Two Families of Bi-Univalent Functions Associating the (p, q)-Derivative with Generalized Bivariate Fibonacci Polynomials

Abstract

Making use of generalized bivariate Fibonacci polynomials, we propose two families of regular functions of the type $\varphi \left(\zeta \right)=\zeta +{\displaystyle \sum _{j=2}^{\infty }}{d}_j{\zeta }^j$, which are bi-univalent in the disc $\{\zeta \in \mathbb{C}:|\zeta |<1\}$ involving the (p, q)-derivative operator. We find estimates on the coefficients $|d_2|$, $|d_3|$ and the of Fekete–Szegö functional for members of these families. Relevant connections to the existing results and new consequences of the main result are presented.

1. Preliminaries

The q-analysis is a generalization of the ordinary analysis that does not employ limit notation. Jackson presented the use and application of the q-calculus in [1]. The q-analogue of the derivative and integral operator were first defined in [2], along with some of their applications. The multifaceted uses of the q-derivative operator make it extremely important in the study of geometric function theory. Initially, Ismail et al. proposed the concept of q-extension of the class of q-starlike functions in [3]. Later, a number of mathematicians studied q-calculus in the context of geometric functions theory: the Ruscheweyh differential operator’s q-analogue was first presented in [4]; some applications for multivalent functions were examined in [5,6]; q-starlike functions associated with the generalized conic domain, by applying the convolution concept was investigated in [7]; an operator associated with q-hypergeometric function was investigated in [8], and so forth. Numerous authors have recently released a number of articles that examine a class of q-starlike functions from different perspectives (see [9,10,11,12,13,14]).

An extension of the q-calculus to the (p, q)-calculus, was taken into consideration by the researchers. The (p, q)-number, was first examined around the same time (1991) and subsequently on its own by [15,16,17,18]. Fibonacci oscillators were studied with the presentation of the (p, q)-number in [15]. The investigation of the (p, q)-number in [16] allows for the construction of a (p, q)-Harmonic oscillator. The (p, q)-number was investigated in [17] as a means of combining different types of q-oscillator algebras and in [18], the (p, q)-number was examined in order to determine the (p, q)-Stirling numbers.

Building on the aforementioned publications, since 1991, a large number of scientists have investigated (p, q)-calculus in a range of research domains. The results in [19] provided a syntax for embedding the q-series into a (p, q)-series. They also investigated (p, q)-hypergeometric series and found some results corresponding to (p, q)-extensions of the known q-identities. The q-identities are extended correspondingly to yield the (p, q)-series (see, e.g., [20]). We provide some basic definitions of the (p, q)-calculus concepts used in this paper. The (p, q)-bracket number is given by ${\left[j\right]}_{p,q}=p^{j-1}+p^{j-2}q+p^{j-3}{q}^2+\cdots +p{q}^{j-2}+q^{j-1}={\displaystyle \frac{p^j-q^j}{p-q}}(p\ne q),$ which is an extension of q-number (see [2]), that is ${\left[j\right]}_q={\displaystyle \frac{1-q^j}{1-q}}(q\ne 1)$. Note that ${\left[j\right]}_{p,q}$ is symmetric and if p = 1, then ${\left[j\right]}_{p,q}={\left[j\right]}_q$.

Let $\{\zeta \in \mathbb{C}:|\zeta |<1\}$ = $\mathfrak{D}$ be the open unit disk, and $\mathbb{C}$ be the complex plane. Let $\mathbb{N}$ represent the natural number set and $\mathbb{R}$ be the real number set.

Definition 1 

([21]). Let φ be a function defined on $\mathbb{C}$ and $0<q<p\le 1$. Then, the (p, q)-derivative of φ is defined by

$$D_{p,q}\phi \left(\zeta \right)={\displaystyle \frac{\phi \left(p\zeta \right)-\phi \left(q\zeta \right)}{(p-q)\zeta }}(\zeta \ne 0),$$

and $D_{p,q}\phi \left(0\right)={\phi }^{\prime }\left(0\right)$, provided ${\phi }^{\prime }\left(0\right)$ exists.

We note that $D_{p,q}{\zeta }^j={\left[j\right]}_{p,q}{\zeta }^{j-1},D_{p,q}ln\left(\zeta \right)={\displaystyle \frac{ln(p/q)}{(p-q)\zeta }}$ and ${\left[j\right]}_{p,q}\to j,\mathrm{if}p=1\mathrm{and}q\to 1^{-}$. Therefore, $D_{p,q}\phi \left(\zeta \right)\to {\phi }^{\prime }\left(\zeta \right)$ as $p=1\mathrm{and}q\to 1^{-}$. For any constants $\delta$ and $\kappa$, it is obvious that $D_{p,q}(\delta {\phi }_1\left(\zeta \right)+\kappa {\phi }_2\left(\zeta \right))=\delta D_{p,q}{\phi }_1\left(\zeta \right)+\kappa D_{p,q}{\phi }_2\left(\zeta \right)$. The product rules and quotient rules are satisfied by the (p, q)-derivative (see [22]). The exponential functions are used to define the (p, q)-analogues of many functions, including sine, cosine, and tangent, in the same way as their well-known Euler expressions. Duran et al. [23] examined the (p, q)-derivatives of these functions.

