A New Pseudo-Type κ-Fold Symmetric Bi-Univalent Function Class

Abstract

We introduce and study a new pseudo-type $\kappa$-fold symmetric bi-univalent function class that meets certain subordination conditions in this article. For functions in the newly formed class, the initial coefficient bounds are obtained. For members in this class, the Fekete–Szegö issue is also estimated. In addition, we uncover pertinent links to previous results and give a few observations.

1. Preliminaries

Let $\{\varsigma \in \mathbb{C}:|\varsigma |<1\}$ = $\mathfrak{D}$, where $\mathbb{C}$ is the set of all complex numbers. Let $\mathcal{A}$ denote the class of all regular functions of the type

$$s\left(\varsigma \right)=\varsigma +\sum _{\mathit{j}=2}^{\infty }{d}_j{\varsigma }^j$$

(1)

with $s\left(0\right)=s^{\prime }\left(0\right)-1=0$, $\varsigma \in \mathfrak{D}$ and $\mathcal{S}$ denote the subfamily of functions $\in \mathcal{A}$ which are univalent in $\mathfrak{D}$. For $\tau \ge 1$, the class of $\tau$-pseudo-convex functions is defined as

$${\mathcal{K}}^{\tau }=\left\{s\in \mathcal{A}:\mathfrak{R}\left(\frac{{\left\{{\left(\varsigma s^{\prime }\left(\varsigma \right)\right)}^{\prime }\right\}}^{\tau }}{s\left(\varsigma \right)}\right)>0,\varsigma \in \mathfrak{D}\right\},$$

the class of $\tau$-pseudo-starlike functions is given by

$${\mathcal{S}}^{\tau }=\left\{s\in \mathcal{A}:\mathfrak{R}\left(\frac{\varsigma {\left\{s^{\prime }\left(\varsigma \right)\right\}}^{\tau }}{s\left(\varsigma \right)}\right)>0,\varsigma \in \mathfrak{D}\right\}$$

and the class of $\tau$-pseudo-bounded turning is introduced as

$${\mathcal{R}}^{\tau }=\left\{s\in \mathcal{A}:\mathfrak{R}{\left(s^{\prime }\left(\varsigma \right)\right)}^{\tau }>0,\right\},(\varsigma \in \mathfrak{D}),$$

The class ${\mathcal{K}}^{\tau }$ was explored by Guney and Murugusundaramoorthy [1] and the class ${\mathcal{S}}^{\tau }$ was examined in [2]. We note that ${\mathcal{S}}^1=\mathcal{S}$. Al-Amiri and Reade [3] presented the class $\mathfrak{M}\left(\nu \right)(\nu <1)$ of functions $s\in \mathcal{A}$ with $s^{\prime }\left(\varsigma \right)\ne 0in\mathfrak{D}$ which satisfy

$$\mathfrak{R}\left(\nu \frac{{\left(\varsigma s^{\prime }\left(\varsigma \right)\right)}^{\prime }}{s^{\prime }\left(\varsigma \right)}+(1-\nu )s^{\prime }\left(\varsigma \right)\right)>0,(\varsigma \in \mathfrak{D}).$$

In [4], Sukhjit Singh and Sushma Gupta gave certain criteria for univalence by proving $\Re \left(s^{\prime }\left(\varsigma \right)\right)>0$, whenever

$$\mathfrak{R}\left(\nu \frac{{\left(\varsigma s^{\prime }\left(\varsigma \right)\right)}^{\prime }}{s^{\prime }\left(\varsigma \right)}+(1-\nu )s^{\prime }\left(\varsigma \right)\right)>\xi ,(0\le \nu <1,0\le \xi <1,\varsigma \in \mathfrak{D}).$$

The Koebe theorem (see [5]) ensures that $s\left(\mathfrak{D}\right)$, $s\in \mathcal{S}$, contains a disc of radius 1/4. Thus, any function s admits an inverse $g=s^{-1}$ defined by $g\left(s\right(\varsigma \left)\right)=\varsigma$, and $s\left(g\left(\varkappa \right)\right)=\varkappa ,\left|\varkappa \right|<r_0\left(s\right),r_0\left(s\right)\ge 1/4$, $\varsigma \in \mathfrak{D}$, $\varkappa \in \mathfrak{D},$ where

$$g\left(\varkappa \right)=\varkappa -d_2{\varkappa }^2+(2d_2^2-d_3){\varkappa }^3-(5d_2^3-5d_2{d}_3+d_4){\varkappa }^4+\cdots$$

(2)

If s$\in \mathcal{S}$ and $s^{-1}$$\in \mathcal{S}$, then a member s of $\mathcal{A}$ given by (1) is called bi-univalent in $\mathfrak{D}$ and the collection of such functions in $\mathfrak{D}$ is symbolized by $\sigma$. For a brief study, and to know some interesting properties of the family $\sigma$, see [6]. Some subfamilies of the family $\sigma$ that are comparable to the well-known subfamilies of the family S have been introduced by Tan [7], Brannan and Taha [8], and Srivastava et al. [9]. In fact, as sequels to the above subfamilies of $\sigma$, a number of different subfamilies of $\sigma$ have since then been explored by many authors (see, for example, [10,11,12,13,14]). Most of these works are devoted to the study of the Fekete–Szegö issue of functions in various subfamilies of $\sigma$.

Let $\mathbb{N}=\{1,2,3,\cdots \}$ and $\mathbb{R}=(-\infty ,+\infty )$.

If, for $\kappa \in \mathbb{N}$, $s\left(e^{\frac{2\pi i}{\kappa }}\varsigma \right)=e^{\frac{2\pi i}{\kappa }}s\left(\varsigma \right)$, $\varsigma \in \mathfrak{D}$, then a regular function s is called a $\kappa$-fold symmetric ($\kappa$-FS). The function s, defined by $s\left(\varsigma \right)={\left(f\left({\varsigma }^{\kappa }\right)\right)}^{1/\kappa }$, $\kappa \in \mathbb{N}$, $f\in \mathcal{S}$, is univalent and maps $\mathfrak{D}$ into a $\kappa$-fold symmetry region. We indicate by ${\mathcal{S}}_{\kappa }$ the class of $\kappa$-fold symmetric univalent ($\kappa$-FSU) functions in $\mathfrak{D}$. A function $s\in {\mathcal{S}}_{\kappa }$ has the following form:

$$s\left(\varsigma \right)=\varsigma +\sum _{\mathit{j}=1}^{\infty }{d}_{\kappa j+1}·{\varsigma }^{\kappa j+1}(\kappa \in \mathbb{N};\varsigma \in \mathfrak{D}).$$

(3)

Clearly ${\mathcal{S}}_1=\mathcal{S}$.

Similar to the idea of ${\mathcal{S}}_{\kappa }$, Srivastava et al. [15] investigated the class ${\sigma }_{\kappa }$ of $\kappa$-fold symmetric bi-univalent ($\kappa$-FSBU) functions. A few intriguing findings were made, including the series

$$\begin{array}{c}{s}^{-1}\left(\varkappa \right)=\varkappa -d_{\kappa +1}{\varkappa }^{\kappa +1}+[(1+\kappa )d_{\kappa +1}^2-d_{2\kappa +1}]{\varkappa }^{2\kappa +1}\hfill \\ -\left[\frac{1}{2}(1+\kappa )(2+3\kappa )d_{\kappa +1}^3-(2+3\kappa )d_{\kappa +1}{d}_{2\kappa +1}+d_{3\kappa +1}\right]{\varkappa }^{3\kappa +1}+\cdots .\hfill \end{array}$$

(4)

when $s\in {\sigma }_{\kappa }$.

