beta ;m)}\right)}^2-3^{n+1}\left(\frac{{\left(\alpha \right)}_2{m}^2}{2{\left(\beta \right)}_{2}F(\alpha ;\beta ;m)}\right)\right]}}\}\hfill \end{array}$$

(36)

and

$$\begin{array}{cc}\hfill \left|a_3\right|& \leqq min\left\{\frac{{\left[\beta \left(\gamma -1\right)F(\alpha ;\beta ;m)\right]}^2}{2^{2(n+1)+3}{\alpha }^2{m}^2}\right[8{\left(\gamma +1\right)}^2+{\left(\gamma -1\right)}^2\left({\left|h^{\prime }\left(0\right)\right|}^2+{\left|p^{\prime }\left(0\right)\right|}^2\right)\hfill \\ \hfill & +4\left({\gamma }^2-1\right)\left(\left|h^{\prime }\left(0\right)\right|+\left|p^{\prime }\left(0\right)\right|\right)]+\frac{\left(\gamma -1\right){\left(\beta \right)}_{2}F(\alpha ;\beta ;m)\left(\left|h^{″}\left(0\right)\right|+\left|p^{″}\left(0\right)\right|\right)}{\left(8\right)\left(3^{n+1}\right){\left(\alpha \right)}_2{m}^2},\hfill \\ \hfill & \frac{\left(\gamma -1\right)\left[3^{n+1}\left(\frac{{\left(\alpha \right)}_2{m}^2}{2{\left(\beta \right)}_{2}F(\alpha ;\beta ;m)}\right)-2^n{\left(\frac{\alpha m}{\beta F(\alpha ;\beta ;m)}\right)}^2\right]\left|h^{″}\left(0\right)\right|+2^n\left(\gamma -1\right){\left(\frac{\alpha m}{\beta F(\alpha ;\beta ;m)}\right)}^2\left|p^{″}\left(0\right)\right|}{8\left(3^{n+1}\right)\left(\frac{{\left(\alpha \right)}_2{m}^2}{2{\left(\beta \right)}_{2}F(\alpha ;\beta ;m)}\right)\left[2^{n+1}{\left(\frac{\alpha m}{\beta F(\alpha ;\beta ;m)}\right)}^2-3^{n+1}\left(\frac{{\left(\alpha \right)}_2{m}^2}{2{\left(\beta \right)}_{2}F(\alpha ;\beta ;m)}\right)\right]}\hfill \\ \hfill & +\frac{\gamma +1}{2\left[2^{n+1}{\left(\frac{\alpha m}{\beta F(\alpha ;\beta ;m)}\right)}^2-3^{n+1}\left(\frac{{\left(\alpha \right)}_2{m}^2}{2{\left(\beta \right)}_{2}F(\alpha ;\beta ;m)}\right)\right]}\}.\hfill \end{array}$$

(37)

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

Corollary 2.

If the function $\mathcal{F}$ given by (1) belongs to the class ${\mathcal{G}}^{\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right),$ and if $\gamma \in (1,2],$ then

$$\begin{array}{cc}\hfill \left|a_2\right|& \leqq min\left\{\right(\frac{{\left[c\left(\gamma +1\right)F(\alpha ;\beta ;m)\right]}^2}{4b^2{m}^2}+\frac{{\left[c\left(\gamma -1\right)F(\alpha ;\beta ;m)\right]}^2\left({\left|h^{\prime }\left(0\right)\right|}^2+{\left|p^{\prime }\left(0\right)\right|}^2\right)}{32b^2{m}^2}\hfill \\ \hfill & +\frac{\left({\gamma }^2-1\right){\left[cF(\alpha ;\beta ;m)\right]}^2\left(\left|h^{\prime }\left(0\right)\right|+\left|p^{\prime }\left(0\right)\right|\right)}{8b^2{m}^2}{)}^{\frac{1}{2}},\hfill \\ \hfill & \sqrt{\frac{8\left(\gamma +1\right)+\left(\gamma -1\right)\left(\left|h^{″}\left(0\right)\right|+\left|p^{″}\left(0\right)\right|\right)}{16\left[2{\left(\frac{bm}{cF(\alpha ;\beta ;m)}\right)}^2-3\left(\frac{{\left(b\right)}_2{m}^2}{2{\left(c\right)}_{2}F(\alpha ;\beta ;m)}\right)\right]}}\}\hfill \end{array}$$

