Initial Coefficient Bounds for Certain New Subclasses of Bi-Univalent Functions Involving Mittag–Leffler Function with Bounded Boundary Rotation

Abstract

By using the Mittag–Leffler function associated with functions of bounded boundary rotation, the authors introduce a few new subclasses of bi-univalent functions involving the Mittag–Leffler function with bounded boundary rotation in the open unit disk $\mathbb{D}.$ For these new classes, the authors establish initial coefficient bounds of $\left|a_2\right|$ and $\left|a_3\right|$. Furthermore, the famous Fekete–Szegö coefficient inequality is also obtained for these new classes of functions.

1. Introduction

The Mittag–Leffler function plays a vital role in a variety of problems in fractional calculus, operator theory, mathematical analysis, and other domains related to science and engineering. The Mittag–Leffler function is commonly encountered in the resolution of integral equations with fractional orders or differential equations with fractional orders. Furthermore, it has continually been a theme that has inspired many researchers [1,2,3,4]. In 1903, Mittag–Leffler [5] introduced the function ${\mathcal{G}}_{\mu }\left(z\right),(z\in \mathbb{D})$, $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ defined by the following power series,

$${\mathcal{G}}_{\mu }\left(z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{z^n}{\mathrm{\Gamma }(\mu n+1)}},\forall \Re \left(\mu \right)>0$$

which is convergent in the whole complex plane. For $\mu >0,$ the Mittag–Leffler function ${\mathcal{G}}_{\mu }\left(z\right)$ is an entire function of order $1/\mu .$ The Mittag–Leffler function satisfies the property

$${\mathcal{G}}_{\mu }\left(z\right)={\displaystyle \frac{1}{z}}\left[{\mathcal{G}}_{\mu ,1-\mu }\left(z\right)-{\displaystyle \frac{1}{\mathrm{\Gamma }(1-\mu )}}\right],$$

where the function ${\mathcal{G}}_{\mu ,\beta }\left(z\right)$ is the generalized Mittag–Leffler function defined by the following power series,

$${\mathcal{G}}_{\mu ,\beta }\left(z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{z^n}{\mathrm{\Gamma }(\mu n+\beta )}},\forall \Re \left(\mu \right)>0,\Re \left(\beta \right)>0,$$

and the integral representation of the Mittag–Leffler function is given by

$${\mathcal{G}}_{\mu }\left(z\right)={\oint }_C{\displaystyle \frac{t^{\mu -1}{e}^t}{t^{\mu }-z}}dt;\Re \left(\mu \right)>0,$$

where the contour C starts and ends at ∞ and circles around the singularities and branch points of the integrand. Wong and Zhao [6] proved that the most interesting properties of the Mittag–Leffler function are associated with its asymptotic expansions as $z\to \infty$ in various sectors of the complex plane. Special values for integer $\mu =k$ are

$${\mathcal{G}}_0\left(z\right)={\displaystyle \frac{1}{1-z}},$$

$${\mathcal{G}}_1\left(z\right)=e^z,$$

$${\mathcal{G}}_2\left(z\right)=cosh\left(\sqrt{z}\right),$$

$${\mathcal{G}}_2(-z^2)=cosz,$$

$${\mathcal{G}}_3\left(z\right)={\displaystyle \frac{1}{3}}\left[e^{\sqrt[3]{z}}+2e^{-{\displaystyle \frac{\sqrt[3]{z}}{2}}}cos\left({\displaystyle \frac{\sqrt{3}}{2}}\sqrt[3]{z}\right)\right],$$

$${\mathcal{G}}_4\left(z\right)={\displaystyle \frac{1}{2}}\left[cos\left(\sqrt[4]{z}\right)+cosh\left(\sqrt[4]{z}\right)\right],$$

and

$${\mathcal{G}}_{{\displaystyle \frac{1}{2}}}\left(z\right)=e^{z^2}erfc(-z)=e^{z^2}[1+erf\left(z\right)]$$

where $erf\left(z\right)$ is the Gauss error function or simply called an error function, which is given by

$$erf\left(z\right)={\displaystyle \frac{2}{\sqrt{\pi }}}{\int }_0^z{e}^{-t^2}dt$$

and $erfc\left(z\right)=1-erf\left(z\right)$ where $erfc$ denotes the complimentary error function.

Srivastava and Tomovski [7] introduced the function ${\mathcal{G}}_{\mu ,\tau }^{\eta ,\delta }$ in the following form:

$${\mathcal{G}}_{\mu ,\tau }^{\eta ,\delta }\left(z\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(\eta \right)}_{n\delta }{z}^n}{\mathrm{\Gamma }(\mu n+\tau )n!}}$$

where $\mu ,\tau ,\eta \in \mathbb{C}$, $\Re \left(\mu \right)>max\{0,\Re (\delta )-1\}$, $\Re \left(\delta \right)>0$ and ${\left(y\right)}_n$ is the Pochhammer symbol defined, in terms of gamma function, by

$${\left(y\right)}_n={\displaystyle \frac{\mathrm{\Gamma }(y+n)}{\mathrm{\Gamma }\left(y\right)}}=\left\{\begin{array}{cc}1\hfill & \mathrm{if} n=0\hfill \\ y(y+1)(y+2)\cdots (y+n-1)\hfill & \mathrm{if} n=1,2,3,\cdots \hfill \end{array}\right..$$

Let $\mathcal{A}$ denote the class of functions f defined in the open unit disk

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

(1)

and normalized by the conditions $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1$, which are holomorphic on the unit disk $\mathbb{D}$. Also, let $\mathcal{S}$ denote the class of functions in $\mathcal{A}$ which are univalent in $\mathbb{D}.$ Some of the important and well-investigated subclasses of the univalent function class $\mathcal{S}$ include the class ${\mathcal{S}}^{\ast }\left(\lambda \right)$, starlike functions of order $\lambda$ and the class $\mathcal{C}\left(\lambda \right)$, and convex functions of order $\lambda .$ The analytic descriptions of the above two classes are, respectively, given by

$${\mathcal{S}}^{\ast }\left(\lambda \right)=\left\{f\in \mathcal{S}:\Re \left({\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right)>\lambda ,\lambda \in [0,1)\right\}$$

and

$$\mathcal{C}\left(\lambda \right)=\left\{f\in \mathcal{S}:\Re \left(1+{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right)>\lambda ,\lambda \in [0,1)\right\}.$$

