1 Introduction and preliminaries

Let \({\mathcal {A}}\) be the class of analytic functions in the open unit disk \({\mathcal {U=}}\left\{ z:\left| z\right| <1\right\} \) with conditions \(f(0)=0\) and \(f^{\prime }(0)=1\) having the form:

$$\begin{aligned} f(z)=z+\overset{\infty }{\underset{k=2}{ {\displaystyle \sum } }}a_{k}z^{k}\qquad (z\in \mathcal {U}). \end{aligned}$$
(1.1)

Further, all functions in \({\mathcal {A}}\) which are univalent in \({\mathcal {U}}\) we will denote it by \({\mathcal {S}}\). So, every function \(f\in {\mathcal {S\ }}\)has an inverse \(f^{-1}\), such that

$$\begin{aligned} f^{-1}(f(z))=z\ \text { and }f(f^{-1}(w))=w\qquad \left( z\in \mathcal {U}\text {, }\left| w\right| <r_{0}(f);\text { }r_{0}(f)\ge \frac{1}{4}\right) \end{aligned}$$

where

$$\begin{aligned} f^{-1}(w)=w-a_{2}w^{2}+(2a_{2}^{2}-a_{3})w^{3}-(5a_{2}^{3}-5a_{2}a_{3} +a_{4})w^{4}+\cdots . \end{aligned}$$
(1.2)

A function \(f\in {\mathcal {A}}\) given by (1.1) is in the class \(\Sigma \) of all bi-univalent functions in \({\mathcal {U}}\) if both f(z) and \(f^{-1}(z)\) are univalent in \({\mathcal {U}}\) (see [1,2,3, 15, 18]).

The class \({\mathcal {S}}^{*}(\varepsilon )\) of starlike functions of order \(\varepsilon \) in \({\mathcal {U}}\) is well-studied and subset of the function class \({\mathcal {S}}\). By definition, we have

$$\begin{aligned} \mathcal {S}^{*}(\varepsilon ):=\left\{ f:\ f\in \mathcal {S}\ \ \text {and} \ \ \hbox {Re}\left\{ \frac{zf^{\prime }(z)}{f(z)}\right\} >\varepsilon ,\quad (z\in \mathcal {U};0\le \varepsilon <1)\right\} . \end{aligned}$$
(1.3)

Ezrohi [7] introduced the class

$$\begin{aligned} \mathcal {U}(\varepsilon )=\left\{ f:\ f\in \mathcal {S}\ \ \text {and} \ \ \hbox {Re}\left\{ f^{\prime }(z)\right\} >\varepsilon ,\quad (z\in \mathcal {U};0\le \varepsilon <1)\right\} \end{aligned}$$

Also, Chen [6] introduced the class

$$\begin{aligned} {\mathcal {ST}}(\varepsilon )=\left\{ f:\ f\in \mathcal {S}\ \ \text {and} \ \ \hbox {Re}\left\{ \frac{f(z)}{z}\right\} >\varepsilon ,\quad (z\in \mathcal {U};0\le \varepsilon <1)\right\} . \end{aligned}$$

It is stated in [4] that a function \(f\in \mathcal {A}\) is in the class \(\mathcal {S}_{\Sigma }^{*}\left[ \varepsilon \right] \) of strongly bi-starlike functions of order \(\varepsilon (0<\varepsilon \le 1)\) if each of the following requirements is met

$$\begin{aligned}&f\in \Sigma \text { and }\left| \arg \left( \frac{zf^{\prime }(z)}{f(z)}\right) \right| <\frac{\varepsilon \pi }{2},~(z\in \mathcal {U}) \end{aligned}$$

and

$$\begin{aligned} \left| \arg \left( \frac{wg^{\prime }(w)}{g(w)}\right) \right| <\frac{\varepsilon \pi }{2}\qquad (w\in \mathcal {U}), \end{aligned}$$

where g \(=f^{-1}\) and given by (1.2).