Note that the functions

$$s_1\left(\varsigma \right)={\left(\frac{1}{2}log\left(\frac{1+{\varsigma }^{\kappa }}{1-{\varsigma }^{\kappa }}\right)\right)}^{1/\kappa },s_2\left(\varsigma \right)={\left(\frac{{\varsigma }^{\kappa }}{1-{\varsigma }^{\kappa }}\right)}^{1/\kappa },s_3\left(\varsigma \right)={(-log(1-{\varsigma }^{\kappa }))}^{1/\kappa },\cdots .$$

with the corresponding inverses

$$g_1\left(\varkappa \right)={\left(\frac{e^{2{\varkappa }^{\kappa }}-1}{e^{2{\varkappa }^{\kappa }}-1}\right)}^{1/\kappa },g_2\left(\varkappa \right)={\left(\frac{{\varkappa }^{\kappa }}{1+{\varkappa }^{\kappa }}\right)}^{1/\kappa },g_3\left(\varkappa \right)={\left(\frac{e^{{\varkappa }^{\kappa }}-1}{e^{{\varkappa }^{\kappa }}}\right)}^{1/m},\cdots .$$

are elements of ${\sigma }_{\kappa }$. We obtain (2) from (4) on taking $\kappa =1$.

The focus on the initial coefficients of functions in some subfamilies of ${\sigma }_{\kappa }$ is an interesting topic and this opened an area for many developments. New subfamilies of ${\sigma }_{\kappa }$ were introduced and examined in depth by many researchers (see, for example, [16,17,18,19]). We mention here some recent works on this topic. Initial coefficient bounds for new subfamilies of ${\sigma }_{\kappa }$ were determined in [20]. The Fekete–Szegö (FS) issue $|d_{2m+1}-\delta d_{m+1}^2|$, $\delta \in \mathbb{R}$ (see [21]) for certain special families of ${\sigma }_{\kappa }$ was examined by Swamy et al. [22,23]; and another special family of ${\sigma }_{\kappa }$ satisfying certain subordination conditions was examined by Aldawish et al. [24]; initial coefficients estimates for elements belonging to certain new families of ${\sigma }_{\kappa }$ were obtained by Breaz and Cotîrlă in [25] (see [26,27,28]), indicating the developments in this domain.

For functions $s_1$ and $s_2$ regular in $\mathfrak{D}$, $s_1$ is said to subordinate $s_2$, if there is a Schwarz function $\psi$ in $\mathfrak{D}$, such that $\psi \left(0\right)=0$, $\left|\psi \right(z\left)\right|<1$ and $s_1\left(z\right)=s_2\left(\psi \left(z\right)\right),z\in \mathfrak{D}$. This subordination is indicated as $s_1\prec s_2$. If $s_2\in \mathcal{S}$, then $s_1\left(z\right)\prec s_2\left(z\right)$ is equivalent to $s_1\left(0\right)=s_2\left(0\right)$ and $s_1\left(\mathfrak{D}\right)\subset s_2\left(\mathfrak{D}\right)$.

Inspired by the efforts of Al-Amiri [3] and the authors of [19], we introduce a new class ${\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\phi )$, $\eta \in {\mathbb{C}}^{\ast }=\mathbb{C}-\left\{0\right\},0\le \nu \le 1,$ and $\phi \left(\varsigma \right)$ is a regular function, such that $\mathfrak{R}\left(\phi \right(\varsigma \left)\right)>0$, ${\phi }^{\prime }\left(0\right)>0$, $\phi \left(0\right)=1$, $\phi \left(\mathfrak{D}\right)$ is symmetric with respect to the real axis. In Section 2, we estimate the upper bounds of $|d_{\kappa +1}|$, $|d_{2\kappa +1}|$ and $|d_{2\kappa +1}-\delta d_{\kappa +1}^2|$ $(\delta \in \mathbb{R})$, for functions that belong to the class ${\displaystyle {\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\phi )}$. We consider two special cases ${\mathfrak{Q}}_{{\sigma }_{\kappa }}^{\varrho }(\eta ,\nu ,\tau )={\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,{\left(\frac{1+\varsigma }{1-\varsigma }\right)}^{\varrho }),0<\varrho \le 1$ and ${\mathfrak{X}}_{{\sigma }_{\kappa }}^{\xi }(\eta ,\nu ,\tau )={\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\frac{1+(1-2\xi )\varsigma }{1-\varsigma }),$$0\le \xi <1$, in Section 3 and Section 4, respectively. We also identify connections to existing results and present a few new observations.

2. The Class ${\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\phi )$

Throughout this paper, $s^{-1}\left(\varkappa \right)=g\left(\varkappa \right)$ is as in (4), $\eta \in {\mathbb{C}}^{*}=\mathbb{C}\setminus \left\{0\right\}$, $\varsigma \in \mathfrak{D}$, $\varkappa \in \mathfrak{D}$ and $\phi \left(\varsigma \right)$ will be a regular function such that $\mathfrak{R}\left(\phi \right(\varsigma \left)\right)>0$, ${\phi }^{\prime }\left(0\right)>0$, $\phi \left(0\right)=1$, and $\phi \left(\mathfrak{D}\right)$ is symmetric with respect to the real axis. An expansion of $\phi \left(\varsigma \right)$ has the form:

$$\phi \left(\varsigma \right)=1+B_1\varsigma +B_2{\varsigma }^2+B_3{\varsigma }^3+\cdots (B_1>0).$$

(5)

Let $\mathbf{P}$ be the class of regular functions of the type $p\left(\varsigma \right)=1+p_1\varsigma +p_2{\varsigma }^2+p_3{\varsigma }^3+\cdots ,$ $\mathfrak{R}\left(p\right(\varsigma \left)\right)>0$. A $\kappa$-FS function $p_{\kappa }$$\in \mathbf{P}$ is of the form $p_{\kappa }\left(\varsigma \right)=1+p_{\kappa }{\varsigma }^{\kappa }+p_{2\kappa }{\varsigma }^{2\kappa }+p_{3\kappa }{\varsigma }^{3\kappa }+\cdots$ (see [29]).

Let $\mathfrak{h}\left(\varsigma \right)$ and $\mathfrak{p}\left(\varkappa \right)$ be regular in $\mathfrak{D}$ with $max\left\{\right|\mathfrak{h}\left(\varsigma \right)|,|\mathfrak{p}\left(\varkappa \right)\left|\right\}$< 1 and $\mathfrak{h}\left(0\right)=0=\mathfrak{p}\left(0\right)$. We suppose that $\mathfrak{h}\left(\varsigma \right)=h_{\kappa }{\varsigma }^{\kappa }+h_{2\kappa }{\varsigma }^{2\kappa }+h_{3\kappa }{\varsigma }^{3\kappa }+\cdots$ and $\mathfrak{p}\left(\varkappa \right)=p_{\kappa }{\varkappa }^{\kappa }+p_{2\kappa }{\varkappa }^{2\kappa }+p_{3\kappa }{\varkappa }^{3\kappa }+\cdots$. Also, we assume that

$$|h_{\kappa }|<1;|h_{2\kappa }|\le 1-|h_{\kappa }{|}^2;|p_{\kappa }|<1;|p_{2\kappa }|\le 1-|p_{\kappa }{|}^2.$$

(6)

After simple computations, using (5), we have

$$\phi \left(\mathfrak{h}\left(\varsigma \right)\right)=1+B_1{h}_{\kappa }{\varsigma }^{\kappa }+(B_1{h}_{2\kappa }+B_2{h}_{\kappa }^2){\varsigma }^{2\kappa }+...$$

(7)

and

$$\phi \left(\mathfrak{p}\left(\varkappa \right)\right)=1+B_1{p}_{\kappa }{\varkappa }^{\kappa }+(B_1{p}_{2\kappa }+B_2{p}_{\kappa }^2){\varkappa }^{2\kappa }+....$$

(8)

Definition 1.