The set of functions $\varphi$, which are regular in $\mathfrak{D}$ and have the following form, is represented by $\mathcal{A}$:

$$\varphi \left(\zeta \right)=\zeta +\sum _{j=2}^{\infty }{d}_j{\zeta }^j,(\zeta \in \mathfrak{D}),$$

(1)

with $\varphi \left(0\right)=0={\varphi }^{\prime }\left(0\right)-1$, and we denote a sub-set of $\mathcal{A}$ containing univalent functions in $\mathfrak{D}$ by $\mathcal{S}$. If $\varphi \in \mathcal{A}$ is of the form (1), then

$$D_{p,q}\varphi \left(\zeta \right)=1+\sum _{j=2}^{\infty }{\left[j\right]}_{p,q}{d}_j{\zeta }^{j-1},(\zeta \in \mathfrak{D}).$$

(2)

The Koebe theorem (see [24]) states that the inverse of each function $\varphi$ in $\mathcal{S}$ is given by

$${\varphi }^{-1}\left(\varpi \right)=\psi \left(\varpi \right)=\varpi -d_2{\varpi }^2+(2d_2^2-d_3){\varpi }^3-(5d_2^3-5d_2{d}_3+d_4){\varpi }^4+\cdots .$$

(3)

satisfying $\zeta ={\varphi }^{-1}\left(\varphi \left(\zeta \right)\right)$ and $\varpi =\varphi \left({\varphi }^{-1}\left(\varpi \right)\right),\left|\varpi \right|<r_0\left(\varphi \right),r_0\left(\varphi \right)\ge 1/4,\zeta ,\varpi \in \mathfrak{D}$. In $\mathfrak{D}$, a member $\varphi$ of $\mathcal{A}$ given by (1) is called bi-univalent if $\varphi$ $\in \mathcal{S}$ and ${\varphi }^{-1}$$\in \mathcal{S}$. The set of such functions in $\mathfrak{D}$ is represented by $\sigma$. ${\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{1+\zeta }{1-\zeta }}\right),-log(1-\zeta )$, and ${\displaystyle \frac{\zeta }{1-\zeta }}$ are some of the functions in the $\sigma$ family. Nevertheless, despite being in $\mathcal{S}$, $\zeta -{\displaystyle \frac{{\zeta }^2}{2}}$, ${\displaystyle \frac{\zeta }{1-{\zeta }^2}}$, and the Koebe function does not belong to $\sigma$. For a concise analysis and to discover some of the remarkable characteristics of the family $\sigma$, see [25,26,27,28] and the citation provided in these papers. Srivastava et al. [29] introduced several subclasses of the family $\sigma$, which are similar to the well-known subclasses of the family $\mathcal{S}$. As follow-ups, numerous authors have since examined a number of different subfamilies of $\sigma$ (see, for instance, [30,31,32,33,34]).

Several subclasses of the class $\sigma$ were studied using the (p, q)-calculus. In [35], the (p, q)-derivative operator and the subordination principle were used to introduce the new generalized classes of (p, q)-starlike and (p, q)-convex functions. The Fekete-Szegö inequalities are also examined and the (p, q)-Bernardi integral operator for analytic functions is defined. The (p, q)-Bernardi integral operator was used to obtain some applications of the main results. Several studies have also introduced and investigated novel subclasses of the class $\sigma$ related to the (p, q)-differential operator (see [36,37,38,39,40,41]).

Let $k(\varkappa ,y)$ and $l(\varkappa ,y)$ be polynomials with real coefficients. For, $j\ge 2$, the generalized bivariate Fibonacci polynomials (GBFP) are defined by the recurrence relation:

$${\mathcal{F}}_j(\varkappa ,y)=k(\varkappa ,y){\mathcal{F}}_{j-1}(\varkappa ,y)+l(\varkappa ,y){\mathcal{F}}_{j-2}(\varkappa ,y),$$

(4)

where ${\mathcal{F}}_0(\varkappa ,y)=0$, ${\mathcal{F}}_1(\varkappa ,y)=1$ and $k^2(\varkappa ,y)+4l(\varkappa ,y)>0.$ The generating function of GBFP is (see [42])

$$\mathcal{F}(\varkappa ,y,\zeta )=\sum _{j=0}^{\infty }{\mathcal{F}}_j(\varkappa ,y){\zeta }^j={\displaystyle \frac{\zeta }{1-k(\varkappa ,y)\zeta -l(\varkappa ,y){\zeta }^2}}.$$

(5)

For specific selections of $k(\varkappa ,y)$ and $l(\varkappa ,y)$, GBFP leads to various known polynomials (see [43]). Readers with an interest in GBFP can find a brief history and extensive information in [44] and its references. Interesting findings regarding coefficient estimates and Fekete–Szegö functional have been presented in [45,46,47] for members of certain subclasses of $\sigma$ associated with GBFP.

For brevity, we write hereafter that $k(\varkappa ,y)=k$ and $l(\varkappa ,y)=l$. ${\mathcal{F}}_2(\varkappa ,y)=k$ and ${\mathcal{F}}_3(\varkappa ,y)=k^2+l$ are evident from (4).

Remark 1. 

By specializing k and l, many polynomial sequences can be inferred from GBFP. They are as follows: (i) the bivariate Fibonacci polynmials are obtained if $k=\varkappa$ and $l=y$; (ii) the Pell polynomials are achieved if $k=2\varkappa$ and $l=1$; (iii) we derive the Jacobsthal polynomials if $k=1$ and $l=\varkappa$; and (iv) we arrive at the Fermat polynomials, if $k=3\varkappa$ and $l=-2$, and so forth.