(38)

and

$$\begin{array}{cc}\hfill \left|a_3\right|& \leqq min\{\frac{{\left[\beta \left(\gamma -1\right)F(\alpha ;\beta ;m)\right]}^2\left[8{\left(\gamma +1\right)}^2+{\left(\gamma -1\right)}^2\left({\left|h^{\prime }\left(0\right)\right|}^2+{\left|p^{\prime }\left(0\right)\right|}^2\right)\right]}{32{\alpha }^2{m}^2}\hfill \\ \hfill & +\frac{{\left[\beta \left(\gamma -1\right)F(\alpha ;\beta ;m)\right]}^2\left[4\left({\gamma }^2-1\right)\left(\left|h^{\prime }\left(0\right)\right|+\left|p^{\prime }\left(0\right)\right|\right)\right]}{32{\alpha }^2{m}^2}\hfill \\ \hfill & +\frac{\left(\gamma -1\right){\left(\beta \right)}_{2}F(\alpha ;\beta ;m)\left(\left|h^{″}\left(0\right)\right|+\left|p^{″}\left(0\right)\right|\right)}{24{\left(\alpha \right)}_2{m}^2},\hfill \\ \hfill & \frac{\left(\gamma -1\right)\left(3\left[\frac{{\left(\alpha \right)}_2{m}^2}{2{\left(\beta \right)}_{2}F(\alpha ;\beta ;m)}-{\left(\frac{\alpha m}{\beta F(\alpha ;\beta ;m)}\right)}^2\right]\right)\left|h^{″}\left(0\right)\right|+\left(\gamma -1\right){\left(\frac{\alpha m}{\beta F(\alpha ;\beta ;m)}\right)}^2\left|p^{″}\left(0\right)\right|}{24\left(\frac{{\left(\alpha \right)}_2{m}^2}{2{\left(\beta \right)}_{2}F(\alpha ;\beta ;m)}\right)\left[2{\left(\frac{\alpha m}{\beta F(\alpha ;\beta ;m)}\right)}^2-3\left(\frac{{\left(\alpha \right)}_2{m}^2}{2{\left(\beta \right)}_{2}F(\alpha ;\beta ;m)}\right)\right]}\hfill \\ \hfill & +\frac{\gamma +1}{2\left[2{\left(\frac{\alpha m}{\beta F(\alpha ;\beta ;m)}\right)}^2-3\left(\frac{{\left(\alpha \right)}_2{m}^2}{2{\left(\beta \right)}_{2}F(\alpha ;\beta ;m)}\right)\right]}\}.\hfill \end{array}$$

(39)

Remark 4.

The functions h and $p,$ which we introduced in Definition 1 and used in Theorem 1, as well as in Corollarys 1 and 2 above, play a significant role in our investigation. For example, by appropriately specializing the functions h and p in Theorem 1, we are led to Corollarys 3 and 4 below.

Corollary 3.

If the functions h and p are given by

$$h\left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\alpha }=1+2\alpha z+2{\alpha }^2{z}^2+\cdots \left(0<\alpha \leqq 1\right)$$

and

$$p\left(z\right)={\left(\frac{1-z}{1+z}\right)}^{\alpha }=1-2\alpha z+2{\alpha }^2{z}^2-\cdots \left(0<\alpha \leqq 1\right),$$

and if the function $\mathcal{F}$ given by (1) belongs to the class ${\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right)$ and $\gamma \in (1,2],$ then

$$\left|a_2\right|\leqq min\left\{\sqrt{\frac{{\left(\gamma +1\right)}^2+{\left(\gamma -1\right)}^2{\alpha }^2+2\left({\gamma }^2-1\right)\alpha }{4{\psi }_2^2}},\sqrt{\frac{\left(\gamma +1\right)+\left(\gamma -1\right){\alpha }^2}{2\left(2{\psi }_2^2-3{\psi }_3\right)}}\right\}$$

(40)

and

$$\begin{array}{cc}\hfill \left|a_3\right|& \leqq min\{\frac{{\left(\gamma +1\right)}^2+{\left(\gamma -1\right)}^2{\alpha }^2+2\left({\gamma }^2-1\right)\alpha }{4{\psi }_2^2}+\frac{\left(\gamma -1\right){\alpha }^2}{6{\psi }_3},\hfill \\ \hfill & \frac{\left|\left(\gamma -1\right)\left(3{\psi }_3-{\psi }_2^2\right)\right|{\alpha }^2+\left(\gamma -1\right){\psi }_2^2{\alpha }^2}{6{\psi }_3\left(2{\psi }_2^2-3{\psi }_3\right)}+\frac{\gamma +1}{2\left(2{\psi }_2^2-3{\psi }_3\right)}\},\hfill \end{array}$$

(41)

where ${\psi }_t$$(t\in \{2,3\left\}\right)$ is given by (9).