Also, a function \(f\in \mathcal {A}\) is in the class \(\mathcal {S}_{\Sigma } ^{*}(\gamma )\) of bi-starlike functions of order \(\gamma (0\le \gamma <1)\) if each of the following requirements is met

$$\begin{aligned} f\in \Sigma \text { and }{\mathrm{Re}}\,\left( \frac{zf^{\prime }(z)}{f(z)}\right) >\gamma \qquad (z\in \mathcal {U}) \end{aligned}$$

and

$$\begin{aligned} {\mathrm{Re}}\,\left( \frac{wg^{\prime }(w)}{g(w)}\right) >\gamma \qquad (w\in \mathcal {U}), \end{aligned}$$

where g \(=f^{-1}\)and given by (1.2).

Now, we introduce the new and Comprehensive subclasses \(\mathbf {B} ^{\varepsilon }(\kappa ,\alpha ,\sigma )\), \(\mathbf {B}^{\gamma }(\kappa ,\alpha ,\sigma )\) and \(\mathbf {B}_{s,t}(\kappa ,\alpha ,\sigma )\).

Definition 1.1

A function f(z) given by (1.1) is said to be in the class \(\mathbf {B}^{\varepsilon }(\kappa ,\alpha ,\sigma )\) where \(\alpha ,\kappa \ge 1,\) \(\sigma \in \mathbb {C}\), \({\mathrm{Re}}\,(\sigma )\ge 0\), and \(0<\varepsilon \le 1\), if the following inequalities are satisfied:

$$\begin{aligned} f\in \Sigma \text { and }\left| \arg \left( (1-\kappa )f^{\prime } (z)+\kappa \left( f^{\prime }(z)\right) ^{\alpha }\left( \frac{f(z)}{z}\right) ^{\sigma -1}\right) \right| <\frac{\varepsilon \pi }{2} \qquad (z\in \mathcal {U}) \end{aligned}$$
(1.4)

and

$$\begin{aligned} \left| \arg \left( (1-\kappa )g^{\prime }(w)+\kappa \left( g^{\prime }(w)\right) ^{\alpha }\left( \frac{g(w)}{w}\right) ^{\sigma -1}\right) \right| <\frac{\varepsilon \pi }{2} \qquad (w\in \mathcal {U} ), \end{aligned}$$
(1.5)

where g is given by (1.2).

Definition 1.2

A function f(z) given by (1.1) is said to be in the class \(\mathbf {B}^{\gamma }(\kappa ,\alpha ,\sigma )\) where \(\alpha ,\kappa \ge 1,\) \(\sigma \in \mathbb {C}\), \({\mathrm{Re}}\,(\sigma )\ge 0,\) and \(0\le \gamma <1,\) if the following inequalities are satisfied:

$$\begin{aligned} f\in \Sigma \text { and }{\mathrm{Re}}\,\left( (1-\kappa )f^{\prime } (z)+\kappa \left( f^{\prime }(z)\right) ^{\alpha }\left( \frac{f(z)}{z}\right) ^{\sigma -1}\right) >\gamma \qquad (z\in \mathcal {U}) \end{aligned}$$
(1.6)

and

$$\begin{aligned} {\mathrm{Re}}\,\left( (1-\kappa )g^{\prime }(w)+\kappa \left( g^{\prime }(w)\right) ^{\alpha }\left( \frac{g(w)}{w}\right) ^{\sigma -1}\right) >\gamma \qquad (w\in \mathcal {U}), \end{aligned}$$
(1.7)

where g is given by (1.2).

Definition 1.3

Let the functions \(s,t:\mathcal {U\rightarrow \mathbb {C}}\) such that

$$\begin{aligned} \min \left\{ {\mathrm{Re}}\,(s(z)),{\mathrm{Re}}\,(t(z))\right\} >0\text { }(z\in \mathcal {U})\ \text { and }s(0)=t(0)=1. \end{aligned}$$