For ${\Omega }_1$, ${\Omega }_2\in \mathcal{A}$ holomorphic in $\mathfrak{D}$, ${\Omega }_1$ is subordinate to ${\Omega }_2$, if there is $h\left(\zeta \right)$ a Schwarz function that is holomorphic in $\mathfrak{D}$ with $h\left(0\right)=0$ and $\left|h\right(\zeta \left)\right|<1$, such that ${\Omega }_1\left(\zeta \right)={\Omega }_2\left(h\left(\zeta \right)\right)$. The notation ${\Omega }_1\prec {\Omega }_2$ or ${\Omega }_1\left(\zeta \right)\prec {\Omega }_2\left(\zeta \right)$ $(\zeta \in \mathfrak{D})$ denotes this subordination. Specifically, when ${\Omega }_2\in \mathcal{S}$, we have

$${\Omega }_1\left(\zeta \right)\prec {\Omega }_2\left(\zeta \right)\iff {\Omega }_1\left(0\right)={\Omega }_2\left(0\right)and{\Omega }_1\left(\mathfrak{D}\right)\subset {\Omega }_2\left(\mathfrak{D}\right).$$

We present two new families of $\sigma$ subordinate to GBFP ${\mathcal{F}}_j(\varkappa ,y)$ as in (4), with the generating function $\mathcal{F}(\varkappa ,y,z)$ as in (5). These families are motivated by the aforementioned trends on coefficient-related problems and the Fekete–Szegö functional [48] on certain subclasses of $\sigma$ linked to GBFP. $\mathcal{F}(\varkappa ,y,z)$ is as in (5), ${\varphi }^{-1}\left(\varpi \right)=\psi \left(\varpi \right)$, an inverse function as in (3),$\varkappa \in \mathbb{R},\zeta \in \mathfrak{D}$, and $\varpi \in \mathfrak{D}$ are assumed throughout this paper, unless otherwise mentioned.

Definition 2. 

A function ϕ given by (1), which is said to be in the set ${\mathfrak{Y}}_{\sigma ,p,q}^{\lambda ,\eta }(\gamma ,F),$$\gamma \ge 1,0<\eta \le 1and\lambda \ge 1$, if

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\gamma {\left[D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\varphi \left(\zeta \right)}}+{\left({\displaystyle \frac{\gamma {\left[D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\varphi \left(\zeta \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\zeta \right)$$

$$={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},$$

and

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\gamma {\left[D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\psi \left(\varpi \right)}}+{\left({\displaystyle \frac{\gamma {\left[D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\psi \left(\varpi \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\varpi \right)$$

$$={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

where

$$F\left(\zeta \right)={\displaystyle \frac{1}{1-k\zeta -l{\zeta }^2}},k\ne 0,and{k}^2+4l>0.$$

(6)

For specific choices of $p,q,\gamma$ and $\lambda$, the family ${\mathfrak{Y}}_{\sigma ,p,q}^{\lambda ,\eta }(\gamma ,F)$ includes many new and existing subfamilies of $\sigma$. This is shown below:

1. let $\lambda =1$. Then ${\mathfrak{H}}_{\sigma ,p,q}^{\eta }(\gamma ,F)\equiv {\mathfrak{Y}}_{\sigma ,p,q}^{1,\eta }(\gamma ,F),\gamma \ge 1,0<\eta \le 1$ is the collection of members $\varphi$ of $\sigma$ that satisfy

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\gamma D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)+(1-\gamma )}{D_{p,q}\varphi \left(\zeta \right)}}+{\left({\displaystyle \frac{\gamma D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)+(1-\gamma )}{D_{p,q}\varphi \left(\zeta \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\zeta \right)$$

$$={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},$$

and

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\gamma D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)+(1-\gamma )}{D_{p,q}\psi \left(\varpi \right)}}+{\left({\displaystyle \frac{\gamma D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)+(1-\gamma )}{D_{p,q}\psi \left(\varpi \right)}}\right)}^{{\displaystyle \frac{1}{\delta }}}\right\}\prec F\left(\varpi \right)$$

$$={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

where $F\left(\zeta \right)$ is as mentioned in (6).

2. Assume that $\gamma =1$. Then ${\mathfrak{I}}_{\sigma ,p,q}^{\lambda ,\eta }\left(F\right)\equiv {\mathfrak{Y}}_{\sigma ,p,q}^{\lambda ,\eta }(1,F),0<\eta \le 1$ and $\lambda \ge 1,$ is the collection of elements $\varphi$ of $\sigma$ that satisfy

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{{\left[D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)\right]}^{\lambda }}{D_{p,q}\varphi \left(\zeta \right)}}+{\left({\displaystyle \frac{{\left[D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)\right]}^{\lambda }}{D_{p,q}\varphi \left(\zeta \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},$$

and

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{{\left[D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)\right]}^{\lambda }}{D_{p,q}\psi \left(\varpi \right)}}+{\left({\displaystyle \frac{{\left[D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)\right]}^{\lambda }}{D_{p,q}\psi \left(\varpi \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

where $F\left(\zeta \right)$ is as mentioned in (6).

3. If $p=1$ and $q\to 1^{-}$ in the class ${\mathfrak{Y}}_{\sigma ,p,q}^{\lambda ,\eta }(\gamma ,F)$, then we obtain a subset ${{\rm Y}}_{\sigma }^{\lambda ,\eta }(\gamma ,F)(\lambda \ge 1,0<\eta \le 1,\gamma \ge 1)$ which is the collection of members $\varphi$ of $\sigma$ that satisfy

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\gamma {\left[{\left(\zeta {\varphi }^{\prime }\left(\zeta \right)\right)}^{\prime }\right]}^{\lambda }+(1-\gamma )}{{\varphi }^{\prime }\left(\zeta \right)}}+{\left({\displaystyle \frac{\gamma {\left[{\left(\zeta {\varphi }^{\prime }\left(\zeta \right)\right)}^{\prime }\right]}^{\lambda }+(1-\gamma )}{{\varphi }^{\prime }\left(\zeta \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},$$

and

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\gamma {\left[{\left(\varpi {\psi }^{\prime }\left(\varpi \right)\right)}^{\prime }\right]}^{\lambda }+(1-\gamma )}{{\psi }^{\prime }\left(\varpi \right)}}+{\left({\displaystyle \frac{\gamma {\left[{\left(\varpi {\psi }^{\prime }\left(\varpi \right)\right)}^{\prime }\right]}^{\lambda }+(1-\gamma )}{{\psi }^{\prime }\left(\varpi \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

where $F\left(\zeta \right)$ is as mentioned in (6).

