Bi-Univalency of m-Fold Symmetric Functions Associated with a Generalized Distribution

Abstract

The m-fold symmetric in terms of a generalized distribution series has not been considered in the literature. In this study, however, the authors investigated the bi-univalency of m-fold symmetric functions for the generalized distribution of two subclasses of analytic functions. The initial few coefficient bounds $\left|\frac{a_m}{S}\right|$ and $\left|\frac{a_{2m}}{S}\right|$ are obtained for the two subclasses of functions defined and the results serve as a new generalization in this direction.

1. Introduction and Preliminaries

Let $\Gamma$ represent the functions of the form

$$\begin{array}{c}\hfill f\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n\end{array}$$

(1)

and $D\subset \Gamma$ as the set of univalent functions in $U=\{z:|z|<1\}$ with conditions $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1$.

The starlike function, the convex function with interpretations $\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}>0,1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}>0$, and many others respectively are subclasses of the set D defined by (1), which have been used in several occasions by erudite scholars in different ways with different perspectives and their generated results are too numerous to discuss. For more information, see [1,2,3,4].

The Koebe one-quarter theorem states that the image of U under every function $f\in D$ contains a disk of radius $\frac{1}{4}$, where the function $f\in D$ has an inverse $f^{-1}$ which satisfies

$$\begin{array}{c}\hfill f^{-1}f\left(z\right)=z\end{array}$$

(2)

and

$$\begin{array}{c}\hfill f\left(f^{-1}\left(\omega \right)\right)=\omega \left(\left|\omega \right|<r_0\left(f\right),r_0\left(f\right)\ge \frac{1}{4}\right).\end{array}$$

(3)

Indeed, the inverse function of (1) is given by

$$\begin{array}{c}\hfill f^{-1}\left(\omega \right)=\omega -a_2{\omega }^2+\left(2a_2^2-a_3\right){\omega }^3-\left(5a_2^3-5a_2{a}_3+a_4\right){\omega }^4+\dots .\end{array}$$

(4)

Many studies have been performed on bi-univalent functions. See [5,6,7,8,9,10] and many other related studies for examples.

Given $f\in D$, there exists $h\left(z\right)={\left(f\left(z^m\right)\right)}^{\frac{1}{m}}$ as a univalent function that maps the unit disc U into a domain with m-fold symmetry, with $m\in \mathbb{N}$ and $z\in \mathbb{C}$. We refer to f as m-fold symmetric if it is of the form

$$\begin{array}{c}\hfill f\left(z\right)=z+\sum _{n=1}^{\infty }{a}_{mn+1}{z}^{mn+1}\end{array}$$

(5)

where $m\in \mathbb{N}$ and $z\in \mathbb{C}$ (see [11]).

Let $S_m$ be the class of the m-fold symmetric given by (5), so that class D becomes one-fold symmetric for $m=1$. Authors like Srivastava et al. [12], Wanas and Majeed [13], Oros and Cotîrlă [14], Sakar and Taşar [15], and Altınkaya and Yalçın [16] have considered different forms of such functions. For the normalized form of f in (5), its inverse is given below:

$$\begin{array}{c}\hfill g\left(w\right)=w-a_{m+1}{w}^{m+1}+\left((m+1)a_{m+1}^2-a_{2m+1}\right)w^{2m+1}\\ \hfill -\left(\frac{(m+1)(3m+2)a_{m+1}^3}{2}-(3m+2)a_{m+1}{a}_{2m+1}+a_{3m+1}\right)w^{3m+1}+\dots .\end{array}$$

(6)

and $f^{-1}=g$. We represent ${\sum }_m$ as the class of m-fold symmetric bi-univalent functions in U. Setting $m=1$ in (5), we obtain (4) of the class ∑. Also, recall that the class of the Caratheodory functions denoted by $\mathcal{P}$ is of the form

$$\begin{array}{c}\hfill p\left(z\right)=1+p_{1}z+p_2{z}^2+p_3{z}^3+\dots \end{array}$$

(7)

such that $Re\left(p\right(z\left)\right)>0$ and $p\left(0\right)=1$ for all $z\in U$. Then, $|p_n|\le 2$ for every $n=1,2,3,\dots$.