Corollary 4.

If the functions h and p are given by

$$h\left(z\right)=\frac{1+\left(1-2\sigma \right)z}{1-z}=1+2\left(1-\sigma \right)z+2\left(1-\sigma \right)z^2+\cdots \left(0\leqq \sigma <1\right)$$

and

$$p\left(z\right)=\frac{1-\left(1-2\sigma \right)z}{1+z}=1-2\left(1-\sigma \right)z+2\left(1-\sigma \right)z^2+\cdots \left(0\leqq \sigma <1\right),$$

and if the function $\mathcal{F}$ given by (1) belongs to the class ${\mathcal{D}}_r^{\delta ,n,\alpha ,\beta ,m}{\mathcal{C}}_0\left(\gamma \right)$ and $\gamma \in (1,2],$ then

$$\begin{array}{cc}\hfill \left|a_2\right|& \leqq min\{\sqrt{\frac{{\left(\gamma +1\right)}^2+{\left(\gamma -1\right)}^2{\left(1-\sigma \right)}^2+2\left({\gamma }^2-1\right)\left(1-\sigma \right)}{4{\psi }_2^2}},\hfill \\ \hfill & \sqrt{\frac{\left(\gamma -1\right)\left(1-\sigma \right)-\left(\gamma +1\right)}{2\left(2{\psi }_2^2-3{\psi }_3\right)}}\}\hfill \end{array}$$

(42)

and

$$\begin{array}{cc}\hfill \left|a_3\right|& \leqq min\{\frac{8{\left(\gamma +1\right)}^2+{\left(\gamma -1\right)}^2\left({\left|h^{\prime }\left(0\right)\right|}^2+{\left|p^{\prime }\left(0\right)\right|}^2\right)}{32{\psi }_2^2}+\frac{\left({\gamma }^2-1\right)\left(\left|h^{\prime }\left(0\right)\right|+\left|p^{\prime }\left(0\right)\right|\right)}{8{\psi }_2^2}\hfill \\ \hfill & +\frac{\left(\gamma -1\right)\left(\left|h^{″}\left(0\right)\right|+\left|p^{″}\left(0\right)\right|\right)}{48{\psi }_3},\frac{\left(\gamma -1\right)\left(3{\psi }_3-{\psi }_2^2\right)\left|h^{″}\left(0\right)\right|+\left(\gamma -1\right){\psi }_2^2\left|p^{″}\left(0\right)\right|}{24{\psi }_3\left(2{\psi }_2^2-3{\psi }_3\right)}\hfill \\ \hfill & +\frac{\gamma +1}{2\left(2{\psi }_2^2-3{\psi }_3\right)}\},\hfill \end{array}$$

(43)

where ${\psi }_t$$(t\in \{2,3\left\}\right)$ is given by (9).

3. Concluding Remarks and Observations

In our present investigation, we have introduced and studied the properties of some new subclasses of the class of analytic and bi-concave functions in the open unit disk $\mathsf{\Lambda }$ by using the combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we have derived the estimates on the initial Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions belonging to the bi-concave function classes, which are introduced in this paper. As some of the interesting consequences of our main results in Theorem 1, we have deduced a set of four corollaries. Each of these results is potentially useful in motivating further research on various other subclasses of the class of normalized analytic (or meromorphic) and univalent (or multivalent) bi-concave functions.

In this article, we have considered the coefficient estimate problems for the initial Taylor-Maclaurin coefficients for functions in some specified subclasses of the class of analytic and bi-concave functions in $\mathsf{\Lambda }$. It would be of interest to investigate the applications of the Faber polynomial expansion method (see [25,26]) in order to tackle the coefficient estimate problems for the general Taylor-Maclaurin coefficients for these and other subclasses of the class of analytic and bi-concave functions (see, for details, [27,28,29]; see also [30] and the references to the earlier literature on the subject, which are cited in each of these works).