4. Let $\eta =1$. Then ${\mathfrak{Z}}_{\sigma ,p,q}^{\lambda }(\gamma ,F)\equiv {\mathfrak{Y}}_{\sigma ,p,q}^{\lambda ,1}(\gamma ,F),\gamma \ge 1and\lambda \ge 1$ is the set of elements $\varphi$ of $\sigma$ that satisfy

$$\left\{{\displaystyle \frac{\gamma {\left[D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\varphi \left(\zeta \right)}}\right\}\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},$$

and

$$\left\{{\displaystyle \frac{\gamma {\left[D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\psi \left(\varpi \right)}}\right\}\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

where $F\left(\zeta \right)$ is as mentioned in (6).

Remark 2. 

(i). ${\mathfrak{H}}_{\sigma ,p,q}^{\eta }(1,F)\equiv {\mathfrak{I}}_{\sigma ,p,q}^{1,\eta }\left(F\right)$, (ii). ${\mathfrak{H}}_{\sigma ,p,q}^1(\gamma ,F)\equiv {\mathfrak{Z}}_{\sigma ,p,q}^1(\gamma ,F)$, and (iii). ${\mathfrak{Z}}_{\sigma ,p,q}^{\lambda }(1,F)\equiv {\mathfrak{I}}_{\sigma ,p,q}^{\lambda ,1}\left(F\right).$

Definition 3. 

A function ϕ given by (1) is said to a member of the set ${\mathfrak{T}}_{\sigma ,p,q}^{\lambda ,\eta }(\nu ,F)$, $0\le \nu \le 1,0<\eta \le 1 and \lambda \ge 1$, if

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\zeta {\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{\lambda }}{\nu \varphi \left(\zeta \right)+(1-\nu )\zeta }}+{\left({\displaystyle \frac{\zeta {\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{\lambda }}{\nu \varphi \left(\zeta \right)+(1-\nu )\zeta }}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},$$

and

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\varpi {\left(D_{p,q}\psi \left(\varpi \right)\right)}^{\lambda }}{\nu \psi \left(\varpi \right)+(1-\nu )\varpi }}+{\left({\displaystyle \frac{\varpi {\left(D_{p,q}\psi \left(\varpi \right)\right)}^{\lambda }}{\nu \psi \left(\varpi \right)+(1-\nu )\varpi }}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

where $F\left(\zeta \right)$ is as mentioned in (6).

The family ${\mathfrak{T}}_{\sigma ,p,q}^{\lambda ,\eta }(\nu ,F)$ contains numerous new and existing subfamilies of $\sigma$ for specific choices of $\nu$ and $\lambda$, as shown below:

• Let $\nu =0$. Then, ${\mathfrak{C}}_{\sigma ,p,q}^{\lambda ,\eta }\left(F\right)\equiv {\mathfrak{T}}_{\sigma ,p,q}^{\lambda ,\eta }(0,F),0<\eta \le 1and\lambda \ge 1,$ is the collection of elements $\varphi$ of $\sigma$ that satisfy

  $${\displaystyle \frac{1}{2}}\left({\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{\lambda }+{\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{{\displaystyle \frac{\lambda }{\eta }}}\right)\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},$$

  and

  $${\displaystyle \frac{1}{2}}\left({\left(D_{p,q}\psi \left(\varpi \right)\right)}^{\lambda }+{\left(D_{p,q}\psi \left(\varpi \right)\right)}^{{\displaystyle \frac{\lambda }{\eta }}}\right)\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

  where $F\left(\zeta \right)$ is as mentioned in (6).
• Let $\nu =1$. Then, ${\mathfrak{D}}_{\sigma ,p,q}^{\lambda ,\eta }\left(F\right)\equiv {\mathfrak{T}}_{\Sigma ,p,q}^{\lambda ,\eta }(1,F),0<\eta \le 1and\lambda \ge 1,$ is the family of members $\varphi$ of $\sigma$ that satisfy

  $${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\zeta {\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{\lambda }}{\varphi \left(\zeta \right)}}+{\left({\displaystyle \frac{\zeta {\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{\lambda }}{\varphi \left(\zeta \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},$$

  and

  $${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\varpi {\left(D_{p,q}\psi \left(\varpi \right)\right)}^{\lambda }}{f\left(\varpi \right)}}+{\left({\displaystyle \frac{\varpi {\left(D_{p,q}\psi \left(\varpi \right)\right)}^{\lambda }}{f\left(\varpi \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

  where $F\left(\zeta \right)$ is as mentioned in (6).
• If $p=1$ and $q\to 1^{-}$ in the class ${\mathfrak{T}}_{\sigma ,p,q}^{\lambda ,\eta }(\nu ,F)$, then we get a subclass ${\Gamma }_{\sigma }^{\lambda ,\eta }(\nu ,F)(\lambda \ge 1,0<\eta \le 1,0\le \nu \le 1)$ of functions $\varphi$$\in \sigma$ satisfying