For two holomorphic functions $h_1,h_2\in \mathcal{A}$, if their Taylor series are given by

$$h_1\left(z\right)=z+\sum _{n=2}^{\infty }{A}_n{z}^{n}and{h}_2\left(z\right)=z+\sum _{n=2}^{\infty }{B}_n{z}^n,$$

then the function $h_1\ast h_2:\mathbb{D}⟶\mathbb{C}$ is defined by

$$(h_1\ast h_2)\left(z\right)=h_1\left(z\right)\ast h_2\left(z\right),$$

where

$$h_1\left(z\right)\ast h_2\left(z\right)=z+\sum _{n=2}^{\infty }{A}_n{B}_n{z}^n,$$

which is called the convolution of $h_1$ and $h_2.$ The function $h_1\ast h_2$ belongs to $\mathcal{A}$, and it is also called the Hadamard product of $h_1$ and $h_2.$

In 2016, Attiya [8] introduced and investigated a linear operator ${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }:\mathcal{A}⟶\mathcal{A}$ by using convolution and defined as follows

$${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }f\left(z\right)={\mathcal{E}}_{\mu ,\tau }^{\eta ,\delta }\left(z\right)\ast f\left(z\right),$$

where

$${\mathcal{E}}_{\mu ,\tau }^{\eta ,\delta }\left(z\right)={\displaystyle \frac{\mathrm{\Gamma }(\mu +n)}{{\left(\eta \right)}_n}}\left[{\mathcal{G}}_{\mu ,\tau }^{\eta ,\delta }\left(z\right)-{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\tau \right)}}\right].$$

Hence, we obtain

$${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }f\left(z\right)=z+\sum _{n=2}^{\infty }{\displaystyle \frac{\mathrm{\Gamma }(\eta +n\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(n\mu +\tau )}}{\displaystyle \frac{a_n}{n!}}{z}^n.$$

Remark 1. 

(i) If we choose $\eta =2$, $\delta =1$ and $\mu =0$, then the operator ${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }f$ reduces to ${\displaystyle {\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }f\left(z\right)\equiv \frac{{\left[z{f}^{\prime }\left(z\right)\right]}^{\prime }}{2}}$, which was defined by Livingston [9].

(ii) 

If we choose $\eta =0$, $\delta =1$ and $\mu =0$, then the operator ${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }f$ reduces to ${\displaystyle {\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }f\left(z\right)\equiv {\int }_0^z\frac{f\left(s\right)}{s}ds}$, which is the Alexander integral [10,11].

In 1975, Padmanabhan and Parvatham [12] introduced class ${\mathcal{P}}_m\left(\lambda \right).$ A function p normalized by $p\left(0\right)=0$ and $p^{\prime }\left(0\right)>1$ belongs to the class ${\mathcal{P}}_m\left(\lambda \right)$ if it satisfies the condition

$$\underset{0}{\overset{2\pi }{\int }}\left|{\displaystyle \frac{\Re \left(p\right(z\left)\right)-\lambda }{1-\lambda }}\right|dt\le m\pi ,$$

where $z=r{e}^{it}\in \mathbb{D}$. The function p is in the class ${\mathcal{P}}_m\left(\lambda \right)$ if and only if there is a non-decreasing function $\varphi$ on $[0,2\pi ]$ with

$${\displaystyle \underset{0}{\overset{2\pi }{\int }}d\varphi \left(t\right)=2\mathrm{and}\underset{0}{\overset{2\pi }{\int }}\left|d\varphi \left(t\right)\right|\le m,m\in [2,\infty ),}$$

(2)

satisfying

$$p\left(z\right)={\int }_0^{2\pi }{\displaystyle \frac{1+(1-2\lambda )z{e}^{-it}}{1-z{e}^{-it}}}d\varphi \left(t\right).$$

Remark 2. 

(i) If we choose $\lambda =0$, the class ${\mathcal{P}}_m\left(\lambda \right)$ reduces to the class ${\mathcal{P}}_m$, which was defined by Pinchuk [13] and studied by Robertson [14].

(ii) 

If we choose $\lambda =0$ and $m=2$, the class ${\mathcal{P}}_m\left(\lambda \right)$ reduces to the class $\mathcal{P}$, which is known as the class of Carathéodory functions.

An interesting observation between class ${\mathcal{P}}_m$ and Carathéodory functions is that a function $p\left(z\right)\in {\mathcal{P}}_m$ if there exist two Carathéodory functions $p_1\left(z\right)$ and $p_2\left(z\right)$ such that

$$p\left(z\right)=\left({\displaystyle \frac{m}{4}}+{\displaystyle \frac{1}{2}}\right)p_1\left(z\right)-\left({\displaystyle \frac{m}{4}}-{\displaystyle \frac{1}{2}}\right)p_2\left(z\right).$$

The function $f\in \mathcal{A}$ is in the class ${\mathcal{R}}_m\left(\lambda \right)$ if and only if there is a non-decreasing function $\varphi \left(t\right)$ as defined in (2) and satisfying

$$f\left(z\right)=zexp\left\{-2(1-\lambda ){\int }_0^{2\pi }log(1-z{e}^{-it})d\varphi \left(t\right)\right\}.$$

If we choose $\lambda =0$, then the class ${\mathcal{R}}_m\left(\lambda \right)$ reduces to the class ${\mathcal{R}}_m$, which is the class of all functions of bounded radius rotation studied by Robertson [14].

The function $f\in \mathcal{A}$ is in the class ${\mathcal{V}}_m\left(\lambda \right)$ if and only if there is a non-decreasing function $\varphi \left(t\right)$ defined in (2) and satisfying

$$f^{\prime }\left(z\right)=exp\left\{-2(1-\lambda ){\int }_0^{2\pi }log(1-z{e}^{-it})d\varphi \left(t\right)\right\}.$$

For $\lambda =0$, the class ${\mathcal{V}}_m\left(\lambda \right)$ reduces to the class ${\mathcal{V}}_m$, which is the class of all functions of bounded boundary rotation studied by Paatero [15] in 1933. The classes ${\mathcal{V}}_m\left(\lambda \right)$ and ${\mathcal{R}}_m\left(\lambda \right)$ are thoroughly investigated by Padmanabhan and Parvatham [12].

Koepf [16] proved that ${\mathcal{V}}_m\left(s\right)$ is a subclass of ${\mathcal{K}}_m\left(\beta \right)$ of the s-fold symmetric close-to-convex of order ${\displaystyle \beta =\frac{m-2}{2s}.}$ Furthermore, for $s\in \mathbb{N}$, ${\mathcal{V}}_m(2s+2)$ consists of close-to-convex functions (hence are univalent). The concept of an odd univalent function with bounded boundary rotation was investigated by Leach [17], and it was extended to s-fold symmetric functions investigated by Leach [18].