In respect of Pommerenke [11], the m-fold symmetric functions in the class $\mathcal{P}$ is of the form

$$\begin{array}{c}\hfill p\left(z\right)=1+p_m{z}^m+p_{2m}{z}^{2m}+p_{3m}{z}^{3m}+\dots .\end{array}$$

(8)

In recent time, generalized discrete probability distribution was introduced by Porwal [17]. Let S be defined by

$$\begin{array}{c}\hfill S=\sum _{n=0}^{\infty }{a}_n,\end{array}$$

where $a_n\ge 0$ for all $n\in \mathbb{N}$. Then, we have that

$$\begin{array}{c}\hfill p\left(n\right)=\frac{a_n}{S}n=0,1,2,\dots ,\end{array}$$

where $p\left(n\right)$ is a probability mass function with $p\left(n\right)\ge 0$ and ${\sum }_{n}p\left(n\right)=1$, see [18].

Let

$$\begin{array}{c}\hfill \gamma \left(x\right)=\sum _{n=0}^{\infty }{a}_n{x}^n.\end{array}$$

Then, from $S={\sum }_{n=0}^{\infty }{a}_n$, series $\gamma$ is convergent for both $\left|x\right|<1$ and $x=1$.

Porwal [17] pointed out the following definitions and derivations:

(i)

If X is a discrete random variable that takes values $x_1,x_2,x_3,\dots$ associated with probabilities $p_1,p_2,p_3,\dots$ then the expected X denoted by $E\left(X\right)$ is defined as

$$\begin{array}{c}\hfill E\left(X\right)=\sum _{n=1}^{\infty }{p}_n{z}^n.\end{array}$$

(ii)

The moment of a discrete probability distribution $\left(r^{t}h\right)$ about $x=0$ is defined by

$$\begin{array}{c}\hfill {\mu }_1^{\prime }=E\left(X^r\right),\end{array}$$

where ${\mu }_1^{\prime }$ is the mean of the distribution and the variance is given as

$$\begin{array}{c}\hfill {\mu }_2^{\prime }-\left({\mu }_1^{\prime }\right).\end{array}$$

(iii)

The moment about the origin is given as

$$\begin{array}{c}\hfill Mean=p_1^{\prime }=\frac{{\gamma }^{\prime }}{S};Variance={\mu }_2^{\prime }-\left({\mu }_1^{\prime }\right)=\frac{1}{S}\left[{{\gamma }^{\prime }}^{\prime }\left(1\right)+{\gamma }^{\prime }\left(1\right)-\frac{{\left({\gamma }^{\prime }\left(1\right)\right)}^2}{S}\right].\end{array}$$

Let

$$\begin{array}{c}\hfill M_x\left(t\right)=E\left(e^{xt}\right)\end{array}$$

and

$$\begin{array}{c}\hfill M_x\left(t\right)=\frac{\gamma \left(e^t\right)}{S}\end{array}$$

as probability functions (see [18]).

Porwal’s interest to give the derivations and definitions above is the introduction of the power series whose coefficients are probabilities of the generalized distribution of the form

$$\begin{array}{c}\hfill K_{\psi }\left(z\right)=z+\sum _{n=2}^{\infty }\frac{a_{n-1}}{S}{z}^n,\end{array}$$

(9)

where $S={\sum }_{n=0}^{\infty }{a}_n$. More details on generalized distribution can be found in [19,20]. Based on the available literature, the authors observed that the m-fold symmetric involving generalized distribution has not been considered. This prompts the research in this direction.