  $${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\zeta {\left({\varphi }^{\prime }\left(\zeta \right)\right)}^{\lambda }}{\nu \varphi \left(\zeta \right)+(1-\nu )\zeta }}+{\left({\displaystyle \frac{\zeta {\left({\varphi }^{\prime }\left(\zeta \right)\right)}^{\lambda }}{\nu \varphi \left(\zeta \right)+(1-\nu )\zeta }}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},$$

  and

  $${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\varpi {\left({\psi }^{\prime }\left(\varpi \right)\right)}^{\lambda }}{\nu \psi \left(\varpi \right)+(1-\nu )\varpi }}+{\left({\displaystyle \frac{\varpi {\left({\psi }^{\prime }\left(\varpi \right)\right)}^{\lambda }}{\nu \psi \left(\varpi \right)+(1-\nu )\varpi }}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

  where $F\left(\zeta \right)$ is as mentioned in (6).
• Suppose $\eta =1$. Then, ${\mathfrak{O}}_{\sigma ,p,q}^{\lambda }(\nu ,F)\equiv {\mathfrak{T}}_{\sigma ,p,q}^{\lambda ,1}(\nu ,F),0\le \nu \le 1and\lambda \ge 1,$ is the set of functions $\varphi$$\in \sigma$ that satisfy

  $${\displaystyle \frac{\zeta {\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{\lambda }}{\nu \varphi \left(\zeta \right)+(1-\nu )\zeta }}\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},and{\displaystyle \frac{\varpi {\left(D_{p,q}\psi \left(\varpi \right)\right)}^{\lambda }}{\nu \psi \left(\varpi \right)+(1-\nu )\varpi }}\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

  where $F\left(\zeta \right)$ is as mentioned in (6).

Remark 3. 

$\left(i\right).{\mathfrak{D}}_{\sigma ,p,q}^{\lambda ,1}\left(F\right)\equiv {\mathfrak{O}}_{\sigma ,p,q}^{\lambda }(1,F),and$ (ii). ${\Gamma }_{\sigma }^{\lambda ,1}(\nu ,F)\equiv {\mathfrak{O}}_{\sigma ,p=1,q\to 1^{-}}^{\lambda }(\nu ,F).$

Estimates for bounds on $|d_2|$, $|d_3|$, and $|d_3-\mu d_2^2|,\mu \in \mathbb{R}$ for functions in the class ${\mathfrak{Y}}_{\sigma ,p,q}^{\lambda ,\eta }(\gamma ,F)$ are found in Section 2. The upper bounds for $|d_2|$, $|d_3|$, and $|d_3-\mu d_2^2|,\mu \in \mathbb{R}$ for functions in the class ${\mathfrak{T}}_{\sigma ,p,q}^{\lambda ,\eta }(\nu ,F)$ are derived in Section 3. Along with pertinent connections to the published research, interesting results are also presented.

2. Results for the Class ${\mathfrak{Y}}_{\mathit{\sigma },\mathit{p},\mathit{q}}^{\mathit{\lambda },\mathit{\eta }}(\mathbf{\gamma },\mathit{F})$

First, for $\varphi \in {\mathfrak{Y}}_{\sigma ,p,q}^{\lambda ,\eta }(\nu ,F)$, the class as defined in Definition 2, we find the coefficient estimates.

Theorem 1. 

Let $\gamma \ge 1,0<\eta \le 1,and\lambda \ge 1$. If ϕ $\in {\mathfrak{Y}}_{\sigma ,p,q}^{\lambda ,\eta }(\gamma ,F)$, then

$$|d_2|\le$$

(7)

$${\displaystyle \frac{2\eta \left|k\right|\sqrt{\left|k\right|}}{\sqrt{|(2\eta (\eta +1)(P{\left[3\right]}_{p,q}+Q{\left[2\right]}_{p,q}^2)+(1-\eta )R^2{\left[2\right]}_{p,q}^2)k^2-{(\eta +1)}^2{R}^2{\left[2\right]}_{p,q}^2(k^2+l)|}}},$$

$$|d_3|\le {\displaystyle \frac{4{\eta }^2{k}^2}{{(\eta +1)}^2{R}^2{\left[2\right]}_{p,q}^2}}+{\displaystyle \frac{2\eta \left|k\right|}{(\eta +1)P{\left[3\right]}_{p,q}}},$$

(8)

and for $\mu \in \mathbb{R}$

$$\begin{array}{ccc}\hfill & |d_3-\mu d_2^2|\hfill & \hfill \le \left\{\begin{array}{cc}{\displaystyle \frac{2\eta \left|k\right|}{(\eta +1)P{\left[3\right]}_{p,q}}};\hfill & |1-\mu |\le J\hfill \\ {\displaystyle \frac{4{\eta }^2{\left|k\right|}^3|1-\mu |}{|(2\eta (\eta +1)(P{\left[3\right]}_{p,q}+Q{\left[2\right]}_{p,q}^2)+{(1-\eta )}^2{R}^2{\left[2\right]}_{p,q}^2)k^2-{(\eta +1)}^2{R}^2{\left[2\right]}_{p,q}^2(k^2+l)|}};\hfill & |1-\mu |\ge J,\hfill \end{array}\right.\end{array}$$

(9)

where

$$J=\left|{\displaystyle \frac{(2\eta (\eta +1)(P{\left[3\right]}_{p,q}+Q{\left[2\right]}_{p,q}^2)+(1-\eta )R^2{\left[2\right]}_{p,q}^2)k^2-{(\eta +1)}^2{R}^2{\left[2\right]}^2(k^2+l)}{2\eta (1+\eta )P{\left[3\right]}_{p,q}{k}^2}}\right|,$$

(10)

$$P=\gamma \lambda {\left[3\right]}_{p,q}-1,$$

(11)

$$Q=1-\gamma \lambda {\left[2\right]}_{p,q}+{\displaystyle \frac{\gamma \lambda (\lambda -1)}{2}}{\left[2\right]}_{p,q}^2,$$

(12)

and

$$R=\gamma \lambda {\left[2\right]}_{p,q}-1.$$

(13)

Proof. 