A function $s\in {\sigma }_{\kappa }$ of the form (3) is said to be in the class ${\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\phi )$ if

$$\frac{1}{\eta }\left(\nu \frac{{\left\{{\left(\varsigma s^{\prime }\left(\varsigma \right)\right)}^{\prime }\right\}}^{\tau }}{s^{\prime }\left(\varsigma \right)}+(1-\nu ){\left(s^{\prime }\left(\varsigma \right)\right)}^{\tau }-1\right)+1\prec \phi \left(\varsigma \right)$$

and

$$\frac{1}{\eta }\left(\nu \frac{{\left\{{\left(\varkappa g^{\prime }\left(\varkappa \right)\right)}^{\prime }\right\}}^{\tau }}{g^{\prime }\left(\varkappa \right)}+(1-\nu ){\left(g^{\prime }\left(\varkappa \right)\right)}^{\tau }-1\right)+1\prec \phi \left(\varkappa \right),$$

where $g=s^{-1}$, $\tau \ge 1$, $\eta \in {\mathbb{C}}^{*}$, and $0\le \nu <1$.

Remark 1. (i) The subclass ${\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,0,\phi )\equiv {\mathcal{H}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\phi )$, and was explored in [24].

(ii) ${\mathfrak{P}}_{{\sigma }_{\kappa }}^1(\eta ,\nu ,\phi )\equiv {\mathcal{I}}_{{\sigma }_{\kappa }}(\eta ,\nu ,\phi )$ is the subclass of functions s $\in {\sigma }_{\kappa }$ satisfying

$$\frac{1}{\eta }\left(\nu \frac{{\left(\varsigma s^{\prime }\left(\varsigma \right)\right)}^{\prime }}{s^{\prime }\left(\varsigma \right)}+(1-\nu )s^{\prime }\left(\varsigma \right)-1\right)+1\prec \phi \left(\varsigma \right)$$

and its inverse $g=s^{-1}$ satisfies

$$\frac{1}{\eta }\left(\nu \frac{{\left(\varkappa g^{\prime }\left(\varkappa \right)\right)}^{\prime }}{g^{\prime }\left(\varkappa \right)}+(1-\nu )g^{\prime }\left(\varkappa \right)-1\right)+1\prec \phi \left(\varkappa \right),$$

where $\eta \in {\mathbb{C}}^{*}$ and $0\le \nu <1$.

Theorem 1.

If the function s given by (3) belongs to the family ${\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\phi )$ and $\delta \in \mathbb{R}$, then

$$\begin{array}{c}|d_{\kappa +1}|\le \hfill \\ \frac{\left|\eta \right|B_1\sqrt{2B_1}}{\sqrt{|\{M(1+\kappa )+[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]{(1+\kappa )}^2\}\eta B_1^2-2L^2{B}_2|+2L^2{B}_1}},\hfill \end{array}$$

(9)

$$\begin{array}{c}|d_{2\kappa +1}|\le \hfill \\ \left\{\begin{array}{c}\frac{B_1\left|\eta \right|}{M};0<B_1<\frac{2L^2}{\left|\eta \right|M(1+\kappa )}\hfill \\ \frac{B_1\left|\eta \right|}{M}+\left(\frac{1+\kappa }{2}-\frac{L^2}{\left|\eta \right|B_{1}M}\right)\frac{2{\eta }^2{B}_1^3}{|\{M(1+\kappa )+[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]{(1+\kappa )}^2\}\eta B_1^2-2L^2{B}_2|+2L^2{B}_1}\hfill \\ ;B_1\ge \frac{2L^2}{\left|\eta \right|M(1+\kappa )},\hfill \end{array}\right.\hfill \end{array}$$

(10)

and

$$|d_{2\kappa +1}-\delta d_{\kappa +1}^2|\le \left\{\begin{array}{c}\frac{B_1\left|\eta \right|}{M};|1+\kappa -2\delta |<J\hfill \\ \frac{{\left|\eta \right|}^2{B}_1^3|\kappa -2\delta +1|}{|\{M(1+\kappa )+[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]{(1+\kappa )}^2\}\eta B_1^2-2L^2{B}_2|};|1+\kappa -2\delta |\ge J,\hfill \end{array}\right.$$

(11)

where

$$J=\left|\frac{\{M(1+\kappa )+[N\tau (\tau -1)+2\nu (1-(1+\kappa )\tau )]{(1+\kappa )}^2\}\eta B_1^2-2L^2{B}_2}{\eta M{B}_1^2}\right|.$$

(12)

$$L=(1+\kappa )\left(\tau \right(1+\nu \kappa )-\nu ),$$

(13)

$$M=\left(\tau \right(1+2\nu \kappa )-\nu )(1+2\kappa )$$

(14)

and

$$N=1+\nu \kappa (2+\kappa ).$$

(15)

Proof.

Let the function s of the form (3) belong to the family ${\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\phi )$. Then, we have regular functions $\mathfrak{h},\mathfrak{p}:\mathfrak{D}⟶\mathfrak{D}$, $\mathfrak{h}\left(0\right)=\mathfrak{p}\left(0\right)=0$ satisfying

$$\frac{1}{\eta }\left(\nu \frac{{\left\{{\left(\varsigma s^{\prime }\left(\varsigma \right)\right)}^{\prime }\right\}}^{\tau }}{s^{\prime }\left(\varsigma \right)}+(1-\nu ){\left(s^{\prime }\left(\varsigma \right)\right)}^{\tau }-1\right)+1=\phi \left(\mathfrak{h}\left(\varsigma \right)\right),$$

(16)

and

$$\frac{1}{\eta }\left(\nu \frac{{\left\{{\left(\varkappa g^{\prime }\left(\varkappa \right)\right)}^{\prime }\right\}}^{\tau }}{g^{\prime }\left(\varkappa \right)}+(1-\nu ){\left(g^{\prime }\left(\varkappa \right)\right)}^{\tau }-1\right)+1=\phi \left(\mathfrak{p}\left(\varkappa \right)\right).$$

(17)

Using (3) in (16) and (17) we obtain:

$$\frac{1}{\eta }\left(\nu \frac{[{\left\{{\left(\varsigma s^{\prime }\left(\varsigma \right)\right)}^{\prime }\right\}}^{\tau }}{s^{\prime }\left(\varsigma \right)}+(1-\nu ){\left(s^{\prime }\left(\varsigma \right)\right)}^{\tau }-1\right)+1=$$

$$\begin{array}{cc}\hfill \frac{1}{\eta }& \left\{L{d}_{\kappa +1}{\varsigma }^{\kappa }+\left[M{d}_{2\kappa +1}+\right.\right.\hfill \\ & \left.\left.{(1+\kappa )}^2\left(\frac{N\tau (\tau -1)}{2}+\nu (1-(1+\kappa )\tau )\right)d_{\kappa +1}^2\right]{\varsigma }^{2\kappa }+\cdots \right\}+1\hfill \\ \end{array}$$