Lemma 1. 

If $p\left(z\right)=1+p_{1}z+p_2{z}^2+p_3{z}^3+\cdots$ belong to class ${\mathcal{P}}_m\left(\lambda \right)$, then for all $n\ge 1$, $|p_n|\le m(1-\lambda )$.

The Koebe one-quarter theorem ensures that the image of $\mathbb{D}$ under every $f\in \mathcal{S}$ contains a disk of radius $1/4.$ Each function f from $\mathcal{S}$ has an inverse $f^{-1},$ which satisfies (see [19,20]) $f^{-1}\left(f\left(z\right)\right)=z,z\in \mathbb{D}$ and

$$f\left(f^{-1}\left(w\right)\right)=w,\left(\left|w\right|\le \rho ;\rho \ge {\displaystyle \frac{1}{4}}\right),$$

where

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

(3)

A function $f\in \mathcal{A}$ is said to be bi-univalent in $\mathbb{D}$ if f and $f^{-1}$ are together univalent in $\mathbb{D}.$ Let us denote $\mathrm{\Sigma }$ as the class of bi-univalent functions in $\mathbb{D}.$ The class $\mathrm{\Sigma }$ is non-empty as

$${\displaystyle f_1\left(z\right)=\frac{z}{1-z},f_2\left(z\right)=log\left(\sqrt{\frac{1+z}{1-z}}\right)and{f}_3\left(z\right)=-log(1-z)}$$

are in the class $\mathrm{\Sigma }$ with the corresponding inverse functions

$$h_1\left(w\right)={\displaystyle \frac{w}{1+w}},h_2\left(w\right)={\displaystyle \frac{e^{2w}-1}{e^{2w}+1}}and{h}_3\left(w\right)=1-e^{-w}$$

respectively. But the functions

$$z-{\displaystyle \frac{z^2}{2}}and{\displaystyle \frac{z}{1-z^2}}$$

do not belong to the class $\mathrm{\Sigma }.$ The class $\mathrm{\Sigma }$ was investigated by Lewin [21] in 1967. Lewin also proved that $|a_2|\le 1.51.$ In 1980, Brannan and Clunie [22] conjectured that $|a_2|\le \sqrt{2}.$ In 1969, Netanyahu [23] proved that $max|a_2|={\displaystyle \frac{4}{3}}$ for a subclass ${\mathrm{\Sigma }}_1$ of $\mathrm{\Sigma },$ where ${\mathrm{\Sigma }}_1$ denotes the class of bi-univalent functions and the range of every function in ${\mathrm{\Sigma }}_1$ contains the disk $\mathbb{D}.$ Further in 1981, Styer and Wright [24] observed that there exist functions in $\mathrm{\Sigma }$ for which $|a_2|>4/3$ and in 1984, Tan [25] proved that $|a_2|\le 1.485.$ The study of operators plays a vibrant role in mathematics. Using the convolution theory, one can define an operator and then can learn its properties. Indeed, it is one of the searing areas of recent ongoing research in the geometric function theory and its related fields.

In this present article, we introduce new subclasses of bi-univalent functions associated with bounded boundary rotation involving the Mittag–Leffler function. For the newly defined classes, the authors determine the estimates on the initial coefficients $|a_2|$ and $|a_3|$ for functions with bounded boundary rotation involving the Mittag–Leffler function. Furthermore, the famous Fekete–Szegö inequality is also obtained for these new subclasses of functions. Apart from new results, few of the bounds improves the existing earlier ones available in the literature.

2. Main Results

Definition 1. 

Let $\lambda \in [0,1)$ and $m\in [2,4].$ A function $f\in \mathcal{S}$ belongs to the class ${\mathcal{R}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ if it satisfies the following conditions:

$${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{f}^{\prime }\left(z\right)\in {\mathcal{P}}_m\left(\lambda \right)$$

and

$${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{h}^{\prime }\left(w\right)\in {\mathcal{P}}_m\left(\lambda \right),$$

where $\mu ,\tau ,\eta \in \mathbb{C}$, $\Re \left(\mu \right)>max\{0,\Re (\delta )-1\}$ and $\Re \left(\delta \right)>0.$

Remark 3. 

(i) If we choose $\mu =0$ and $\tau =\eta =\delta =1$, then the class ${\mathcal{R}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{H}}_{\mathrm{\Sigma }}(m,\lambda )$ defined earlier by Sharma et al. [26].

(ii) 

If we choose $\mu =0$, $\tau =\eta =\delta =1$ and $m=2$, then the class ${\mathcal{R}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{H}}_{\mathrm{\Sigma }}\left(\lambda \right)$ defined by Srivastava et al. [27].

(iii) 

If we choose $m=2$, then the class ${\mathcal{R}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{R}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }\left(\lambda \right)$, which is the class of functions whose derivatives have a positive real part involving Mittag–Leffler function of order $\lambda .$

Theorem 1. 

Let $\lambda \in [0,1)$ and $m\in [2,4].$ If a function $f\in {\mathcal{R}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda ),$ then

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}}$$

(4)

$$\left|a_3\right|\le {\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}$$

(5)

and

$$|a_3-\mathrm{\Delta }{a}_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )(1-\mathrm{\Delta })}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}\hfill & \mathit{for}\mathrm{\Delta }\le 0,\hfill \\ {\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}\hfill & \mathit{for}0\le \mathrm{\Delta }\le 2,\hfill \\ {\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )(\mathrm{\Delta }-1)}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}\hfill & \mathit{for}\mathrm{\Delta }\ge 2,\hfill \end{array}\right.$$

(6)

where $\mathrm{\Delta }\in \mathbb{R}$, $\mu ,\tau ,\eta \in \mathbb{C}$, $\Re \left(\mu \right)>max\{0,\Re (\delta )-1\}$ and $\Re \left(\delta \right)>0.$

Proof. 