Let $\varphi \in {\mathfrak{Y}}_{\sigma ,p,q}^{\lambda ,\eta }(\gamma ,\varkappa )$. Then, due to Definition 2, we obtain

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\gamma {\left[D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\varphi \left(\zeta \right)}}+{\left({\displaystyle \frac{\gamma {\left[D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\varphi \left(\zeta \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}=F\left(\mathfrak{m}\left(\zeta \right)\right)$$

(14)

and

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\gamma {\left[D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\psi \left(\varpi \right)}}+{\left({\displaystyle \frac{\gamma {\left[D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\psi \left(\varpi \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}=F\left(\mathfrak{n}\left(\varpi \right)\right)$$

(15)

where

$$\mathfrak{m}\left(\zeta \right)=m_1\zeta +m_2{\zeta }^2+m_3{\zeta }^3+\cdots and\mathfrak{n}\left(\varpi \right)=n_1\varpi +n_2{\varpi }^2+n_3{\varpi }^3+\cdots ,$$

(16)

are some holomorphic functions with $\left|\mathfrak{m}\right(\zeta \left)\right|<1,\left|\mathfrak{n}\right(\varpi \left)\right|<1,\zeta ,\varpi \in \mathfrak{D}$ and $\mathfrak{m}\left(0\right)=0=\mathfrak{n}\left(0\right)$. It is known that

$$|{\mathfrak{m}}_i|\le 1,and|{\mathfrak{n}}_i|\le 1,(i\in \mathbb{N}).$$

(17)

From (14)–(16), it follows that

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\gamma {\left[D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\varphi \left(\zeta \right)}}+{\left({\displaystyle \frac{\gamma {\left[D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\varphi \left(\zeta \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}=\\ \hfill 1+{\mathcal{F}}_2(\varkappa ,y)\mathfrak{m}\left(\zeta \right)+{\mathcal{F}}_3(\varkappa ,y){\mathfrak{m}}^2\left(\zeta \right)+\cdots ,\end{array}$$

(18)

and

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\gamma {\left[D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\psi \left(\varpi \right)}}+{\left({\displaystyle \frac{\gamma {\left[D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\psi \left(\varpi \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}=\\ \hfill 1+{\mathcal{F}}_2(\varkappa ,y)\mathfrak{n}\left(\varpi \right)+{\mathcal{F}}_3(\varkappa ,y){\mathfrak{n}}^2\left(\varpi \right)+\cdots .\end{array}$$

(19)

In light of (4), Equations (18) and (19) can be expressed as follows.

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\gamma {\left[D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\varphi \left(\zeta \right)}}+{\left({\displaystyle \frac{\gamma {\left[D_{p,q}\left(\zeta D_{p,q}\varphi \left(\zeta \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\varphi \left(\zeta \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}=\\ \hfill 1+{\mathcal{F}}_2(\varkappa ,y){\mathfrak{m}}_1\zeta +[{\mathcal{F}}_2(\varkappa ,y){\mathfrak{m}}_2+{\mathcal{F}}_3(\varkappa ,y){\mathfrak{m}}_1^2]{\zeta }^2+\cdots ,\end{array}$$

(20)

and

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\gamma {\left[D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\psi \left(\varpi \right)}}+{\left({\displaystyle \frac{\gamma {\left[D_{p,q}\left(\varpi D_{p,q}\psi \left(\varpi \right)\right)\right]}^{\lambda }+(1-\gamma )}{D_{p,q}\psi \left(\varpi \right)}}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}=\\ \hfill 1+{\mathcal{F}}_2(\varkappa ,y){\mathfrak{n}}_1\varpi +[{\mathcal{F}}_2(\varkappa ,y){\mathfrak{n}}_2+{\mathcal{F}}_3(\varkappa ,y){\mathfrak{n}}_1^2]{\varpi }^2+\cdots .\end{array}$$

(21)

Comparing (20) and (21), we have

$${\displaystyle \frac{(\eta +1)R{\left[2\right]}_{p,q}}{2\eta }}{d}_2={\mathcal{F}}_2(\varkappa ,y){\mathfrak{m}}_1,$$

(22)

$$\left({\displaystyle \frac{\eta +1}{2\eta }}\right)\left(P{\left[3\right]}_{p,q}{d}_3+Q{\left[2\right]}_{p,q}^2{d}_2^2\right)+\left({\displaystyle \frac{1-\eta }{4{\eta }^2}}\right)R^2{\left[2\right]}_{p,q}^2{d}_2^2={\mathcal{F}}_2(\varkappa ,y){\mathfrak{m}}_2+{\mathcal{F}}_3(\varkappa ,y){\mathfrak{m}}_1^2,$$

(23)

$$-{\displaystyle \frac{(\eta +1)R{\left[2\right]}_{p,q}}{2\eta }}{d}_2={\mathcal{F}}_2(\varkappa ,y){\mathfrak{n}}_1,$$