Also let \(f\in \mathcal {A}\), defined by (1.1). We say that \(f\in \mathbf {B}_{s,t}(\kappa ,\delta ,\mu )\) where \(\alpha ,\kappa \ge 1,\) \(\sigma \in \mathbb {C},\) \({\mathrm{Re}}\,(\sigma )\ge 0\) if the following inequalities are satisfied:

$$\begin{aligned} f\in \Sigma \text { and }{\mathrm{Re}}\,\left( (1-\kappa )f^{\prime } (z)+\kappa \left( f^{\prime }(z)\right) ^{\alpha }\left( \frac{f(z)}{z}\right) ^{\sigma -1}\right) \in t(\mathcal {U})\qquad (z\in \mathcal {U}) \end{aligned}$$
(1.8)

and

$$\begin{aligned} {\mathrm{Re}}\,\left( (1-\kappa )g^{\prime }(w)+\kappa \left( g^{\prime }(w)\right) ^{\alpha }\left( \frac{g(w)}{w}\right) ^{\sigma -1}\right) \in s(\mathcal {U})\qquad (w\in \mathcal {U}), \end{aligned}$$
(1.9)

where g is given by (1.2).

Remark 1.4

By taking specific values of the functions s(z) and t(z) in Definition 1.3 we get various well known subclasses of \(\mathcal {A}\), for example, if

$$\begin{aligned} s(z)=t(z)=\left( \frac{1+z}{1-z}\right) ^{\varepsilon } \ \ \ \ (0<\varepsilon \le 1;z\in \mathcal {U}) \end{aligned}$$

or

$$\begin{aligned} s(z)=t(z)=\frac{1+(1-2\gamma )z}{1-z} \ \ \ \ (0\le \gamma <1;z\in \mathcal {U}) \end{aligned}$$

it is simple to verify that s(z) and t(z) satisfy the Definition 1.3. If \(f\in \mathbf {B}_{s,t}(\kappa ,\alpha ,\sigma ),\) then the function f satisfied the inequalities (1.4) and (1.5) or (1.6) and (1.7), where g is given by (1.2). This means that, \(f\in \mathbf {B}^{\varepsilon }(\kappa ,\alpha ,\sigma )\) or \(f\in \mathbf {B} ^{\gamma }(\kappa ,\alpha ,\sigma ),\) where \(\alpha ,\kappa \ge 1,\) \(0<\varepsilon \le 1,\) \(0\le \gamma <1,\) \(\sigma \in \mathbb {C} \) and \({\mathrm{Re}}\,(\sigma )\ge 0.\)

The purpose of this paper is to introduce qualitative subclasses \(\mathbf {B}^{\varepsilon }(\kappa ,\alpha ,\sigma )\), \(\mathbf {B}^{\gamma } (\kappa ,\alpha ,\sigma )\) and \(\mathbf {B}_{s,t}(\kappa ,\alpha ,\sigma )\) of the function class \(\Sigma .\) Motivated by the earlier work of, Bulut [5], Frasin et al. [8,9,10], Li and Wang [11], Siregar and Raman [14], and Yousef et al. [17, 19,20,21], we find estimates on the coefficients \(|a_{2}|\), \(|a_{3}|\) and \(\left| a_{3}-\varsigma a_{2}^{2}\right| \). Furthermore, variety of new results will follow by specializing cases in our main results.

To proof our theorem we will need the following lemma:

Lemma 1.5

[12] If \(h\in \mathcal {H}\), then \(\left| h_{i}\right| \le 2\) for each i,  where \(\mathcal {H}\) is the family of all functions h analytic in \(\mathcal {U}\) for which

$$\begin{aligned} \hbox {Re}\left( h(z)\right) >0, h(z)=1+h_{1}z+h_{2}z^{2}+\cdots \ (z\in \mathcal {U}). \end{aligned}$$

2 Coefficient bounds for subclass \(\mathbf {B}_{s,t}(\kappa ,\alpha ,\sigma )\)

In this section we state and prove the main results for subclass\( {\mathbf {B}} _{s,t}(\kappa ,\alpha ,\sigma )\) given by Definition 1.3.