Using the analogue of the m-fold symmetric in (5) and (8), we want to establish that the m-fold symmetric in terms of generalized distribution is given as

$$\begin{array}{c}\hfill K_{\psi }\left(z\right)=z+\sum _{n=1}^{\infty }\frac{a_{mn}}{S}{z}^{mn+1}\end{array}$$

(10)

and its inverse gives

$$\begin{array}{c}\hfill G_{\psi }\left(\omega \right)=\omega -\frac{a_m}{S}{\omega }^{m+1}+\left(\frac{(m+1)a_m^2}{S^2}-\frac{a_{2m}}{S}\right){\omega }^{2m+1}\\ \hfill -\left(\frac{(m+1)(3m+2)a_m^3}{2S^3}-\frac{(3m+2)a_m{a}_{2m}}{S^2}+\frac{a_{3m}}{S}\right){\omega }^{3m+1}+\dots .\end{array}$$

(11)

Definition and Example of an m-Fold Symmetric Bi-Univalent Function in Geometric Function Theory

Consequently, we give the definition and an example of an m-fold symmetric bi-univalent function in Geometric Function Theory and its corresponding inverse function.

A function is said to be bi-univalent if it is univalent (injective) and its inverse is also univalent. An m-fold symmetric bi-univalent function is a special type of bi-univalent function that possesses a specific rotational symmetry of order m.

Let us consider the example of a 3-fold symmetric bi-univalent function:

Example: Let the function $f\left(z\right)=z^3$ be defined in the unit disk $\left|z\right|<1$. In this example, $f\left(z\right)=z^3$ has a 3-fold rotational symmetry. To see this, by substituting $z=e^{i\theta }$, it will be observed that $f\left(e^{i\theta }\right)=e^{i\left(3\theta \right)}$, which means that the function maps in the unit disk point to themselves in a 3-fold symmetric fashion. To find the inverse of $f\left(z\right)=z^3$, we take the cube root as follows:

Inverse: The inverse function

$$f^{-1}\left(z\right)=\sqrt[3]{z^3},$$

defined in the unit disk maps that point back to the original unit disk in a 3-fold symmetric manner. Thus, $f\left(z\right)=z^3$ and its inverse $f^{-1}\left(z\right)=\sqrt[3]{z^3}$ form a 3-fold symmetric bi-univalent pair in the context of Geometric Function Theory.

2. Definitions

The authors employ the Wanas definition in [21] to establish the following:

Definition 1.

A function $K_{\psi }\in {\sum }_m$ given by (10) is said to be in the class $Z_{{\sum }_m}\left(\beta ,\varphi ,S\right)$ if it satisfies the conditions

$$\begin{array}{c}\hfill \left|arg\left[1+\frac{z{K}_{\psi }^{\prime }\left(z\right)}{K_{\psi }\left(z\right)}+\frac{z{K}_{\psi }^{″}\left(z\right)}{K_{\psi }^{\prime }\left(z\right)}-\frac{\beta z^2{K}_{\psi }^{″}\left(z\right)+z{K}_{\psi }^{\prime }\left(z\right)}{\beta z{K}_{\psi }^{\prime }\left(z\right)+(1-\beta )K_{\psi }\left(z\right)}\right]\right|<\frac{\varphi \pi }{2}z\in U\end{array}$$

(12)

and

$$\begin{array}{c}\hfill \left|arg\left[1+\frac{\omega G_{\psi }^{\prime }\left(\omega \right)}{G_{\psi }\left(\omega \right)}+\frac{\omega G_{\psi }^{″}\left(\omega \right)}{G_{\psi }^{\prime }\left(\omega \right)}-\frac{\beta {\omega }^2{G}_{\psi }^{″}\left(\omega \right)+\omega G_{\psi }^{\prime }\left(\omega \right)}{\beta \omega G_{\psi }^{\prime }\left(\omega \right)+(1-\beta )G_{\psi }\left(\omega \right)}\right]\right|<\frac{\varphi \pi }{2}\omega \in U,\end{array}$$

(13)

where $0\le \beta \le 1,0<\varphi \le 1$ and $m\in \mathbb{N}$.