(24)

and

$$\left({\displaystyle \frac{\eta +1}{2\eta }}\right)\left(P{\left[3\right]}_{p,q}(2d_2^2-d_3)+Q{\left[2\right]}_{p,q}^2{d}_2^2\right)+\left({\displaystyle \frac{1-\eta }{4{\eta }^2}}\right)R^2{\left[2\right]}_{p,q}^2{d}_2^2={\mathcal{F}}_2(\varkappa ,y){\mathfrak{n}}_2+{\mathcal{F}}_3(\varkappa ,y){\mathfrak{n}}_1^2,$$

(25)

where P, Q, and R are as mentioned in (11), (12) and (13), respectively. From (22) and (24), we easily obtain

$${\mathfrak{m}}_1=-{\mathfrak{n}}_1$$

(26)

and also

$${\displaystyle \frac{{(\eta +1)}^2{R}^2{\left[2\right]}_{p,q}^2}{2{\eta }^2}}{d}_2^2=({\mathfrak{m}}_1^2+{\mathfrak{n}}_1^2){\mathcal{F}}_2^2(\varkappa ,y).$$

(27)

By adding (23) and (25), we obtain

$$\left[\left({\displaystyle \frac{\eta +1}{\eta }}\right)(P{\left[3\right]}_{p,q}+Q{\left[2\right]}_{p,q}^2)+\left({\displaystyle \frac{1-\eta }{2{\eta }^2}}\right)R^2{\left[2\right]}_{p,q}^2\right]d_2^2={\mathcal{F}}_2(\varkappa ,y)({\mathfrak{m}}_2+{\mathfrak{n}}_2)+{\mathcal{F}}_3(\varkappa ,y)({\mathfrak{m}}_1^2+{\mathfrak{n}}_1^2).$$

(28)

The value of ${\mathfrak{m}}_1^2+{\mathfrak{n}}_1^2$ from (27) is substituted in (28), yielding

$$d_2^2={\displaystyle \frac{2{\eta }^2{\mathcal{F}}_2^3(\varkappa ,y)({\mathfrak{m}}_2+{\mathfrak{n}}_2)}{(2\eta (\eta +1)(P{\left[3\right]}_{p,q}+Q{\left[2\right]}_{p,q}^2)+(1-\eta )R^2{\left[2\right]}_{p,q}^2){\mathcal{F}}_2^2(\varkappa ,y)-{(\eta +1)}^2{R}^2{\left[2\right]}_{p,q}^2{\mathcal{F}}_3(\varkappa ,y)}}.$$

(29)

Applying (17) to the coefficients ${\mathfrak{m}}_2$ and ${\mathfrak{n}}_2$ yields (7).

We deduct (25) from (23) to get the bound on $|d_3|$:

$$d_3=d_2^2+{\displaystyle \frac{{\mathcal{F}}_2(\varkappa ,y)({\mathfrak{m}}_2-{\mathfrak{n}}_2)}{\left(\frac{\eta +1}{\eta }\right)P{\left[3\right]}_{p,q}}}.$$

(30)

Then in view of (26) and (27), (30) becomes

$$d_3={\displaystyle \frac{2{\eta }^2{\mathcal{F}}_2^2(\varkappa ,y)({\mathfrak{m}}_1^2+{\mathfrak{n}}_1^2)}{{(\eta +1)}^2{R}^2{\left[2\right]}_{p,q}^2}}+{\displaystyle \frac{\eta {\mathcal{F}}_2(\varkappa ,y)({\mathfrak{m}}_2-{\mathfrak{n}}_2)}{(\eta +1)P{\left[3\right]}_{p,q}}},$$

and applying (17) for the coefficients ${\mathfrak{m}}_1$, ${\mathfrak{m}}_2$, ${\mathfrak{n}}_1$ and ${\mathfrak{n}}_2$ we obtain (8).

From (29) and (30), for $\mu \in \mathbb{R}$, we obtain

$$|d_3-\mu d_2^2|=|{\mathcal{F}}_2(\varkappa ,y)|\left|\left(\mathfrak{B}\left(\mu \right)+{\displaystyle \frac{\eta }{(\eta +1)P{\left[3\right]}_{p,q}}}\right){\mathfrak{m}}_2+\left(\mathfrak{B}\left(\mu \right)-{\displaystyle \frac{\eta }{(\eta +1)P{\left[3\right]}_{p,q}}}\right){\mathfrak{n}}_2\right|,$$

where

$$\mathfrak{B}\left(\mu \right)={\displaystyle \frac{2{\eta }^2(1-\mu ){\mathcal{F}}_2^2(\varkappa ,y)}{(2\eta (\eta +1)(P{\left[3\right]}_{p,q}+Q{\left[2\right]}_{p,q}^2)+(1-\eta )R^2{\left[2\right]}_{p,q}^2){\mathcal{F}}_2^2(\varkappa ,y)-{(\eta +1)}^2{R}^2{\left[2\right]}_{p,q}^2{\mathcal{F}}_3(\varkappa ,y)}}.$$

Clearly

$$|d_3-\mu d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{2\eta |{\mathcal{F}}_2(\varkappa ,y)|}{(\eta +1)P{\left[3\right]}_{p,q}}}\hfill & ;0\le \left|\mathfrak{B}\left(\mu \right)\right|\le {\displaystyle \frac{\eta }{(\eta +1)P{\left[3\right]}_{p,q}}}\hfill \\ 2|{\mathcal{F}}_2(\varkappa ,y)\left|\right|\mathfrak{B}\left(\mu \right)|\hfill & ;\left|\mathfrak{B}\left(\mu \right)\right|\ge {\displaystyle \frac{\eta }{(\eta +1)P{\left[3\right]}_{p,q}}},\hfill \end{array}\right.$$

which results in (9) and J as in (10). □

Corollary 1. 