Since $f\in {\mathcal{R}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda ),$ then from Definition 1, there exist two holomorphic functions $u\left(z\right)$ and $v\left(w\right)$ belonging to ${\mathcal{P}}_m\left(\lambda \right)$ such that

$${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{f}^{\prime }\left(z\right)=u\left(z\right)$$

(7)

$${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{h}^{\prime }\left(w\right)=v\left(w\right)$$

(8)

with

$$u\left(z\right)=1+u_{1}z+u_2{z}^2+\cdots$$

and

$$v\left(w\right)=1+v_{1}w+v_2{w}^2+\cdots .$$

Hence, from (7) and (8), we obtain

$${\displaystyle \frac{\mathrm{\Gamma }(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(2\mu +\tau )}}{a}_2=u_1,$$

(9)

$${\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{2\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{a}_3=u_2,$$

(10)

$$-{\displaystyle \frac{\mathrm{\Gamma }(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(2\mu +\tau )}}{a}_2=v_1$$

(11)

and

$${\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{2\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}(2a_2^2-a_3)=v_2.$$

(12)

From, (10) and (12), we obtain

$${\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{a}_2^2=u_2+v_2.$$

(13)

Applying Lemma 1 in (13) for $u_2$ and $v_2$, we obtain

$${\left|a_2\right|}^2\le {\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}.$$

(14)

Equation (14) gives $\left|a_2\right|$ given in (4). Now, applying Lemma 1 in (10), we obtain

$$\left|a_3\right|\le {\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}.$$

(15)

From, (10) and (14) and for any $\mathrm{\Delta }\in \mathbb{R}$, we obtain

$$a_3-\mathrm{\Delta }{a}_2^2={\displaystyle \frac{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}[\{2-\mathrm{\Delta }\}{u}_2-\mathrm{\Delta }{v}_2].$$

(16)

Applying Lemma 1 in (16) for $u_2$ and $v_2$, we obtain

$$|a_3-\mathrm{\Delta }{a}_2^2|\le {\displaystyle \frac{m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}\left[\right|2-\mathrm{\Delta }|+|\mathrm{\Delta }\left|\right].$$

(17)

Equation (17) gives (6), which completes the proof of Theorem 1. □

If we choose $\mu =0$ and $\tau =\eta =\delta =1,$ then the class ${\mathcal{R}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{H}}_{\mathrm{\Sigma }}(m,\lambda ).$ Hence, we obtain the following result for the functions belonging to ${\mathcal{H}}_{\mathrm{\Sigma }}(m,\lambda ).$

Corollary 1. 

For $\lambda \in [0,1)$ and $m\in [2,4]$, if a function $f\in {\mathcal{H}}_{\mathrm{\Sigma }}(m,\lambda ),$ then

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{m(1-\lambda )}{3}}}$$

$$\left|a_3\right|\le {\displaystyle \frac{m(1-\lambda )}{3}}$$

and

$$|a_3-\mathrm{\Delta }{a}_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{m(1-\lambda )(1-\mathrm{\Delta })}{3}}\hfill & \mathit{for}\mathrm{\Delta }<0,\hfill \\ {\displaystyle \frac{m(1-\lambda )}{3}}\hfill & \mathit{for}0\le \mathrm{\Delta }\le 2,\hfill \\ {\displaystyle \frac{m(1-\lambda )(\mathrm{\Delta }-1)}{3}}\hfill & \mathit{for}\mathrm{\Delta }\ge 2.\hfill \end{array}\right.$$

where $\mathrm{\Delta }\in \mathbb{R}.$

If we choose $\mu =0$, $\tau =\eta =\delta =1$ and $m=2,$ then the class ${\mathcal{R}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{H}}_{\mathrm{\Sigma }}\left(\lambda \right).$ Hence, we obtain the following result for the functions belonging to ${\mathcal{H}}_{\mathrm{\Sigma }}\left(\lambda \right).$

Corollary 2. 

Let $\lambda \in [0,1).$ If a function $f\in {\mathcal{H}}_{\mathrm{\Sigma }}\left(\lambda \right)$, then

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{2(1-\lambda )}{3}}}$$

$$\left|a_3\right|\le {\displaystyle \frac{2(1-\lambda )}{3}}$$

and

$$|a_3-\mathrm{\Delta }{a}_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{2(1-\lambda )(1-\mathrm{\Delta })}{3}}\hfill & \mathit{for}\mathrm{\Delta }<0,\hfill \\ {\displaystyle \frac{2(1-\lambda )}{3}}\hfill & \mathit{for}0\le \mathrm{\Delta }\le 2,\hfill \\ {\displaystyle \frac{2(1-\lambda )(\mathrm{\Delta }-1)}{3}}\hfill & \mathit{for}\mathrm{\Delta }\ge 2,\hfill \end{array}\right.$$

where $\mathrm{\Delta }\in \mathbb{R}.$

If we choose $m=2$, the class ${\mathcal{R}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{H}}_{\mathrm{\Sigma },\mu ,\tau }^{\eta ,\delta }\left(\lambda \right).$ A function $f\in {\mathcal{H}}_{\mathrm{\Sigma },\mu ,\tau }^{\eta ,\delta }\left(\lambda \right)$ if it satisfies

$$\Re \left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{f}^{\prime }\left(z\right)\right)>\lambda$$

and

$$\Re \left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{h}^{\prime }\left(w\right)\right)>\lambda .$$

Corollary 3. 

Let $\lambda \in [0,1).$ If the function $f\in {\mathcal{H}}_{\mathrm{\Sigma },\mu ,\tau }^{\eta ,\delta }\left(\lambda \right),$ then

$$\left|a_2\right|\le 2\sqrt{{\displaystyle \frac{(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}}$$

$$\left|a_3\right|\le {\displaystyle \frac{4(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}$$

and

$$|a_3-\mathrm{\Delta }{a}_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{4(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )(1-\mathrm{\Delta })}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}\hfill & \mathit{for}\mathrm{\Delta }\le 0,\hfill \\ {\displaystyle \frac{4(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}\hfill & \mathit{for}0\le \mathrm{\Delta }\le 2,\hfill \\ {\displaystyle \frac{4(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )(\mathrm{\Delta }-1)}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}\hfill & \mathit{for}\mathrm{\Delta }\ge 2,\hfill \end{array}\right.$$

where $\mathrm{\Delta }\in \mathbb{R}$.

Definition 2. 

Let $\lambda \in [0,1)$ and $m\in [2,4].$ A function $f\in \mathcal{S}$ is said to belong to the class ${\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ if it satisfies the conditions

$${\displaystyle \frac{z{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{f}^{\prime }\left(z\right)}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }f\left(z\right)}}\in {\mathcal{P}}_m\left(\lambda \right)$$

and

$${\displaystyle \frac{w{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{h}^{\prime }\left(w\right)}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }h\left(w\right)}}\in {\mathcal{P}}_m\left(\lambda \right),$$

where $\mu ,\tau ,\eta \in \mathbb{C}$, $\Re \left(\mu \right)>max\{0,\Re (\delta )-1\}$ and $\Re \left(\delta \right)>0.$

Remark 4. 

(i) If we choose $\mu =0$ and $\tau =\eta =\delta =1$, the class ${\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{S}}_{\mathrm{\Sigma }}^{\ast }(m,\lambda )$, which was defined by Li et al. [28].