(18)

and

$$\frac{1}{\eta }\left(\nu \frac{{\left\{{\left(\varkappa g^{\prime }\left(\varkappa \right)\right)}^{\prime }\right\}}^{\tau }}{g^{\prime }\left(\varkappa \right)}+(1-\nu ){\left(g^{\prime }\left(\varkappa \right)\right)}^{\tau }-1\right)+1=$$

$$\begin{array}{cc}\hfill \frac{1}{\eta }& \left\{-L{d}_{\kappa +1}{\varkappa }^{\kappa }+\left[M((1+\kappa )d_{\kappa +1}^2-d_{2\kappa +1})+\right.\right.\hfill \\ & \left.\left.{(1+\kappa )}^2\left(\frac{N\tau (\tau -1)}{2}+\nu (1-(1+\kappa )\tau )\right)d_{\kappa +1}^2\right]{\varkappa }^{2\kappa }+\cdots \right\}+1,\hfill \\ \end{array}$$

(19)

where L, M, and N are as in (13), (14), and (15), respectively.

Comparing (7) and (18), we obtain

$$L{d}_{\kappa +1}=\eta B_1{h}_{\kappa }$$

(20)

and

$$M{d}_{2\kappa +1}+\left(\frac{N\tau (\tau -1)}{2}+\nu (1-(1+\kappa )\tau )\right){(1+\kappa )}^2{d}_{\kappa +1}^2=\eta [B_1{h}_{2\kappa }+B_2{h}_{\kappa }^2].$$

(21)

Comparing (8) and (19), we obtain

$$-L{d}_{\kappa +1}=\eta B_1{p}_{\kappa }$$

(22)

and

$$\begin{array}{c}\hfill M((\kappa +1)d_{\kappa +1}^2-d_{2\kappa +1})+\left(\frac{N\tau (\tau -1)}{2}+\nu (1-(1+\kappa )\tau )\right){(1+\kappa )}^2{d}_{\kappa +1}^2\\ \hfill =\eta [B_1{p}_{2\kappa }+B_2{p}_{\kappa }^2],\end{array}$$

(23)

From (20) and (22), we obtain

$$h_{\kappa }=-p_{\kappa }$$

(24)

and

$$2L^2{d}_{\kappa +1}^2={\eta }^2{B}_1^2(h_{\kappa }^2+p_{\kappa }^2).$$

(25)

We add (21) and (23) and then use (25) to obtain

$$\begin{array}{c}\hfill [\{M(1+\kappa )+[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]{(1+\kappa )}^2\}\eta B_1^2-2L^2{B}_2]d_{\kappa +1}^2\\ \hfill ={\eta }^2{B}_1^3{h}_{2\kappa }+p_{2\kappa }\end{array}$$

(26)

By using (6) and (20) in (26) for the coefficients $h_{2\kappa }$ and $p_{2\kappa }$, we obtain

$$\begin{array}{c}\hfill \left[\right|\{M(1+\kappa )+[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]{(1+\kappa )}^2\}\eta B_1^2-2L^2{B}_2|+2L^2{B}_1\left]\right|d_{\kappa +1}{|}^2\\ \hfill \le 2{\eta }^2{B}_1^3\end{array},$$

(27)

which implies (9).

We subtract (23) from (21) to find the bound on $|d_{2\kappa +1}|$:

$$d_{2\kappa +1}=\frac{\eta B_1(h_{2\kappa }-p_{2\kappa })}{2M}+\left(\frac{1+\kappa }{2}\right)d_{\kappa +1}^2.$$

(28)

In view of (20), (24), (28) and applying (6), we obtain

$$|d_{2\kappa +1}|\le \frac{\left|\eta \right|B_1}{M}+\left(\frac{1+\kappa }{2}-\frac{L^2}{\left|\eta \right|B_{1}M}\right)\times$$

(29)

$$\times \frac{2{\eta }^2{B}_1^3}{|\{M(1+\kappa )+[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]{(1+\kappa )}^2\}\eta B_1^2-2L^2{B}_2|+2L^2{B}_1},$$

which obtains (10), the desired assessment.

From (26) and (28), for $\delta \in \mathbb{R}$, we obtain

$$d_{2\kappa +1}-\delta d_{\kappa +1}^2=\frac{\eta B_1}{2}\left[\left(\mathfrak{Y}\left(\delta \right)+\frac{1}{M}\right)h_{2\kappa }+\left(\mathfrak{Y}\left(\delta \right)-\frac{1}{M}\right)p_{2\kappa }\right],$$

where

$$\mathfrak{Y}\left(\delta \right)=\frac{\eta B_1^2(\kappa -2\delta +1)}{\{M(1+\kappa )+[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]{(\kappa +1)}^2\}\eta B_1^2-2L^2{B}_2}.$$

In view of (6), we conclude that

$$|d_{2\kappa +1}-\delta d_{\kappa +1}^2|\le \left\{\begin{array}{cc}\frac{\left|\eta \right|B_1}{M}\hfill & ;0\le \left|\mathfrak{Y}\left(\delta \right)\right|<\frac{1}{M}\hfill \\ \left|\eta \right|B_1\left|\mathfrak{Y}\left(\delta \right)\right|\hfill & ;\left|\mathfrak{Y}\left(\delta \right)\right|\ge \frac{1}{M},\hfill \end{array}\right.$$

form which we obtain (11) with J as in (12). So the proof is completed.

Remark 2.

We obtain Corollary 1 of [24] if $\nu =0$ in Theorem 1.

Choosing $\tau =1$ in ${\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\phi )$, we have the corollary given below:

Corollary 1.

Let $\delta \in \mathbb{R}$ and let the function s given by (3) be in the family ${\mathcal{I}}_{{\sigma }_{\kappa }}(\eta ,\nu ,\phi )$. Then,

$$\begin{array}{c}|d_{\kappa +1}|\le \frac{\left|\eta \right|B_1\sqrt{2B_1}}{\sqrt{|\{(1+\kappa )M_1-2\nu \kappa {(1+\kappa )}^2\}\eta B_1^2-2L_1^2{B}_2|+2L_1^2{B}_1}},\hfill \\ |d_{2\kappa +1}|\le \hfill \\ \left\{\begin{array}{c}\frac{\left|\eta \right|B_1}{M_1};0<B_1<\frac{2L_1^2}{(1+\kappa )M_1\left|\eta \right|}\hfill \\ \frac{\left|\eta \right|B_1}{M_1}+\left(\frac{1+\kappa }{2}-\frac{L_1^2}{B_1{M}_1\left|\eta \right|}\right)\frac{2{\eta }^2{B}_1^3}{|\{(1+\kappa )M_1-2\nu \kappa {(1+\kappa )}^2\}\eta B_1^2-2L_1^2{B}_2|+2L_1^2{B}_1};B_1\ge \frac{2L_1^2}{(1+\kappa )M_1\left|\eta \right|}\hfill \end{array}\right.\hfill \end{array}$$

and

$$|d_{2\kappa +1}-\delta d_{\kappa +1}^2|\le \left\{\begin{array}{c}\frac{\left|\eta \right|B_1}{M_1};|\kappa -2\delta +1|<J_1\hfill \\ \frac{B_1^3{|\kappa -2\delta +1|\left|\eta \right|}^2}{|\{(\kappa +1)M_1-2\nu \kappa {(\kappa +1)}^2\}\eta B_1^2-2L_1^2{B}_2|};|\kappa -2\delta +1|\ge J_1,\hfill \end{array}\right.$$

where

$$J_1=\left|\frac{\{(1+\kappa )M_1-2\nu \kappa {(1+\kappa )}^2\}\eta B_1^2-2L_1^2{B}_2}{M_1{B}_1^2\eta }\right|,$$

$$L_1=(1+\kappa )((\kappa -1)\nu +1)$$

(30)

and

$$M_1=(1+2\kappa )((2\kappa -1)\nu +1),$$

(31)

Remark 3.