Definition 2.

A function $K_{\psi }\in {\sum }_m$ given by (10) is said to be in the class $Z_{{\sum }_m}\left(\beta ,\alpha ,S\right)$ if it satisfies the conditions

$$\begin{array}{c}\hfill \left\{1+\frac{z{K}_{\psi }^{\prime }\left(z\right)}{K_{\psi }\left(z\right)}+\frac{z{K}_{\psi }^{″}\left(z\right)}{K_{\psi }^{\prime }\left(z\right)}-\frac{\beta z^2{K}_{\psi }^{″}\left(z\right)+z{K}_{\psi }^{\prime }\left(z\right)}{\beta z{K}_{\psi }^{\prime }\left(z\right)+(1-\beta )K_{\psi }\left(z\right)}\right\}>\alpha z\in U\end{array}$$

(14)

and

$$\begin{array}{c}\hfill \left\{1+\frac{\omega G_{\psi }^{\prime }\left(\omega \right)}{G_{\psi }\left(\omega \right)}+\frac{\omega G_{\psi }^{″}\left(\omega \right)}{G_{\psi }^{\prime }\left(\omega \right)}-\frac{\beta {\omega }^2{G}_{\psi }^{″}\left(\omega \right)+\omega G_{\psi }^{\prime }\left(\omega \right)}{\beta \omega G_{\psi }^{\prime }\left(\omega \right)+(1-\beta )G_{\psi }\left(\omega \right)}\right\}>\alpha \omega \in U,\end{array}$$

(15)

where $0\le \beta \le 1,0\le \alpha <1$ and $m\in \mathbb{N}$.

Theorem 1.

Let $K_{\psi }\in Z_{{\sum }_m}\left(\beta ,\varphi ,S\right)\left(0\le \beta \le 1,0<\varphi \le 1,m\in \mathbb{N}\right)$ be given by (10). Then

$$\begin{array}{c}\hfill \left|\frac{a_m}{S}\right|\le \frac{\sqrt{2\varphi }}{m\sqrt{m\beta (\varphi \beta -2)+m^2{(1-\beta )}^2(1-\varphi )+4\varphi \beta (m+1)+m(2-\varphi )}}\end{array}$$

(16)

and

$$\begin{array}{c}\hfill \left|\frac{a_{2m}}{S}\right|\le \frac{2{\varphi }^2(m+1)}{m^2{(1+m(1-\beta ))}^2}+\frac{\varphi }{m(1+2m(1-\beta \left)\right)}.\end{array}$$

(17)

Proof. 

It follows from conditions (12) and (13) that

$$\begin{array}{c}\hfill 1+\frac{z{K}_{\psi }^{\prime }\left(z\right)}{K_{\psi }\left(z\right)}+\frac{z{K}_{\psi }^{″}\left(z\right)}{K_{\psi }^{\prime }\left(z\right)}-\frac{\beta z^2{K}_{\psi }^{″}\left(z\right)+z{K}_{\psi }^{\prime }\left(z\right)}{\beta z{K}_{\psi }^{\prime }\left(z\right)+(1-\beta )K_{\psi }\left(z\right)}={\left[p\left(z\right)\right]}^{\varphi }\end{array}$$

(18)

and

$$\begin{array}{c}\hfill 1+\frac{\omega G_{\psi }^{\prime }\left(\omega \right)}{G_{\psi }\left(\omega \right)}+\frac{\omega G_{\psi }^{″}\left(\omega \right)}{G_{\psi }^{\prime }\left(\omega \right)}-\frac{\beta {\omega }^2{G}_{\psi }^{″}\left(\omega \right)+\omega G_{\psi }^{\prime }\left(\omega \right)}{\beta \omega G_{\psi }^{\prime }\left(\omega \right)+(1-\beta )G_{\psi }\left(\omega \right)}={\left[q\left(\omega \right)\right]}^{\varphi },\end{array}$$