Theorem 2.1

Let f(z) given by (1.1) be in the class \({\mathbf {B}}_{s,t}(\kappa ,\alpha ,\sigma ),~\)where \(\alpha ,\kappa \ge 1,\) \(\sigma \in \mathbb {C}\) and \({\mathrm{Re}}\,(\sigma )\ge 0.\) Then

$$\begin{aligned}&\left| a_{2}\right| \le \min \left\{ \sqrt{\frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{2\left| \kappa \left( 2\alpha +\sigma -3\right) +2\right| ^{2}}},\sqrt{\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{2\left| \kappa \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right| }}\right\} ,\nonumber \\&\left| a_{3}\right| \le \min \left\{ \frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{2\left| \kappa \left( 2\alpha +\sigma -3\right) +2\right| ^{2}}+\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{4\left| \kappa \left( 3\alpha +\sigma -4\right) +3\right| }\right. ,\nonumber \\&\quad \left. \frac{ \begin{array} [c]{c} \left| \kappa \left( \sigma ^{2}+\sigma +4\alpha \left( \sigma -1\right) +4\alpha (\alpha +2)-14\right) +12\right| \left| s^{\prime \prime }(0)\right| \\ +\left| \kappa \left( \left( \sigma -1\right) \left( \sigma -2\right) +4\alpha \left( \sigma -1\right) +4\alpha (\alpha -1)\right) \right| \left| t^{\prime \prime }(0)\right| \end{array} }{4\left| \kappa \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right| \left| \kappa \left( 3\alpha +\sigma -4\right) +3\right| }\right\} , \end{aligned}$$
(2.1)

and

$$\begin{aligned} \left| a_{3}-\varsigma a_{2}^{2}\right| \le \frac{\left| t^{\prime \prime }(0)\right| }{\left| \kappa \left( 3\alpha +\sigma -4\right) +3\right| }, \end{aligned}$$

where

$$\begin{aligned} \varsigma =\frac{\kappa \left( \frac{\left( \sigma -2\right) \left( \sigma +3\right) }{2}+2\alpha \left( \sigma -1\right) +2\alpha (\alpha +2)-4\right) +6}{\kappa \left( 3\alpha +\sigma -4\right) +3}. \end{aligned}$$

Proof

First, we write the equivalent forms for inequalities (1.6) and (1.7) as follows:

$$\begin{aligned} (1-\kappa )f^{\prime }(z)+\kappa \left( f^{\prime }(z)\right) ^{\alpha }\left( \frac{f(z)}{z}\right) ^{\sigma -1}=s(z) \end{aligned}$$
(2.2)

and

$$\begin{aligned} (1-\kappa )g^{\prime }(w)+\kappa \left( g^{\prime }(w)\right) ^{\alpha }\left( \frac{g(w)}{w}\right) ^{\sigma -1}=t(w) \end{aligned}$$
(2.3)

where s(z) and t(w) are in \(\mathcal {H}\) and satisfy the conditions of Definition 1.3 and have the forms

$$\begin{aligned} s(z)=1+s_{1}z+s_{2}z^{2}+s_{3}z^{3}+\cdots \quad \text {and}\quad t(w)=1+t_{1}w+t_{2} w^{2}+t_{3}w^{3}+\cdots . \end{aligned}$$

Now, equating coefficients in (2.2) and (2.3), yields

$$\begin{aligned}&\left( \kappa \left( 2\alpha +\sigma -3\right) +2\right) a_{2}=s_{1}, \end{aligned}$$
(2.4)
$$\begin{aligned}&\left[ \kappa \left( \frac{\left( \sigma -1\right) \left( \sigma -2\right) }{2}+2\alpha \left( \sigma -1\right) +2\alpha (\alpha -1)\right) \right] a_{2}^{2}+\left[ \kappa \left( 3\alpha +\sigma -4\right) +3\right] a_{3}=s_{2}\nonumber \\ \end{aligned}$$
(2.5)
$$\begin{aligned}&-\left( \kappa \left( 2\alpha +\sigma -3\right) +2\right) a_{2}=t_{1} \end{aligned}$$
(2.6)

and

$$\begin{aligned}&\left[ \kappa \left( \frac{\left( \sigma -2\right) \left( \sigma +3\right) }{2}+2\alpha \left( \sigma -1\right) +2\alpha (\alpha +2)-4\right) +6\right] a_{2}^{2}-\left[ \kappa \left( 3\alpha +\sigma -4\right) +3\right] a_{3}\nonumber \\&\quad =t_{2}. \end{aligned}$$
(2.7)