If $\nu =0$ and $\eta =1$ in Corollary 1 are allowed, then the first and second theorems of Tang et al. [19] are obtained.

Choosing $\kappa =1$ in Theorem 1, we have

Corollary 2.

If s $\in {\mathfrak{P}}_{{\sigma }_1}^{\tau }(\eta ,\nu ,\phi )$ is given by (1) and $\delta \in \mathbb{R}$, then

$$\begin{array}{c}|d_2|\le \frac{\left|\eta \right|B_1\sqrt{B_1}}{\sqrt{|\{M_2+2(N_2\tau (\tau -1)+2\nu (1-2\tau ))\}\eta B_1^2-L_2^2{B}_2|+L_2^2{B}_1}},\hfill \\ |d_3|\le \left\{\begin{array}{l}\frac{\left|\eta \right|B_1}{M_2};0<B_1<\frac{L_2^2}{\left|\eta \right|M_2}\hfill \\ \frac{\left|\eta \right|B_1}{M_2}+\left(1-\frac{L_2^2}{\left|\eta \right|B_1{M}_2}\right)\frac{{\eta }^2{B}_1^3}{|\{M_2+2(N_2\tau (\tau -1)+2\nu (1-2\tau ))\}\eta B_1^2-L_2^2{B}_2|+L_2^2{B}_1};B_1\ge \frac{L_2^2}{\left|\eta \right|M_2},\hfill \end{array}\right.\hfill \end{array}$$

and

$$|d_3-\delta d_2^2|\le \left\{\begin{array}{c}\frac{\left|\eta \right|B_1}{M_2};|1-\delta |<J_2\hfill \\ \frac{{\left|\eta \right|}^2{B}_1^3|1-\delta |}{|\{M_2+2(N_2\tau (\tau -1)+2\nu (1-2\tau ))\}\eta B_1^2-L_2^2{B}_2|};|1-\delta |\ge J_2,\hfill \end{array}\right.$$

where

$$J_2=\left|\frac{\{M_2+2(N_2\tau (\tau -1)+2\nu (1-2\tau ))\}\eta B_1^2-L_2^2{B}_2}{\eta M_2{B}_1^2}\right|,$$

$$L_2=2((1+\nu )\tau -\nu ),$$

(32)

$$M_2=3((1+2\nu )\tau -\nu )$$

(33)

and

$$N_2=3\nu +1.$$

(34)

Setting $\eta =\tau =1$ in Corollary 2, we obtain the following.

Corollary 3.

If $s\in$${\mathfrak{P}}_{{\sigma }_1}^1(1,\nu ,\phi )$ is given by (1) and $\delta \in \mathbb{R}$, then

$$\begin{array}{c}|d_2|\le \frac{B_1\sqrt{B_1}}{\sqrt{|(3-\nu )B_1^2-4B_2|+4B_1}},\hfill \\ |d_3|\le \left\{\begin{array}{c}\frac{B_1}{3(\nu +1)};0<B_1<\frac{4}{3(\nu +1)}\hfill \\ \frac{B_1}{3(\nu +1)}+\left(1-\frac{4}{3(\nu +1)B_1}\right)\frac{B_1^3}{|(3-\nu )B_1^2-4B_2|+4B_1};B_1\ge \frac{4}{3(\nu +1)},\hfill \end{array}\right.\hfill \end{array}$$

and

$$|d_3-\delta d_2^2|\le \left\{\begin{array}{c}\frac{B_1}{3(\nu +1)};|1-\delta |<\left|\frac{(3-\nu )B_1^2-4B_2}{3(\nu +1)B_1^2}\right|\hfill \\ \frac{B_1^3|1-\delta |}{|(3-\nu )B_1^2-4B_2|};|1-\delta |\ge \left|\frac{(3-\nu )B_1^2-4B_2}{3(\nu +1)B_1^2}\right|.\hfill \end{array}\right.$$

Remark 4.

When $\nu =0$ is selected in Corollary 3, we obtain Corollaries 1 and 4 of Tang et al. [19] (also see [30]).

3. The Class ${\mathfrak{Q}}_{{\mathbf{\sigma }}_{\mathbf{\kappa }}}^{{}^{\mathbf{\tau }}}(\mathbf{\eta },\mathbf{\nu },\mathbf{\varrho })$

Let $\phi \left(\varsigma \right)=1+2\varrho \varsigma +2{\varrho }^2{\varsigma }^2+\cdots ={\left(\frac{1+\varsigma }{1-\varsigma }\right)}^{\varrho }$ in Definition 1. Then, we have the subclass of all $s\in {\sigma }_{\kappa }$ satisfying

$$\left|arg\left[\frac{1}{\eta }\left(\nu \frac{{\left\{{\left(\varsigma s^{\prime }\left(\varsigma \right)\right)}^{\prime }\right\}}^{\tau }}{s^{\prime }\left(\varsigma \right)}+(1-\nu ){\left(s^{\prime }\left(\varsigma \right)\right)}^{\tau }-1\right)+1\right]\right|<\frac{\varrho \pi }{2}$$

and

$$\left|arg\left[\frac{1}{\eta }\left(\nu \frac{{\left\{{\left(\varkappa g^{\prime }\left(\varkappa \right)\right)}^{\prime }\right\}}^{\tau }}{g^{\prime }\left(\varkappa \right)}+(1-\nu ){\left(g^{\prime }\left(\varkappa \right)\right)}^{\tau }-1\right)+1\right]\right|<\frac{\varrho \pi }{2},$$

where $g=s^{-1}$, $0<\varrho \le 1$, $\eta \in {\mathbb{C}}^{*},\tau \ge 1$, and $0\le \nu <1$. We denote this class by ${\mathfrak{Q}}_{{\sigma }_{\kappa }}^{{}^{\tau }}(\eta ,\nu ,\varrho )={\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,{\left(\frac{1+\varsigma }{1-\varsigma }\right)}^{\varrho })$.

Remark 5. (i) The family ${\mathfrak{Q}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,0,\varrho )\equiv {\mathcal{B}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\varrho )$, and was explored in [24], where $\eta \in {\mathbb{C}}^{*}$, $\tau \ge 1$ and $0<\varrho \le 1$.