(19)

where $G_{\psi }=K_{\psi }^{-1}$ with $p,q\in \mathcal{P}$ and

$$\begin{array}{c}\hfill p\left(z\right)=1+p_m{z}^m+p_{2}m{z}^{2m}+\dots \end{array}$$

(20)

and

$$\begin{array}{c}\hfill q\left(\omega \right)=1+q_m{\omega }^m+q_{2m}{\omega }^{2m}+\dots .\end{array}$$

(21)

Comparing the corresponding coefficients of (18) and (19) gives

$$\begin{array}{c}\hfill m(1+m(1-\beta ))\frac{a_m}{S}=\varphi p_m\end{array}$$

(22)

$$\begin{array}{c}\hfill m\left[2(1+2m(1-\beta ))\frac{a_{2m}}{S}-(1+{(m+1)}^2-{(m\beta +1)}^2)\frac{a_m^2}{S^2}\right]\\ \hfill =\varphi p_{2m}+\frac{\varphi (\varphi -1)}{2}{p}_m^2\end{array}$$

(23)

$$\begin{array}{c}\hfill -m(1+m(1-\beta ))\frac{a_m}{S}=\varphi q_m\end{array}$$

(24)

and

$$\begin{array}{c}\hfill m\left[(m^2(3-4\beta )+4m(1-\beta )+{(m\beta +1)}^2)\frac{a_m^2}{S^2}-2(1+2m(1-\beta ))\frac{a_{2m}}{S}\right]\\ \hfill =\varphi q_{2m}+\frac{\varphi (\varphi -1)}{2}{q}_m^2.\end{array}$$

(25)

Using (22) and (24), we obtain

$$\begin{array}{c}\hfill p_m=-q_m\end{array}$$

(26)

and

$$\begin{array}{c}\hfill 2m^2{(1+m(1-\beta ))}^2\frac{a_m^2}{S^2}={\varphi }^2(p_m^2+q_m^2).\end{array}$$

(27)

Also, from (23), (25), and (27), we find that

$$\begin{array}{c}m[-1-{(m+1)}^2+m^2(3-4\beta )+4m(1-\beta )+2{(m\beta +1)}^2]\frac{a_m^2}{S^2}\\ =\varphi (p_{2m}+q_{2m})+\frac{\varphi (\varphi -1)(p_m^2+q_m^2)}{2}=\varphi (p_{2m}+q_{2m})+\frac{2m^2(\varphi -1){(1+2m(1-\beta ))}^2}{\varphi }\frac{a_m^2}{S^2}.\end{array}$$

(28)

Therefore, we have

$$\begin{array}{c}\hfill \frac{a_m^2}{S^2}=\frac{{\varphi }^2(p_{2m}+q_{2m})}{2m^2\left[\beta m(\varphi \beta -2)+m^2{(1-\beta )}^2(1-\varphi )+4\varphi \beta (m+1)+m(2-\varphi )\right]}.\end{array}$$

(29)

Now, taking the absolute value of (29) and applying (8) for $p_{2m}$ and $q_{2m}$, we obtain

$$\begin{array}{c}\hfill \left|\frac{a_m}{S}\right|\le \frac{\sqrt{2\varphi }}{m\sqrt{\beta m(\varphi \beta -2)+m^2{(1-\beta )}^2(1-\varphi )+4\varphi \beta (m+1)+m(2-\varphi )}}.\end{array}$$

In order to find the bound on $|\frac{a_{2m}}{S}|$, by subtracting (25) from (23), we obtain

$$\begin{array}{c}\hfill 2m(1+2m(1-\beta ))[\frac{2a_{2m}}{S}-\frac{(m+1)a_m^2}{S^2}]=\varphi (p_{2m}-q_{2m})+\frac{\varphi (\varphi -1)(p_m^2-q_m^2)}{2}.\end{array}$$