From (2.4) and (2.6), we get

$$\begin{aligned} s_{1}=-t_{1} \end{aligned}$$
(2.8)

and

$$\begin{aligned} 2\left( \kappa \left( 2\alpha +\sigma -3\right) +2\right) ^{2}a_{2}^{2} =s_{1}^{2}+t_{1}^{2}. \end{aligned}$$
(2.9)

Also, adding (2.5) to (2.7), we find that

$$\begin{aligned} \left[ \kappa \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] a_{2}^{2}=s_{2}+t_{2}. \end{aligned}$$
(2.10)

From Eqs. (2.9) and (2.10), we have

$$\begin{aligned} \left| a_{2}^{2}\right| \le \frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{2\left| \kappa \left( 2\alpha +\sigma -3\right) +2\right| ^{2}} \end{aligned}$$
(2.11)

and

$$\begin{aligned} \left| a_{2}^{2}\right| \le \frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{2\left| \kappa \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right| }, \end{aligned}$$
(2.12)

respectively. So we get the inequality (2.1).

Next, to find the bound on \(\left| a_{3}\right| \), by subtracting (2.7) from (2.5), we get

$$\begin{aligned} 2\left[ \kappa \left( 3\alpha +\sigma -4\right) +3\right] \left( a_{3} -a_{2}^{2}\right) =s_{2}-t_{2}. \end{aligned}$$
(2.13)

Further, in view of (2.9) in Eq. (2.13), it follows that

$$\begin{aligned} a_{3}=\frac{s_{1}^{2}+t_{1}^{2}}{2(\kappa \left( 2\alpha +\sigma -3\right) +2)^{2}}+\frac{s_{2}-t_{2}}{2\kappa \left( 3\alpha +\sigma -4\right) +6}. \end{aligned}$$
(2.14)

We thus find that

$$\begin{aligned} \left| a_{3}\right| \le \frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{2\left| \kappa \left( 2\alpha +\sigma -3\right) +2\right| ^{2}}+\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{4\left| \kappa \left( 3\alpha +\sigma -4\right) +3\right| }. \end{aligned}$$

On other hand, by using (2.10) in (2.13), we get

$$\begin{aligned} a_{3}=\frac{ \begin{array} [c]{c} \left[ \kappa \left( \sigma ^{2}+\sigma +4\alpha \left( \sigma -1\right) +4\alpha (\alpha +2)-14\right) +12\right] s_{2}\\ -\left[ \kappa \left( \left( \sigma -1\right) \left( \sigma -2\right) +4\alpha \left( \sigma -1\right) +4\alpha (\alpha -1)\right) \right] t_{2} \end{array} }{\left[ \kappa \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right] \left[ 2\kappa \left( 3\alpha +\sigma -4\right) +6\right] }. \nonumber \\ \end{aligned}$$
(2.15)

Consequently, we have

$$\begin{aligned} \left| a_{3}\right| \le \frac{ \begin{array} [c]{c} \left| \kappa \left( \sigma ^{2}+\sigma +4\alpha \left( \sigma -1\right) +4\alpha (\alpha +2)-14\right) +12\right| \left| s^{\prime \prime }(0)\right| \\ +\left| \kappa \left( \left( \sigma -1\right) \left( \sigma -2\right) +4\alpha \left( \sigma -1\right) +4\alpha (\alpha -1)\right) \right| \left| t^{\prime \prime }(0)\right| \end{array} }{4\left| \kappa \left( \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)\right) +6\right| \left| \kappa \left( 3\alpha +\sigma -4\right) +3\right| }. \end{aligned}$$

Also, from (2.7) we find that

$$\begin{aligned} \frac{\kappa \left( \frac{\left( \sigma -2\right) \left( \sigma +3\right) }{2}+2\alpha \left( \sigma -1\right) +2\alpha (\alpha +2)-4\right) +6}{\kappa \left( 3\alpha +\sigma -4\right) +3}a_{2}^{2}-a_{3}=\frac{t_{2}}{\kappa \left( 3\alpha +\sigma -4\right) +3}. \end{aligned}$$