(ii) ${\mathfrak{Q}}_{{\sigma }_{\kappa }}^1(\eta ,\nu ,\varrho )\equiv {\mathcal{D}}_{{\sigma }_{\kappa }}(\eta ,\nu ,\varrho )$ is the subfamily of all s $\in {\sigma }_{\kappa }$ satisfying

$$\left|arg\left[\frac{1}{\eta }\left(\nu \frac{{\left(\varsigma s^{\prime }\left(\varsigma \right)\right)}^{\prime }}{s^{\prime }\left(\varsigma \right)}+(1-\nu )s^{\prime }\left(\varsigma \right)-1\right)+1\right]\right|<\frac{\varrho \pi }{2}$$

and its inverse $g=s^{-1}$ satisfies

$$\left|arg\left[\frac{1}{\eta }\left(\nu \frac{{\left(\varkappa g^{\prime }\left(\varkappa \right)\right)}^{\prime }}{g^{\prime }\left(\varkappa \right)}+(1-\nu )g^{\prime }\left(\varkappa \right)-1\right)+1\right]\right|<\frac{\varrho \pi }{2},$$

where $0<\varrho \le 1$, $\eta \in {\mathbb{C}}^{*}$ and $0\le \nu <1$.

Taking $\phi \left(\varsigma \right)={\left(\frac{1+\varsigma }{1-\varsigma }\right)}^{\varrho }$ in Theorem 1, we obtain

Corollary 4.

If the function s given by (3) belongs to the family $\in {\mathfrak{Q}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\varrho )$ and $\delta \in \mathbb{R}$, then

$$\begin{array}{c}|d_{\kappa +1}|\le \frac{2\varrho \left|\eta \right|}{\sqrt{\varrho |\{M(1+\kappa )+[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]{(1+\kappa )}^2\}\eta -L^2|+L^2}},\hfill \\ |d_{2\kappa +1}|\le \hfill \\ \left\{\begin{array}{c}\frac{2\varrho \left|\eta \right|}{M};0<\varrho <\frac{L^2}{M(1+\kappa )\left|\eta \right|}\hfill \\ \frac{2\varrho \left|\eta \right|}{M}+\left(1+\kappa -\frac{L^2}{\varrho M\left|\eta \right|}\right)\frac{2{\varrho }^2{\eta }^2}{\varrho |\{(1+\kappa )M+[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]{(1+\kappa )}^2\}\eta -L^2|+L^2};\varrho \ge \frac{L^2}{M(1+\kappa )\left|\eta \right|},\hfill \end{array}\right.\hfill \end{array}$$

and

$$|d_{2\kappa +1}-\delta d_{\kappa +1}^2|\le \left\{\begin{array}{c}\frac{2\varrho \left|\eta \right|}{M};|\kappa -2\delta +1|<J_3\hfill \\ \frac{{2\varrho |\kappa -2\delta +1|\left|\eta \right|}^2}{|\{(1+\kappa )M+[\tau (\tau -1)N+(1-(\kappa +1)\tau )2\nu ]{(\kappa +1)}^2\}\eta -L^2|};|\kappa -2\delta +1|\ge J_3,\hfill \end{array}\right.$$

where

$$J_3=\left|\frac{\{M(1+\kappa )+[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]{(1+\kappa )}^2\}\eta -L^2}{\eta M}\right|,$$

L, M, and N are as in (13), (14), and (15), respectively.

Remark 6.

For $\nu =0$ in Corollary 4, we obtain Corollary 4 in [24].

Choosing $\tau =1$ in ${\mathfrak{Q}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\phi )$, we obtain the corollary given below:

Corollary 5.

If the function s given by (3) belongs to the family ${\mathcal{D}}_{{\sigma }_{\kappa }}(\eta ,\nu ,\varrho )$ and $\delta \in \mathbb{R}$, then

$$\begin{array}{c}|d_{\kappa +1}|\le \frac{2\varrho \left|\eta \right|}{\sqrt{\varrho |\{M_1(1+\kappa )-2\nu \kappa {(1+\kappa )}^2\}\eta -L_1^2|+L_1^2}},\hfill \\ |d_{2\kappa +1}|\le \hfill \\ \left\{\begin{array}{c}\frac{2\varrho \left|\eta \right|}{M_1};0<\varrho <\frac{L_1^2}{M_1(1+\kappa )\left|\eta \right|}\hfill \\ \frac{2\varrho \left|\eta \right|}{M_1}+\left(1+\kappa -\frac{L_1^2}{\varrho M_1\left|\eta \right|}\right)\frac{2{\varrho }^2{\eta }^2}{\varrho |\{M_1(1+\kappa )-2\nu \kappa {(1+\kappa )}^2\}\eta -L_1^2|+L^2};\varrho \ge \frac{L_1^2}{(1+\kappa )M_1\left|\eta \right|}\hfill \end{array}\right.\hfill \end{array}$$

and

$$|d_{2\kappa +1}-\delta d_{\kappa +1}^2|\le \left\{\begin{array}{c}\frac{2\varrho \left|\eta \right|}{M_1};|\kappa -2\delta +1|<J_4\hfill \\ \frac{{2\varrho |\kappa -2\delta +1|\left|\eta \right|}^2}{|\{M_1(\kappa +1)-2\nu \kappa {(\kappa +1)}^2\}\eta -L_1^2|};|\kappa -2\delta +1|\ge J_4,\hfill \end{array}\right.$$

where

$$J_4=\left|\frac{\{M_1(1+\kappa )-2\nu \kappa {(1+\kappa )}^2\}\eta -L_1^2}{\eta M_1}\right|,$$

$L_1$ and $M_1$ are as in (30) and (31), respectively.

Corollary 4 yields the following if $\kappa =1$:

Corollary 6.

If s $\in {\mathfrak{Q}}_{{\sigma }_1}^{\tau }(\eta ,\nu ,\varrho )$ is given by (1) and $\delta \in \mathbb{R}$, then

$$\begin{array}{c}|d_2|\le \frac{2\left|\eta \right|\varrho }{\sqrt{\varrho |\{2M_2+4(N_2\tau (\tau -1)+2\nu (1-2\tau ))\}\eta -L_2^2|+L_2^2}},\hfill \\ \left|d_3\right|\le \left\{\begin{array}{c}\frac{2\left|\eta \right|\varrho }{M_2};0<\varrho <\frac{L_2^2}{2\left|\eta \right|M_2}\hfill \\ \frac{2\left|\eta \right|\varrho }{M_2}+\left(2-\frac{L_2^2}{\left|\eta \right|\varrho M_2}\right)\frac{2{\eta }^2{\varrho }^2}{\varrho |\{2M_2+4(N_2\tau (\tau -1)+2\nu (1-2\tau ))\}\eta -L_2^2|+L_2^2};\varrho \ge \frac{L_2^2}{2\left|\eta \right|M_2},\hfill \end{array}\right.\hfill \end{array}$$

and

$$|d_3-\delta d_2^2|\le \left\{\begin{array}{c}\frac{2\left|\eta \right|\varrho }{M_2};|1-\delta |<J_5\hfill \\ \frac{{2\left|\eta \right|}^2\varrho |1-\delta |}{|\{2M_2+4(N_2\tau (\tau -1)+2\nu (1-2\tau ))\}\eta -L_2^2|};|1-\delta |\ge J_5,\hfill \end{array}\right.$$

where

$$J_5=\left|\frac{\{2M_2+4(N_2\tau (\tau -1)+2\nu (1-2\tau ))\}\eta -L_2^2}{2\eta M_2}\right|,$$

$L_2,M_2$, and $N_2$ are as in (32), (33), and (34), respectively.