(30)

It follows from (27), (28), and (30) that

$$\begin{array}{c}\hfill \frac{a_{2m}}{S}=\frac{{\varphi }^2(m+1)(p_m^2+q_m^2)}{4m^2{(1+m(1-\beta ))}^2}+\frac{\varphi (p_{2m}-q_{2m})}{4m(1+2m(1-\beta \left)\right)}.\end{array}$$

(31)

Taking the absolute values of (31) and applying (8) once again for the coefficients $p_m,q_m,p_{2m}$, and $q_{2m}$, we have

$$\begin{array}{c}\hfill \left|\frac{a_{2m}}{S}\right|\le \frac{2{\varphi }^2(m+1)}{m^2{(1+m(1-\beta ))}^2}+\frac{\varphi }{m(1+2m(1-\beta \left)\right)},\end{array}$$

which is the proof of Theorem 1.  □

Setting $\varphi =1$ in Theorem 1, we have

Corollary 1.

Let $K_{\psi }\in Z_{{\sum }_m}\left(\beta ,1,S\right)\left(0\le \beta \le 1\right)$ and $m\in \mathbb{N}$ be given by (10). Then

$$\begin{array}{c}\hfill \left|\frac{a_m}{S}\right|\le \frac{\sqrt{2}}{m\sqrt{m\beta (\beta -2)+4\beta (m+1)+m}}\end{array}$$

(32)

and

$$\begin{array}{c}\hfill \left|\frac{a_{2m}}{S}\right|\le \frac{2(m+1)}{m^2{(1+m(1-\beta ))}^2}+\frac{1}{m(1+2m(1-\beta \left)\right)}.\end{array}$$

(33)

Taking $\beta =1$ in Corollary 1, we obtain

Corollary 2.

Let $K_{\psi }\in Z_{{\sum }_m}\left(1,1,S\right)$ and $m\in \mathbb{N}$ be given by (10). Then

$$\begin{array}{c}\hfill \left|\frac{a_m}{S}\right|\le \frac{1}{m\sqrt{2(m+1)}}\end{array}$$

(34)

and

$$\begin{array}{c}\hfill \left|\frac{a_{2m}}{S}\right|\le \frac{1}{m}\left(1+\frac{2(m+1)}{m}\right)\end{array}$$

(35)

Putting $m=1$ in Corollary 2, we obtain

Corollary 3.

Let $K_{\psi }\in Z_{{\sum }_1}\left(1,1,S\right)$ be given by (10). Then

$$\begin{array}{c}\hfill \left|\frac{a_1}{S}\right|\le \frac{1}{2}\end{array}$$

(36)

and

$$\begin{array}{c}\hfill \left|\frac{a_2}{S}\right|\le 5.\end{array}$$

(37)

Theorem 2.

Let $K_{\psi }\in Z_{{\sum }_m}\left(\beta ,\alpha ,S\right)\left(0\le \beta \le 1,0\le \alpha <1,m\in \mathbb{N}\right)$ be given by (10). Then

$$\begin{array}{c}\hfill \left|\frac{a_m}{S}\right|\le \sqrt{\frac{2(1-\alpha )}{m[{(m\beta +1)}^2+m(m+1)(1-2\beta )-1]}}\end{array}$$

(38)

and

$$\begin{array}{c}\hfill \left|\frac{a_{2m}}{S}\right|\le \frac{2(m+1){(1-\alpha )}^2}{m^2{(1+m(1-\beta ))}^2}+\frac{(1-\alpha )}{m(1+2m(1-\beta \left)\right)}.\end{array}$$

(39)

Proof. 