Consequently, we have

$$\begin{aligned} \left| a_{3}-\varsigma a_{2}^{2}\right| \le \frac{\left| t^{\prime \prime }(0)\right| }{\left| \kappa \left( 3\alpha +\sigma -4\right) +3\right| }, \end{aligned}$$

where

$$\begin{aligned} \varsigma =\frac{\kappa \left( \frac{\left( \sigma -2\right) \left( \sigma +3\right) }{2}+2\alpha \left( \sigma -1\right) +2\alpha (\alpha +2)-4\right) +6}{\kappa \left( 3\alpha +\sigma -4\right) +3}. \end{aligned}$$

Which completes the proof. \(\square \)

3 Corollaries and consequences

Choosing \(\kappa =1\) in Theorem 2.1, we obtain the following Corollary:

Corollary 3.1

Let f(z) given by (1.1) be in the class \(\mathbf {B} _{s,t}(1,\alpha ,\sigma ),~\)where \(\alpha \ge 1,\) \(\sigma \in \mathbb {C} \) and \({\mathrm{Re}}\,(\sigma )\ge 0.\) Then

$$\begin{aligned} \left| a_{2}\right|\le & {} \min \left\{ \sqrt{\frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{2\left| 2\alpha +\sigma -1\right| ^{2}}},\sqrt{\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{2\left| \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)+6\right| }}\right\} ,\\ \left| a_{3}\right|\le & {} \min \left\{ \frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{2\left| 2\alpha +\sigma -1\right| ^{2}}+\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{4\left| 3\alpha +\sigma -1\right| }\right. ,\\&\quad \left. \frac{ \begin{array} [c]{c} \left| \sigma ^{2}+\sigma +4\alpha \left( \sigma -1\right) +4\alpha (\alpha +2)-2\right| \left| s^{\prime \prime }(0)\right| \\ +\left| \left( \sigma -1\right) \left( \sigma -2\right) +4\alpha \left( \sigma -1\right) +4\alpha (\alpha -1)\right| \left| t^{\prime \prime }(0)\right| \end{array} }{4\left| \left( \sigma +2\right) \left( \sigma -3\right) +4\alpha \left( \sigma -1\right) +2\alpha (2\alpha +1)+6\right| \left| 3\alpha +\sigma -1\right| }\right\} , \end{aligned}$$

and

$$\begin{aligned} \left| a_{3}-\varsigma a_{2}^{2}\right| \le \frac{\left| t^{\prime \prime }(0)\right| }{\left| 3\alpha +\sigma -1\right| }, \end{aligned}$$

where

$$\begin{aligned} \varsigma =\frac{\frac{\left( \sigma -2\right) \left( \sigma +3\right) }{2}+2\alpha \left( \sigma -1\right) +2\alpha (\alpha +2)+2}{3\alpha +\sigma -1}. \end{aligned}$$

Putting \(\alpha =1\) in Corollary 3.1, we obtain the following Corollary:

Corollary 3.2

Let f(z) given by (1.1) be in the class \(\mathbf {B} _{s,t}(1,1,\sigma ),~\)where \(\sigma \in \mathbb {C} \) and \({\mathrm{Re}}\,(\sigma )\ge 0.\) Then

$$\begin{aligned}&\left| a_{2}\right| \le \min \left\{ \sqrt{\frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{2\left| \sigma +1\right| ^{2}}},\sqrt{\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{2\left| \left( \sigma +1\right) \left( \sigma +2\right) \right| }}\right\} ,\\&\left| a_{3}\right| \le \min \left\{ \frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{2\left| \sigma +1\right| ^{2}}+\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{4\left| \sigma +2\right| },\frac{\left| \left( \sigma +3\right) \right| \left| s^{\prime \prime }(0)\right| +\left| \left( \sigma -1\right) \right| \left| t^{\prime \prime }(0)\right| }{4\left| \left( \sigma +1\right) \left( \sigma +2\right) \right| }\right\} , \end{aligned}$$

and

$$\begin{aligned} \left| a_{3}-\varsigma a_{2}^{2}\right| \le \frac{\left| t^{\prime \prime }(0)\right| }{\left| \sigma +2\right| }, \end{aligned}$$

where

$$\begin{aligned} \varsigma =\frac{\sigma +3}{2}. \end{aligned}$$

Putting \(\sigma =0\) in Corollary 3.1, we obtain the following Corollary:

Corollary 3.3

Let f(z) given by (1.1) be in the class \(\mathbf {B} _{s,t}(1,\alpha ,0),~\)where \(\alpha \ge 1.\) Then

$$\begin{aligned}&\left| a_{2}\right| \le \min \left\{ \sqrt{\frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{2\left| 2\alpha -1\right| ^{2}}},\sqrt{\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{2\left| 2\alpha (2\alpha -1)\right| }}\right\} ,\\&\left| a_{3}\right| \le \min \left\{ \frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{2\left| 2\alpha -1\right| ^{2}}+\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{4\left| 3\alpha -1\right| }\right. ,\\&\quad \left. \frac{\left| 4\alpha (\alpha +1)-2\right| \left| s^{\prime \prime }(0)\right| +\left| 4\alpha (\alpha -2)+2\right| \left| t^{\prime \prime }(0)\right| }{4\left| 2\alpha (2\alpha -1)\right| \left| 3\alpha -1\right| }\right\} , \end{aligned}$$

and

$$\begin{aligned} \left| a_{3}-\varsigma a_{2}^{2}\right| \le \frac{\left| t^{\prime \prime }(0)\right| }{\left| 3\alpha -1\right| }, \end{aligned}$$

where

$$\begin{aligned} \varsigma =\frac{2\alpha ^{2}+2\alpha -1}{3\alpha -1}. \end{aligned}$$

Putting \(\alpha =1\) in Corollary 3.3, we obtain the following Corollary:

Corollary 3.4

Let f(z) given by (1.1) be in the class \(\mathbf {B} _{s,t}(1,1,0).\) Then

$$\begin{aligned}&\left| a_{2}\right| \le \min \left\{ \sqrt{\frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{2}},\sqrt{\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{4}}\right\} ,\\&\left| a_{3}\right| \le \min \left\{ \frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{2} +\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{8},\frac{3\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{8}\right\} , \end{aligned}$$

and

$$\begin{aligned} \left| a_{3}-\varsigma a_{2}^{2}\right| \le \frac{\left| t^{\prime \prime }(0)\right| }{2}, \end{aligned}$$

where

$$\begin{aligned} \varsigma =\frac{3}{2}. \end{aligned}$$

Remark 3.5

The estimates for coefficients \(|a_{2}|\) and \(|a_{3}|\) in Corollary 3.4 obtained by Bulut [5]

Putting \(\sigma =1\) in Corollary 3.2, we obtain the following Corollary:

Corollary 3.6

Let f(z) given by (1.1) be in the class \(\mathbf {B} _{s,t}(1,1,1).\) Then

$$\begin{aligned}&\left| a_{2}\right| \le \min \left\{ \sqrt{\frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{8}},\sqrt{\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{12}}\right\} ,\\&\left| a_{3}\right| \le \min \left\{ \frac{\left| s^{\prime }(0)\right| ^{2}+\left| t^{\prime }(0)\right| ^{2}}{8} +\frac{\left| s^{\prime \prime }(0)\right| +\left| t^{\prime \prime }(0)\right| }{12},\frac{\left| s^{\prime \prime }(0)\right| }{6}\right\} , \end{aligned}$$

and

$$\begin{aligned} \left| a_{3}-2a_{2}^{2}\right| \le \frac{\left| t^{\prime \prime }(0)\right| }{3}. \end{aligned}$$

Remark 3.7

The estimates for coefficients \(|a_{2}|\) and \(|a_{3}|\) in Corollary 3.6 obtained by Xu et al. [16]