Corollary 6 would yield the following if $\eta =\tau =1$.

Corollary 7.

If the function s of the form (1) $\in {\mathfrak{Q}}_{{\sigma }_1}^1(1,\nu ,\phi )$ and $\delta \in \mathbb{R}$, then

$$\begin{array}{c}|d_2|\le \frac{\varrho \sqrt{2}}{\sqrt{\varrho (1-\nu )+2}},\hfill \\ \left|d_3\right|\le \left\{\begin{array}{c}\frac{2\varrho }{3(\nu +1)};0<\varrho <\frac{2}{3(\nu +1)}\hfill \\ \frac{2\varrho }{3(\nu +1)}+\left(1-\frac{2}{3\varrho (\nu +1)}\right)\frac{2{\varrho }^2}{\varrho (1-\nu )+2};\varrho \ge \frac{2}{3(\nu +1)},\hfill \end{array}\right.\hfill \end{array}$$

and

$$|d_3-\delta d_2^2|\le \left\{\begin{array}{c}\frac{2\varrho }{3(\nu +1)};|1-\delta |<\frac{1-\nu }{3(\nu +1)}\hfill \\ \frac{\varrho |1-\delta |}{1-\nu };|1-\delta |\ge \frac{1-\nu }{3(\nu +1)}.\hfill \end{array}\right.$$

Remark 7.

Letting $\nu =0$ in Corollary 7, we obtain Corollary 2 of Tang et al. [19]. The estimate obtained here for $|d_3|$ is more accurate when compared to that in Theorem 2 of Srivastava et al. [9].

4. The Class ${\mathfrak{X}}_{{\mathbf{\sigma }}_{\mathbf{\kappa }}}^{\mathbf{\tau }}(\mathbf{\eta },\mathbf{\nu },\mathbf{\xi })$

If $\phi \left(\varsigma \right)=1+2(1-\xi )\varsigma +2(1-\xi ){\varsigma }^2+\cdots =\frac{1+(1-2\xi )\varsigma }{1-\varsigma }$ in Definition 1, then we have the subset of all $s\in {\sigma }_{\kappa }$ satisfying

$$\mathfrak{R}\left[\left(\nu \frac{{\left\{{\left(\varsigma s^{\prime }\left(\varsigma \right)\right)}^{\prime }\right\}}^{\tau }}{s^{\prime }\left(\varsigma \right)}+(1-\nu ){\left(s^{\prime }\left(\varsigma \right)\right)}^{\tau }-1\right)\frac{1}{\eta }+1\right]>\xi$$

and

$$\mathfrak{R}\left[\left(\nu \frac{{\left\{{\left(\varkappa g^{\prime }\left(\varkappa \right)\right)}^{\prime }\right\}}^{\tau }}{g^{\prime }\left(\varkappa \right)}+(1-\nu ){\left(g^{\prime }\left(\varkappa \right)\right)}^{\tau }-1\right)\frac{1}{\eta }+1\right]>\xi ,$$

where $g=s^{-1},\eta \in {\mathbb{C}}^{*},0\le \xi <1,\tau \ge 1$, and $0\le \nu <1$. We denote this set by ${\mathfrak{X}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\xi )={\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\left(\frac{1+(1-2\xi )\varsigma }{1-\varsigma }\right)$.

Remark 8. (i). The family ${\mathfrak{X}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,0,\xi )\equiv {\mathcal{E}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\xi )$, $\tau \ge 1,0\le \xi <1$, and was studied in [24].(ii). ${\mathfrak{X}}_{{\sigma }_{\kappa }}^1(\eta ,\nu ,\xi )\equiv {\mathcal{F}}_{{\sigma }_{\kappa }}(\eta ,\nu ,\xi )$ is a set of all s $\in {\sigma }_{\kappa }$ satisfying

$$\mathfrak{R}\left[\left(\nu \frac{{\left(\varsigma s^{\prime }\left(\varsigma \right)\right)}^{\prime }}{s^{\prime }\left(\varsigma \right)}+(1-\nu )s^{\prime }\left(\varsigma \right)-1\right)\frac{1}{\eta }+1\right]>\xi$$

and its inverse $g=s^{-1}$ satisfies

$$\mathfrak{R}\left[\left(\nu \frac{{\left(\varkappa g^{\prime }\left(\varkappa \right)\right)}^{\prime }}{g^{\prime }\left(\varkappa \right)}+(1-\nu )g^{\prime }\left(\varkappa \right)-1\right)\frac{1}{\eta }+1\right]>\xi ,$$

where $\eta \in {\mathbb{C}}^{*},0\le \xi <1$, and $0\le \nu <1$.

Allowing $\phi \left(\varsigma \right)=\frac{1+(1-2\xi )\varsigma }{1-\varsigma }$, $0\le \xi <1,$ in Theorem 1, we obtain

Corollary 8.

Let the function s of the form (3) belong to the class ${\mathfrak{X}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\xi )$ and $\delta \in \mathbb{R}$. Then,

$$|d_{\kappa +1}|\le \frac{(1-\xi )2\left|\eta \right|}{\sqrt{|\{(1+\kappa )M+{(1+\kappa )}^2[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]\}(1-\xi )\eta -L^2|+L^2}},$$

$$|d_{2\kappa +1}|\le$$

$$\left\{\begin{array}{c}\frac{(1-\xi )2\left|\eta \right|}{M};1-\frac{L^2}{(1+\kappa )M\left|\eta \right|}<\xi <1\hfill \\ \frac{(1-\xi )2\left|\eta \right|}{M}+\left(1+\kappa -\frac{L^2}{(1-\xi )M\left|\eta \right|}\right)\frac{2{(1-\xi )}^2{\left|\eta \right|}^2}{|\{(1+\kappa )M+[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]{(1+\kappa )}^2\}(1-\xi )\eta -L^2|+L^2}\hfill \\ ;0\le \xi \le 1-\frac{L^2}{(1+\kappa )M\left|\eta \right|}\hfill \end{array}\right.$$

and

$$|d_{2\kappa +1}-\delta d_{\kappa +1}^2|\le \left\{\begin{array}{c}\frac{(1-\xi )2\left|\eta \right|}{M};|\kappa -2\delta +1|<J_6\hfill \\ \frac{2{(1-\xi )}^2{\left|\eta \right|}^2|\kappa -2\delta +1|}{|\{(1+\kappa )M+[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]{(1+\kappa )}^2\}(1-\xi )\eta -L^2|};|\kappa -2\delta +1|\ge J_6,\hfill \end{array}\right.$$

where

$$J_6=\left|\frac{\{(1+\kappa )M+{(1+\kappa )}^2[N\tau (\tau -1)+(1-(1+\kappa )\tau )2\nu ]\}(1-\xi )\eta -L^2}{M(1-\xi )\eta }\right|.$$

L, M, and N are as in (13), (14), and (15), respectively.

Remark 9.

We obtain Corollary 7 of Aldawish et al. [24] if $\nu =0$ in Corollary 8. In addition, we obtain Corollary 11 of Swamy et al. [22] when $\eta =\tau =1$.

Corollary 9.