It follows from conditions (14) and (15) that

$$\begin{array}{c}\hfill 1+\frac{z{K}_{\psi }^{\prime }\left(z\right)}{K_{\psi }\left(z\right)}+\frac{z{K}_{\psi }^{″}\left(z\right)}{K_{\psi }^{\prime }\left(z\right)}-\frac{\beta z^2{K}_{\psi }^{″}\left(z\right)+z{K}_{\psi }^{\prime }\left(z\right)}{\beta z{K}_{\psi }^{\prime }\left(z\right)+(1-\beta )K_{\psi }\left(z\right)}=\alpha +(1-\alpha )p\left(z\right)\end{array}$$

(40)

and

$$\begin{array}{c}\hfill 1+\frac{\omega G_{\psi }^{\prime }\left(\omega \right)}{G_{\psi }\left(\omega \right)}+\frac{\omega G_{\psi }^{″}\left(\omega \right)}{G_{\psi }^{\prime }\left(\omega \right)}-\frac{\beta {\omega }^2{G}_{\psi }^{″}\left(\omega \right)+\omega G_{\psi }^{\prime }\left(\omega \right)}{\beta \omega G_{\psi }^{\prime }\left(\omega \right)+(1-\beta )G_{\psi }\left(\omega \right)}=\alpha +(1-\alpha )q\left(\omega \right),\end{array}$$

(41)

where $G_{\psi }=K_{\psi }^{-1}$ and $p,q\in \mathcal{P}$ have the expansions

$$\begin{array}{c}\hfill p\left(z\right)=1+p_m{z}^m+p_{2m}{z}^{2m}+\dots \end{array}$$

and

$$\begin{array}{c}\hfill q\left(\omega \right)=1+q_m{\omega }^m+q_{2m}{\omega }^{2m}+\dots .\end{array}$$

Comparing the corresponding coefficients of (40) and (41) gives

$$\begin{array}{c}\hfill m(1+m(1-\beta ))\frac{a_m}{S}=(1-\alpha )p_m\end{array}$$

(42)

$$\begin{array}{c}\hfill m\left[2(1+2m(1-\beta ))\frac{a_{2m}}{S}-(1+{(m+1)}^2-{(m\beta +1)}^2)\frac{a_m^2}{S^2}\right]\\ \hfill =(1-\alpha )p_{2m}\end{array}$$

(43)

$$\begin{array}{c}\hfill -m(1+m(1-\beta ))\frac{a_m}{S}=(1-\alpha )q_m\end{array}$$

(44)

and

$$\begin{array}{c}\hfill m\left[(m^2(3-4\beta )+4m(1-\beta )+{(m\beta +1)}^2)\frac{a_m^2}{S^2}-2(1+2m(1-\beta ))\frac{a_{2m}}{S}\right]\\ \hfill =(1-\alpha )q_{2m}.\end{array}$$

(45)

Using (42) and (44), we obtain

$$\begin{array}{c}\hfill p_m=-q_m\end{array}$$

(46)

and

$$\begin{array}{c}\hfill 2m^2{(1+2m(1-\beta ))}^2\frac{a_m^2}{S^2}={(1-\alpha )}^2(p_m^2+q_m^2).\end{array}$$

(47)

Adding (43) and (45), we obtain

$$\begin{array}{c}\hfill 2m[{(m\beta +1)}^2+m(m+1)(1-2\beta )-1]\frac{a_m^2}{S^2}=(1-\alpha )(p_{2m}+q_{2m}).\end{array}$$

(48)

Therefore, we have

$$\begin{array}{c}\hfill \frac{a_m^2}{S^2}=\frac{(1-\alpha )(p_{2m}+q_{2m})}{2m[{(m\beta +1)}^2+m(m+1)(1-2\beta )-1]}.\end{array}$$

(49)

Applying (8) for coefficients $p_{2m}$ and $q_{2m}$, we obtain

$$\begin{array}{c}\hfill \left|\frac{a_m}{S}\right|\le \sqrt{\frac{2(1-\alpha )}{m[{(m\beta +1)}^2+m(m+1)(1-2\beta )-1]}}.\end{array}$$