(ii) 

If we choose $\mu =0$, $\tau =\eta =\delta =1$ and $m=2$, the class ${\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{S}}_{\mathrm{\Sigma }}^{\ast }\left(\lambda \right)$, which was defined by Brannan and Taha. [29].

(iii) 

If we choose $m=2$, the class ${\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }\left(\lambda \right)$ which is the class of bi-starlike functions involving the Mittag–Leffler function of order $\lambda .$

Theorem 2. 

Let $\lambda \in [0,1)$ and $m\in [2,4].$ If a function $f\in {\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda ),$ then

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Phi }}_1}}}$$

(18)

$$\left|a_3\right|\le {\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Phi }}_1}}$$

(19)

and for any $\mathrm{\Delta }\in \mathbb{R}$

$$|a_3-\mathrm{\Delta }{a}_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )(1-\mathrm{\Delta })}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Phi }}_1}}\hfill & \mathit{for}\mathrm{\Delta }<{\mathrm{\Phi }}_2,\hfill \\ {\displaystyle \frac{m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}\hfill & \mathit{for}{\mathrm{\Phi }}_2\le \mathrm{\Delta }\le 2-{\mathrm{\Phi }}_2,\hfill \\ {\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )(\mathrm{\Delta }-1)}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Phi }}_1}}\hfill & \mathit{for}\mathrm{\Delta }>2-{\mathrm{\Phi }}_2,\hfill \end{array}\right.$$

(20)

where

$${\mathrm{\Phi }}_1={\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )}{3\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{2{\mathrm{\Gamma }}^2(2\mu +\tau )\mathrm{\Gamma }(\eta +\delta )}},$$

$${\mathrm{\Phi }}_2={\displaystyle \frac{3{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )\mathrm{\Gamma }(3\mu +\tau )}{4{\mathrm{\Gamma }}^2(2\mu +\tau )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(\eta +3\delta )}},$$

$\mu ,\tau ,\eta \in \mathbb{C}$, $\Re \left(\mu \right)>max\{0,\Re (\delta )-1\}$ and $\Re \left(\delta \right)>0.$

Proof. 

Since $f\in {\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda ),$ then from Definition 2, there exist two holomorphic functions $u\left(z\right)$ and $v\left(w\right)$ belonging to ${\mathcal{P}}_m\left(\lambda \right)$ such that

$${\displaystyle \frac{z{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{f}^{\prime }\left(z\right)}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }f\left(z\right)}}=u\left(z\right)$$

(21)

$${\displaystyle \frac{w{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{h}^{\prime }\left(w\right)}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }h\left(w\right)}}=v\left(w\right),$$

(22)

where

$$u\left(z\right)=1+u_{1}z+u_2{z}^2+\cdots$$

and

$$v\left(w\right)=1+v_{1}w+v_2{w}^2+\cdots .$$

Now, from (21) and (22), we obtain

$${\displaystyle \frac{\mathrm{\Gamma }(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{2\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(2\mu +\tau )}}{a}_2=u_1,$$

(23)

$${\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{3\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{a}_3-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta ){\mathrm{\Gamma }}^2(\mu +\tau )}{4{\mathrm{\Gamma }}^2(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}{a}_2^2=u_2,$$

(24)

$$-{\displaystyle \frac{\mathrm{\Gamma }(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{2\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(2\mu +\tau )}}{a}_2=v_1$$

(25)

and

$$\left[{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{3\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta ){\mathrm{\Gamma }}^2(\mu +\tau )}{4{\mathrm{\Gamma }}^2(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}\right]a_2^2-{\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{3\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{a}_3=v_2.$$

(26)

Hence, from (24) and (26), we obtain

$$\left[{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{3\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta ){\mathrm{\Gamma }}^2(\mu +\tau )}{2{\mathrm{\Gamma }}^2(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}\right]a_2^2=u_2+v_2.$$

(27)

Applying Lemma 1 in (27) for $u_2$ and $v_2$, we obtain

$${\mathrm{\Phi }}_1{\displaystyle \frac{\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )}}{a}_2^2=2m(1-\lambda ),$$

(28)

where

$${\mathrm{\Phi }}_1={\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )}{3\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{2{\mathrm{\Gamma }}^2(2\mu +\tau )\mathrm{\Gamma }(\eta +\delta )}}.$$

Equation (28) gives $\left|a_2\right|$ given in (18). Now, again from from (24) and (26), we obtain

$$\left[{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{3\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}\right](a_3-a_2^2)=u_2-v_2.$$

(29)

Now, using (28) in (29), we obtain

$${\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{3\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{a}_3={\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{3\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{a}_2^2+u_2+v_2.$$

(30)

Now, by using (28) in (30), we obtain

$${\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{3\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{\mathrm{\Phi }}_1{a}_3=\left[{\displaystyle \frac{4\mathrm{\Gamma }(\eta +3\delta )}{3\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{2\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}\right]u_2+{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{2\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}{v}_2.$$

(31)

Applying Lemma 1 in (31) for $u_2$ and $v_2$, we obtain

$$\left|a_3\right|\le {\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Phi }}_1}}.$$

(32)

From (28) and (30) and for any $\mathrm{\Delta }\in \mathbb{R}$, we obtain

$$\begin{array}{c}{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{3\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{\mathrm{\Phi }}_1[a_3-\mathrm{\Delta }{a}_2^2]=\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left[{\displaystyle \frac{4\mathrm{\Gamma }(\eta +3\delta )}{3\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{2\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}-\mathrm{\Delta }{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )}{3\mathrm{\Gamma }(3\mu +\tau )}}\right]u_2+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left[{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{2\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}-\mathrm{\Delta }{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )}{3\mathrm{\Gamma }(3\mu +\tau )}}\right]v_2.\hfill \end{array}$$

(33)

Applying Lemma 1 in (33) for $u_2$ and $v_2$, we obtain

$$\begin{array}{c}{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{3\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{\mathrm{\Phi }}_1[a_3-\mathrm{\Delta }{a}_2^2]\le \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}m(1-\lambda )\left|{\displaystyle \frac{4\mathrm{\Gamma }(\eta +3\delta )}{3\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{2\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}-\mathrm{\Delta }{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )}{3\mathrm{\Gamma }(3\mu +\tau )}}\right|+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}m(1-\lambda )\left|{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{2\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}-\mathrm{\Delta }{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )}{3\mathrm{\Gamma }(3\mu +\tau )}}\right|.\hfill \end{array}$$

(34)

Equation (34) gives (20). This completes the proof of Theorem 2. □

If we choose $m=2$, then the class ${\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }\left(\lambda \right)$, which is called the bi-starlike functions involving the Mittag–Leffler function of order $\lambda$. A function $f\in {\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }\left(\lambda \right)$ if it satisfies the conditions

$$\Re \left({\displaystyle \frac{z{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{f}^{\prime }\left(z\right)}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }f\left(z\right)}}\right)>\lambda$$

and

$$\Re \left({\displaystyle \frac{w{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{h}^{\prime }\left(w\right)}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }h\left(w\right)}}\right)>\lambda .$$

For the special choices mentioned, Theorem 2 gives the following corollary.