Let the function s of the form (3) belong to the class ${\mathcal{F}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\xi )$ and $\delta \in \mathbb{R}$. Then,

$$\begin{array}{c}|d_{\kappa +1}|\le \frac{(1-\xi )2\left|\eta \right|}{\sqrt{|\{(1+\kappa )M_1-2\nu \kappa {(1+\kappa )}^2\}(1-\xi )\eta -L_1^2|+L_1^2}},\hfill \\ \left|d_{2\kappa +1}\right|\le \\ \{\begin{array}{c}\frac{(1-\xi )2\left|\eta \right|}{M_1};1-\frac{L_1^2}{(\kappa +1)M_1\left|\eta \right|}<\xi <1\hfill \\ \frac{(1-\xi )2\left|\eta \right|}{M_1}+\left(1+\kappa -\frac{L_1^2}{(1-\xi )M_1\left|\eta \right|}\right)\frac{2{(1-\xi )}^2{\left|\eta \right|}^2}{|\{(1+\kappa )M_1-2\nu \kappa {(1+\kappa )}^2\}\eta (1-\xi )-L_1^2|+L_1^2}\hfill \\ ;0\le \xi \le 1-\frac{L_1^2}{(1+\kappa )M_1\left|\eta \right|}\hfill \end{array}\end{array}$$

and

$$|d_{2\kappa +1}-\delta d_{\kappa +1}^2|\le \left\{\begin{array}{c}\frac{2\left|\eta \right|(1-\xi )}{M_1};|\kappa -2\delta +1|<J_7\hfill \\ \frac{{2\left|\eta \right|}^2{(1-\xi )}^2|\kappa -2\delta +1|}{|\{(1+\kappa )M_1-2\nu \kappa {(1+\kappa )}^2\}(1-\xi )\eta -L_1^2|};|\kappa -2\delta +1|\ge J_7,\hfill \end{array}\right.$$

where

$$J_7=\left|\frac{\{(1+\kappa )M_1-2\nu \kappa {(1+\kappa )}^2\}(1-\xi )\eta -L_1^2}{M_1(1-\xi )\eta }\right|,$$

$L_1$ and $M_1$ are as in (30) and (31), respectively.

If we let $\kappa =1$ in Corollary 8, then we have

Corollary 10.

Let the function s of the form (1) belong to the class ${\mathfrak{X}}_{{\sigma }_1}^{\tau }(\eta ,\nu ,\xi )$ and $\delta \in \mathbb{R}$. Then,

$$\begin{array}{c}|d_2|\le \frac{2(1-\xi )\left|\eta \right|}{\sqrt{|\{2M_2+4(N_2\tau (\tau -1)+2\nu (1-2\tau ))\}(1-\xi )\eta -L_2^2|+L_2^2}},\hfill \\ \left|d_3\right|\le \left\{\begin{array}{c}\frac{2(1-\xi )\left|\eta \right|}{M_2};1-\frac{L_2^2}{2M_2\left|\eta \right|}<\xi <1\hfill \\ \frac{2(1-\xi )\left|\eta \right|}{M_2}+\left(2-\frac{L_2^2}{(1-\xi )M_2\left|\eta \right|}\right)\frac{2{(1-\xi )}^2{\left|\eta \right|}^2}{|\{2M_2+4(N\tau (\tau -1)+2\nu (1-2\tau ))\}(1-\xi )\eta -L_2^2|+L_2^2}\hfill \\ ;0\le \xi \le 1-\frac{L_2^2}{2M_2\left|\eta \right|}\hfill \end{array}\right.\hfill \end{array}$$

and

$$|d_3-\delta d_2^2|\le \left\{\begin{array}{c}\frac{2(1-\xi )\left|\eta \right|}{3(\nu +1)};|1-\delta |<J_8\hfill \\ \frac{2{(1-\xi )}^2{\left|\eta \right|}^2|1-\delta |}{|\{2M_2+4(N_2\tau (\tau -1)+2\nu (1-2\tau ))\}(1-\xi )\eta -L_2^2|};|1-\delta |\ge J_8,\hfill \end{array}\right.$$

where

$$J_8=\left|\frac{\{2M_2+4(N\tau (\tau -1)+2\nu (1-2\tau ))\}(1-\xi )\eta -L_2^2}{2M_2(1-\xi )\eta }\right|,$$

$L_2,M_2$, and $N_2$ are as in (32), (33), and (34), respectively.

If $\eta =\tau =1$ in Corollary 10, then we obtain

Corollary 11.

If s $\in {\mathfrak{X}}_{{\sigma }_1}^1(1,\nu ,\phi )$ is of the form (1) and $\delta \in \mathbb{R}$, then

$$\begin{array}{c}|d_2|\le \frac{\sqrt{2}(1-\xi )}{\sqrt{\left|\right(3-\nu \left)\right(1-\xi )-2|+2}},\hfill \\ \left|d_3\right|\le \left\{\begin{array}{c}\frac{2(1-\xi )}{3(1+\nu )};\frac{3\nu +1}{3(1+\nu )}<\xi <1\hfill \\ \frac{2(1-\xi )}{3(1+\nu )}+\left(1-\frac{2}{3(1-\xi )(1+\nu )}\right)\frac{2{(1-\xi )}^2}{\left|\right(1-\xi \left)\right(3-\nu )-2|+2};0\le \xi \le \frac{3\nu +1}{3(1+\nu )},\hfill \end{array}\right.\hfill \end{array}$$

and

$$|d_3-\delta d_2^2|\le \left\{\begin{array}{c}\frac{2(1-\xi )}{3(1+\nu )};|1-\delta |<\left|\frac{(1-\xi )(3-\nu )-2}{3(1-\xi )(1+\nu )}\right|\hfill \\ \frac{{2|1-\delta |(1-\xi )}^2}{\left|\right(1-\xi \left)\right(3-\nu )-2|};|1-\delta |\ge \left|\frac{(3-\nu )(1-\xi )-2}{3(1-\xi )(1+\nu )}\right|,\hfill \end{array}\right.$$

Remark 10.

Putting $\nu =0$ in Corollary 11, we obtain Corollary 3 of Tang et al. [19]. The estimates obtained here for $|d_2|$ and $|d_3|$ are more accurate when compared to those estimates of Theorem 2 in [9].

5. Conclusions

In this paper, a new class ${\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\phi )$ is explored and the upper bounds of $|d_{\kappa +1}|$, $|d_{2\kappa +1}|$, and $|d_{2\kappa +1}-\delta d_{\kappa +1}^2|$ $\delta \in \Re ,$ are estimated for elements in ${\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\phi )$. Two special cases ${\mathfrak{Q}}_{{\sigma }_{\kappa }}^{\varrho }(\eta ,\nu ,\tau )={\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,{\left(\frac{1+\varsigma }{1-\varsigma }\right)}^{\varrho }),0<\varrho \le 1$ and ${\mathfrak{X}}_{{\sigma }_{\kappa }}^{\xi }(\eta ,\nu ,\tau )={\mathfrak{P}}_{{\sigma }_{\kappa }}^{\tau }(\eta ,\nu ,\frac{1+(1-2\xi )\varsigma }{1-\varsigma })$, $0\le \xi <1$, have been considered. In addition, we have uncovered pertinent links to previous results and given a few observations. This paper could inspire researchers towards further investigations using the (i) integro-differential operator [31], (ii) q-differential operator [32], (iii) q-integral operator [33], and (iv) Hohlov operator [34].