In order to find the bound on $|\frac{a_{2m}}{S}|$, by subtracting (45) from (43), we obtain

$$\begin{array}{c}\hfill \frac{a_{2m}}{S}=\frac{(m+1)a_m^2}{2S^2}+\frac{(1-\alpha )(p_{2m}-q_{2m})}{4m(1+2m(1-\beta \left)\right)}.\end{array}$$

(50)

Substituting the values of $\frac{a_m^2}{S^2}$ from (49), it follows that

$$\begin{array}{c}\hfill \frac{a_{2m}}{S}=\frac{(m+1){(1-\alpha )}^2(p_m^2+q_m^2)}{4m^2{(1+m(1-\beta ))}^2}+\frac{(1-\alpha )(p_{2m}-q_{2m})}{4m(1+2m(1-\beta \left)\right)}.\end{array}$$

(51)

Applying (8) once again for the coefficients $p_m,q_m,p_{2m}$, and $q_{2m}$, we have

$$\begin{array}{c}\hfill \left|\frac{a_{2m}}{S}\right|\le \frac{2(m+1){(1-\alpha )}^2}{m^2{(1+m(1-\beta ))}^2}+\frac{(1-\alpha )}{m(1+2m(1-\beta \left)\right)},\end{array}$$

which completes the proof of Theorem 2.  □

Setting $\alpha =0$ in Theorem 2, we have

Corollary 4.

Let $K_{\psi }\in Z_{{\sum }_m}\left(\beta ,0,S\right)$, $0\le \beta \le 1$ and $m\in \mathbb{N}$ be given by (10). Then

$$\begin{array}{c}\hfill \left|\frac{a_m}{S}\right|\le \sqrt{\frac{2}{m[{(m\beta +1)}^2+m(m+1)(1-2\beta )-1]}}\end{array}$$

(52)

and

$$\begin{array}{c}\hfill \left|\frac{a_{2m}}{S}\right|\le \frac{2(m+1)}{m^2{(1+m(1-\beta ))}^2}+\frac{1}{m(1+2m(1-\beta \left)\right)}.\end{array}$$

(53)

Putting $\beta =1$ in Corollary 4, we obtain

Corollary 5.

Let $K_{\psi }\in Z_{{\sum }_m}\left(1,1,S\right)$ and $m\in \mathbb{N}$ be given by (10). Then

$$\begin{array}{c}\hfill \left|\frac{a_m}{S}\right|\le \frac{\sqrt{2}}{m}\end{array}$$

(54)

and

$$\begin{array}{c}\hfill \left|\frac{a_{2m}}{S}\right|\le \frac{1}{m}\left(1+\frac{2(m+1)}{m}\right)\end{array}$$

(55)

Putting $m=1$ in Corollary 5, we obtain

Corollary 6.

Let $K_{\psi }\in Z_{{\sum }_1}\left(1,1,S\right)$ be given by (10). Then

$$\begin{array}{c}\hfill \left|\frac{a_1}{S}\right|\le \sqrt{2}\end{array}$$

(56)

and

$$\begin{array}{c}\hfill \left|\frac{a_2}{S}\right|\le 5.\end{array}$$

(57)

3. Conclusions

Various forms of bi-univalent functions have been studied by many authors. However, little studies have been carried out on bi-univalent functions with respect to a generalized distribution. In this study, therefore, the authors introduced an m-fold symmetric univalent function associated with generalized distribution. The two subclasses of functions, which are $Z_{{\sum }_m}\left(\beta ,\varphi ,S\right)$ and $Z_{{\sum }_m}\left(\beta ,\alpha ,S\right)$, were studied in the work with the initial coefficients obtained. The study provides a new direction in the literature for authors in this field of study. Two results were obtained in the work on coefficient bounds $\frac{a_m}{S}$ and $\frac{a_{2}m}{S}$ as Theorems 1 and 2.