Corollary 4. 

Let $\lambda \in [0,1).$ If a function $f\in {\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }\left(\lambda \right),$ then

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{4(1-\lambda )\mathrm{\Gamma }(\eta +\delta )}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Phi }}_1}}},$$

$$\left|a_3\right|\le {\displaystyle \frac{4(1-\lambda )\mathrm{\Gamma }(\eta +\delta )}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Phi }}_1}},$$

and for any $\mathrm{\Delta }\in \mathbb{R}$

$$|a_3-\mathrm{\Delta }{a}_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{4(1-\lambda )\mathrm{\Gamma }(\eta +\delta )(1-\mathrm{\Delta })}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Phi }}_1}}\hfill & \mathit{for}\mathrm{\Delta }<{\mathrm{\Phi }}_2,\hfill \\ {\displaystyle \frac{2(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}\hfill & \mathit{for}{\mathrm{\Phi }}_2\le \mathrm{\Delta }\le 2-{\mathrm{\Phi }}_2,\hfill \\ {\displaystyle \frac{4(1-\lambda )\mathrm{\Gamma }(\eta +\delta )(\mathrm{\Delta }-1)}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Phi }}_1}}\hfill & \mathit{for}\mathrm{\Delta }>2-{\mathrm{\Phi }}_2\hfill \end{array}\right.$$

where

$${\mathrm{\Phi }}_1={\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )}{3\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{2{\mathrm{\Gamma }}^2(2\mu +\tau )\mathrm{\Gamma }(\eta +\delta )}},$$

and

$${\mathrm{\Phi }}_2={\displaystyle \frac{3{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )\mathrm{\Gamma }(3\mu +\tau )}{4{\mathrm{\Gamma }}^2(2\mu +\tau )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(\eta +3\delta )}}$$

If we choose $\mu =0$ and $\tau =\eta =\delta =1,$ then the class ${\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{S}}_{\mathrm{\Sigma }}^{\ast }(m,\lambda ).$ Hence, we obtain the following result for the functions belonging to ${\mathcal{S}}_{\mathrm{\Sigma }}^{\ast }(m,\lambda ).$

Corollary 5. 

For $\lambda \in [0,1)$, if a function $f\in {\mathcal{S}}_{\mathrm{\Sigma }}^{\ast }(m,\lambda ),$ then

$$\left|a_2\right|\le \sqrt{m(1-\lambda )},$$

$$\left|a_3\right|\le m(1-\lambda )$$

and for any $\mathrm{\Delta }\in \mathbb{R}$

$$|a_3-\mathrm{\Delta }{a}_2^2|\le \left\{\begin{array}{cc}m(1-\lambda )(1-\mathrm{\Delta })\hfill & {\displaystyle \mathit{for}\mathrm{\Delta }<\frac{1}{2},}\hfill \\ {\displaystyle \frac{m(1-\lambda )}{6}}\hfill & {\displaystyle \mathit{for}\frac{1}{2}\le \mathrm{\Delta }\le \frac{3}{2},}\hfill \\ m(1-\lambda )(\mathrm{\Delta }-1)\hfill & {\displaystyle \mathit{for}\mathrm{\Delta }>\frac{3}{2}.}\hfill \end{array}\right.$$

If we choose $\mu =0$, $\tau =\eta =\delta =1$ and $m=2,$ then the class ${\mathcal{S}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{S}}_{\mathrm{\Sigma }}^{\ast }\left(\lambda \right).$ Hence, we obtain the following result for functions belonging to ${\mathcal{S}}_{\mathrm{\Sigma }}^{\ast }\left(\lambda \right).$

Corollary 6. 

If $\lambda \in [0,1)$ and if the function $f\in {\mathcal{S}}_{\mathrm{\Sigma }}^{\ast }\left(\lambda \right),$ then

$$\left|a_2\right|\le \sqrt{2(1-\lambda )},$$

$$\left|a_3\right|\le 2(1-\lambda )$$

and for any $\mathrm{\Delta }\in \mathbb{R}$,

$$|a_3-\mathrm{\Delta }{a}_2^2|\le \left\{\begin{array}{cc}2(1-\lambda )(1-\mathrm{\Delta })\hfill & {\displaystyle \mathit{for}\mathrm{\Delta }<\frac{1}{2},}\hfill \\ {\displaystyle \frac{1-\lambda }{3}}\hfill & {\displaystyle \mathit{for}\frac{1}{2}\le \mathrm{\Delta }\le \frac{3}{2},}\hfill \\ 2(1-\lambda )(\mathrm{\Delta }-1)\hfill & {\displaystyle \mathit{for}\mathrm{\Delta }>\frac{3}{2}.}\hfill \end{array}\right.$$

Definition 3. 

Let $\lambda \in [0,1)$ and $m\in [2,4].$ A function $f\in \mathcal{S}$ is said to belong to the class ${\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ if it satisfies the conditions

$$1+{\displaystyle \frac{z{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{f}^{″}\left(z\right)}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{f}^{\prime }\left(z\right)}}\in {\mathcal{P}}_m\left(\lambda \right)$$

and

$$1+{\displaystyle \frac{w{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{h}^{″}\left(w\right)}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{h}^{\prime }\left(w\right)}}\in {\mathcal{P}}_m\left(\lambda \right),$$

where $\mu ,\tau ,\eta \in \mathbb{C}$, $\Re \left(\mu \right)>max\{0,\Re (\delta )-1\}$ and $\Re \left(\delta \right)>0.$

Remark 5. 

(i) If we choose $\mu =0$ and $\tau =\eta =\delta =1$, then the class ${\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{C}}_{\mathrm{\Sigma }}^{\ast }(m,\lambda )$, which was defined by Li et al. [28].

(ii) 

If we choose $\mu =0$, $\tau =\eta =\delta =1$ and $m=2$, then the class ${\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{C}}_{\mathrm{\Sigma }}^{\ast }\left(\lambda \right)$, which was defined by Brannan and Taha. [29].

(iii) 

If we choose $m=2$, then the class ${\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }\left(\lambda \right)$, which was the class of bi-convex functions involving the Mittag–Leffler function of order $\lambda .$

Theorem 3. 

Let $\lambda \in [0,1)$ and $m\in [2,4].$ If a function $f\in {\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda ),$ then

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Psi }}_1}}},$$

(35)

$$\left|a_3\right|\le {\displaystyle \frac{m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Psi }}_1}}$$

(36)

and for any $\mathrm{\Delta }\in \mathbb{R}$

$$|a_3-\mathrm{\Delta }{a}_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )(1-\mathrm{\Delta })}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Psi }}_1}}\hfill & \mathit{for}\mathrm{\Delta }<{\mathrm{\Psi }}_2,\hfill \\ {\displaystyle \frac{2m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}\hfill & \mathit{for}{\mathrm{\Psi }}_2\le \mathrm{\Delta }\le 2-{\mathrm{\Psi }}_2,\hfill \\ {\displaystyle \frac{m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )(\mathrm{\Delta }-1)}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Psi }}_1}}\hfill & \mathit{for}\mathrm{\Delta }>2-{\mathrm{\Psi }}_2,\hfill \end{array}\right.$$

(37)

where

$${\mathrm{\Psi }}_1={\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )}{\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}},$$

$${\mathrm{\Psi }}_2={\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(3\mu +\tau )\mathrm{\Gamma }(\mu +\tau )}{{\mathrm{\Gamma }}^2(2\mu +\tau )\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\eta +\delta )}},$$

$\mu ,\tau ,\eta \in \mathbb{C}$, $\Re \left(\mu \right)>max\{0,\Re (\delta )-1\}$ and $\Re \left(\delta \right)>0.$

Proof. 

Since $f\in {\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda ),$ then from Definition 3, there exist two holomorphic functions $u\left(z\right)$ and $v\left(w\right)$ belonging to ${\mathcal{P}}_m\left(\lambda \right)$ such that

$$1+{\displaystyle \frac{z{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{f}^{″}\left(z\right)}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{f}^{\prime }\left(z\right)}}=u\left(z\right)$$

(38)

$$1+{\displaystyle \frac{w{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{h}^{″}\left(w\right)}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{h}^{\prime }\left(w\right)}}=v\left(w\right)$$

(39)

where

$$u\left(z\right)=1+u_{1}z+u_2{z}^2+\cdots$$

and

$$v\left(w\right)=1+v_{1}w+v_2{w}^2+\cdots .$$

Now, from (38) and (39), we obtain

$${\displaystyle \frac{\mathrm{\Gamma }(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(2\mu +\tau )}}{a}_2=u_1,$$

(40)

$${\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{a}_3-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta ){\mathrm{\Gamma }}^2(\mu +\tau )}{{\mathrm{\Gamma }}^2(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}{a}_2^2=u_2$$

(41)

$$-{\displaystyle \frac{\mathrm{\Gamma }(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(2\mu +\tau )}}{a}_2=v_1$$

(42)

and

$$\left[{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta ){\mathrm{\Gamma }}^2(\mu +\tau )}{{\mathrm{\Gamma }}^2(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}\right]a_2^2-{\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{a}_3=v_2.$$

(43)

Hence, from (41) and (43), we obtain

$$\left[{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{2{\mathrm{\Gamma }}^2(\eta +2\delta ){\mathrm{\Gamma }}^2(\mu +\tau )}{{\mathrm{\Gamma }}^2(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}\right]a_2^2=u_2+v_2.$$

(44)

Applying Lemma 1 in (44) for $u_2$ and $v_2$, we obtain

$${\mathrm{\Psi }}_1{\displaystyle \frac{\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )}}{\left|a_2\right|}^2\le m(1-\lambda ),$$

(45)

where

$${\mathrm{\Psi }}_1={\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )}{\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}.$$

Equation (45) gives $\left|a_2\right|$ given in (35). Now, again from from (41) and (43), we obtain

$${\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{a}_3={\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(3\mu +\tau )}}{a}_2^2+u_2-v_2.$$

(46)

Now, by using (45) in (46), we obtain

$${\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{\mathrm{\Psi }}_1{a}_3=\left[{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )}{\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}\right]u_2+{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}{v}_2.$$

(47)

Applying Lemma 1 in (47) for $u_2$ and $v_2$, we obtain

$$\left|a_3\right|\le {\displaystyle \frac{m(1-\lambda )\mathrm{\Gamma }(\eta +\delta )}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Psi }}_1}}.$$

(48)

From, (45) and (47) and for any $\mathrm{\Delta }\in \mathbb{R}$, we obtain

$$\begin{array}{c}{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{\mathrm{\Psi }}_1[a_3-\mathrm{\Delta }{a}_2^2]=\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left[{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )}{\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}-\mathrm{\Delta }{\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )}{\mathrm{\Gamma }(3\mu +\tau )}}\right]u_2+\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left[{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}-\mathrm{\Delta }{\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )}{\mathrm{\Gamma }(3\mu +\tau )}}\right]v_2.\hfill \end{array}$$

(49)

Applying Lemma 1 in (50) for $u_2$ and $v_2$, we obtain

$$\begin{array}{c}{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}}{\mathrm{\Psi }}_1[a_3-\mathrm{\Delta }{a}_2^2]\le \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left|{\displaystyle \frac{2\mathrm{\Gamma }(\eta +3\delta )}{\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}-\mathrm{\Delta }{\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )}{\mathrm{\Gamma }(3\mu +\tau )}}\right|m(1-\lambda )+\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left|{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}-\mathrm{\Delta }{\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )}{\mathrm{\Gamma }(3\mu +\tau )}}\right|m(1-\lambda ).\end{array}$$

(50)

Equation (50) gives (37), which completes the proof of Theorem 3. □

If we choose $m=2$, then the class ${\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\eta ,\delta }\left(\lambda \right)$, which is called the bi-convex functions involving the Mittag–Leffler function of order $\lambda$. A function $f\in {\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\eta ,\delta }\left(\lambda \right)$ if it satisfies the conditions,

$$\Re \left(1+{\displaystyle \frac{z{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{f}^{″}\left(z\right)}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{f}^{\prime }\left(z\right)}}\right)>\lambda$$

and

$$\Re \left(1+{\displaystyle \frac{w{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{h}^{″}\left(w\right)}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }{h}^{\prime }\left(w\right)}}\right)>\lambda .$$

Corollary 7. 

For $\lambda \in [0,1),$ if a function $f\in {\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\eta ,\delta }\left(\lambda \right),$ then

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{2(1-\lambda )\mathrm{\Gamma }(\eta +\delta )}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Psi }}_1}}},$$

$$\left|a_3\right|\le {\displaystyle \frac{2(1-\lambda )\mathrm{\Gamma }(\eta +\delta )}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Psi }}_1}},$$

and for any $\mathrm{\Delta }\in \mathbb{R}$

$$|a_3-\mathrm{\Delta }{a}_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{2(1-\lambda )\mathrm{\Gamma }(\eta +\delta )(1-\mathrm{\Delta })}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Psi }}_1}}\hfill & \mathit{for}\mathrm{\Delta }<{\mathrm{\Psi }}_2,\hfill \\ {\displaystyle \frac{4(1-\lambda )\mathrm{\Gamma }(\eta +\delta )\mathrm{\Gamma }(3\mu +\tau )}{\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\mu +\tau )}}\hfill & \mathit{for}{\mathrm{\Psi }}_2\le \mathrm{\Delta }\le 2-{\mathrm{\Psi }}_2,\hfill \\ {\displaystyle \frac{2(1-\lambda )\mathrm{\Gamma }(\eta +\delta )(\mathrm{\Delta }-1)}{\mathrm{\Gamma }(\mu +\tau ){\mathrm{\Psi }}_1}}\hfill & \mathit{for}\mathrm{\Delta }>2-{\mathrm{\Psi }}_2.\hfill \end{array}\right.$$

where

$${\mathrm{\Psi }}_1={\displaystyle \frac{\mathrm{\Gamma }(\eta +3\delta )}{\mathrm{\Gamma }(3\mu +\tau )}}-{\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(\mu +\tau )}{\mathrm{\Gamma }(\eta +\delta ){\mathrm{\Gamma }}^2(2\mu +\tau )}}$$

and

$${\mathrm{\Psi }}_2={\displaystyle \frac{{\mathrm{\Gamma }}^2(\eta +2\delta )\mathrm{\Gamma }(3\mu +\tau )\mathrm{\Gamma }(\mu +\tau )}{{\mathrm{\Gamma }}^2(2\mu +\tau )\mathrm{\Gamma }(\eta +3\delta )\mathrm{\Gamma }(\eta +\delta )}}.$$

If we choose $\mu =0$ and $\tau =\eta =\delta =1,$ then the class ${\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{C}}_{\mathrm{\Sigma }}(m,\lambda ).$ Hence, we obtain the following result for the functions belonging to ${\mathcal{C}}_{\mathrm{\Sigma }}(m,\lambda ).$

Corollary 8. 

For $\lambda \in [0,1)$, if a function $f\in {\mathcal{C}}_{\mathrm{\Sigma }}(m,\lambda ),$ then

$$\left|a_2\right|\le \sqrt{{\displaystyle \frac{m(1-\lambda )}{2}}},$$

$$\left|a_3\right|\le {\displaystyle \frac{m(1-\lambda )}{2}},$$

and for any $\mathrm{\Delta }\in \mathbb{R}$

$$|a_3-\mathrm{\Delta }{a}_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{m(1-\lambda )(1-\mathrm{\Delta })}{2}}\hfill & {\displaystyle \mathit{for}\mathrm{\Delta }<\frac{2}{3},}\hfill \\ {\displaystyle \frac{m(1-\lambda )}{3}}\hfill & {\displaystyle \mathit{for}\frac{2}{3}\le \mathrm{\Delta }\le \frac{4}{3},}\hfill \\ {\displaystyle \frac{m(1-\lambda )(\mathrm{\Delta }-1)}{2}}\hfill & {\displaystyle \mathit{for}\mathrm{\Delta }>\frac{4}{3}.}\hfill \end{array}\right.$$

If we choose $\mu =0$, $\tau =\eta =\delta =1$ and $m=2,$ then the class ${\mathcal{C}}_{\mathrm{\Sigma },\mu ,\tau }^{\ast ,\eta ,\delta }(m,\lambda )$ reduces to the class ${\mathcal{C}}_{\mathrm{\Sigma }}\left(\lambda \right).$ Hence, we obtain the following result for the functions belonging to the class ${\mathcal{C}}_{\mathrm{\Sigma }}\left(\lambda \right).$

Corollary 9. 

For $\lambda \in [0,1)$, if a function $f\in {\mathcal{C}}_{\mathrm{\Sigma }}\left(\lambda \right),$ then

$$\left|a_2\right|\le \sqrt{1-\lambda },$$

$$\left|a_3\right|\le 1-\lambda$$

and for any $\mathrm{\Delta }\in \mathbb{R}$

$$|a_3-\mathrm{\Delta }{a}_2^2|\le \left\{\begin{array}{cc}(1-\lambda )(1-\mathrm{\Delta })\hfill & {\displaystyle \mathit{for}\mathrm{\Delta }<\frac{2}{3},}\hfill \\ {\displaystyle \frac{2(1-\lambda )}{3}}\hfill & {\displaystyle \mathit{for}\frac{2}{3}\le \mathrm{\Delta }\le \frac{4}{3},}\hfill \\ (1-\lambda )(\mathrm{\Delta }-1)\hfill & {\displaystyle \mathit{for}\mathrm{\Delta }>\frac{4}{3}.}\hfill \end{array}\right.$$

Remark 6. 

(i) Corollary 1 verifies the bounds of $|a_2|$ and $|a_3|$, which were obtained by Sharma et al. [26].

(ii) 

Corollary 2 verifies the bound of $|a_2|$ and improves the bound of $|a_3|$, which was obtained by Srivastava et al. [27].

(iii) 

Corollary 5 and Corollary 8 verifies the bound of $|a_2|$ and improves the bound of $|a_3|$, which was obtained by Li et al. [21].

(iv) 

Corollary 6 and Corollary 9 verifies the bound of $|a_2|$ and improves the bound of $|a_3|$, which was obtained by Brannan and Taha [29].

3. Conclusions

In the present work, the authors first found the two initial Taylor–Maclaurin’s coefficients for new subclasses of bi-univalent functions with bounded boundary rotation in the open unit disk $\mathbb{D}$ involving the Mittag–Leffler function. Also, the famous Fekete–Szegö inequality is obtained for these new subclasses. Interesting remarks on the main results including improvements of the earlier bounds are also given.

Furthermore, the study considered in this article can be extended by taking the generalized Mittag–Leffler-type function, Legendre polynomial, and Chebyshev polynomial with bounded boundary rotation. However, those interesting details and observations are not stated. Also, the same type of results can be worked out for interesting other special functions existing in the literature.