Let $\lambda =1$ in the above theorem. Then, for any function $\varphi \in {\mathfrak{H}}_{\sigma ,p,q}^{\eta }(\gamma ,F)$ the upper bounds of $|d_2|,|d_3|$, and $|d_3-\mu d_2^2|,\mu \in \Re ,$ are given by (7), (8), and (9), respectively, with $P=P_1=\gamma {\left[3\right]}_{p,q}-1,Q=Q_1=1-\gamma {\left[2\right]}_{p,q}andR=R_1=-Q_1$ in (11), (12), and (13), respectively. $P_1,Q_1,and{R}_1$ are to be used in place of $P,Q,andR$ for J in (10).

Corollary 2. 

Let $\gamma =1$ in the above theorem. Then, for any function $\varphi \in {\mathfrak{I}}_{\sigma ,p,q}^{\lambda ,\eta }\left(\varkappa \right)$ the upper bounds of $|d_2|,|d_3|,$ and $|d_3-\mu d_2^2|,\mu \in \Re ,$ are given by (7), (8), and (9), respectively, where $P=P_2=\lambda {\left[3\right]}_{p,q}-1,Q=Q_2=1-\lambda {\left[2\right]}_{p,q}+{\displaystyle \frac{\lambda (\lambda -1)}{2}}{\left[2\right]}_{p,q}^2,andR=R_2=\lambda {\left[2\right]}_{p,q}-1$. $P_2,Q_2,and{R}_2$ are to be used in place of $P,Q,$ and R for J in (10).

From Theorem 1, taking $p=1$ and $q\to 1^{-}$, we obtain

Corollary 3. 

Let $\gamma \ge 1,0<\eta \le 1,and\lambda \ge 1$. If $\varphi \in$${{\rm Y}}_{\sigma }^{\lambda ,\eta }(\gamma ,F)$, then

$$|d_2|\le$$

$${\displaystyle \frac{\eta \left|k\right|\sqrt{2\left|k\right|}}{\sqrt{|(\eta (\eta +1)(8\gamma {\lambda }^2-7\gamma \lambda +1)+2(1-\eta ){(2\gamma \lambda -1)}^2)k^2-2{(\eta +1)}^2{(2\gamma \lambda -1)}^2(k^2+l)|}}},$$

$$|d_3|\le {\left({\displaystyle \frac{\eta k}{(\eta +1)(2\gamma \lambda -1)}}\right)}^2+{\displaystyle \frac{2\eta \left|k\right|}{3(\eta +1)(3\gamma \lambda -1)}},$$

and for $\mu \in \mathbb{R}$

$$\begin{array}{ccc}\hfill & |d_3-\mu d_2^2|\hfill & \hfill \le \left\{\begin{array}{cc}{\displaystyle \frac{2\eta \left|k\right|}{3(\eta +1)(3\gamma \lambda -1)}};\hfill & |1-\mu |\le J_1\hfill \\ {\displaystyle \frac{2{\eta }^2{\left|k\right|}^3|1-\mu |}{|(\eta (\eta +1)(8\gamma {\lambda }^2-7\gamma \lambda +1)+2(1-\eta ){(2\gamma \lambda -1)}^2)k^2-2{(\eta +1)}^2{(2\gamma \lambda -1)}^2(k^2+l)|}};\hfill & |1-\mu |\ge J_1,\hfill \end{array}\right.\end{array}$$

where

$$J_1=$$

$$\left|{\displaystyle \frac{(\eta (\eta +1)(8\gamma {\lambda }^2-7\gamma \lambda +1)+2(1-\eta ){(2\gamma \lambda -1)}^2)k^2-2{(\eta +1)}^2{(2\gamma \lambda -1)}^2(k^2+l)}{\eta (\eta +1)(3\gamma \lambda -1)k^2}}\right|,$$

Here are a few examples of the above corollary’s special cases.

Example 1. 

Letting $\eta =1$ in the class ${{\rm Y}}_{\sigma }^{\lambda ,\eta }(\gamma ,F)$, we obtain a subclass ${\chi }_{\sigma }^{\lambda }(\gamma ,F)\equiv {{\rm Y}}_{\sigma }^{\lambda ,1}(\gamma ,F)$ of functions ϕ$\in \sigma$ satisfying

$${\displaystyle \frac{\gamma {\left[{\left(\zeta {\varphi }^{\prime }\left(\zeta \right)\right)}^{\prime }\right]}^{\lambda }+(1-\gamma )}{{\varphi }^{\prime }\left(\zeta \right)}}\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }}$$

and

$${\displaystyle \frac{\gamma {\left[{\left(\varpi {\psi }^{\prime }\left(\varpi \right)\right)}^{\prime }\right]}^{\lambda }+(1-\gamma )}{{\psi }^{\prime }\left(\varpi \right)}}\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

where $F\left(\zeta \right)$ is as mentioned in (6).

The class ${\chi }_{\sigma }^{\lambda }(\gamma ,F)$ is not an empty set, which is an important point to note. Consider the following example, where $\gamma =\lambda =1,k=1andl=0$. The function $u\left(\zeta \right)$=$\zeta +{\displaystyle \frac{{\zeta }^2}{6}}+{\displaystyle \frac{{\zeta }^3}{27}}+\cdots =-3log(1-{\displaystyle \frac{\zeta }{3}})\in \mathcal{S}$ is in the class ${\chi }_{\sigma }^1(1,F)$. Because, its inverse $v\left(\varpi \right)=\varpi -{\displaystyle \frac{{\varpi }^2}{6}}+{\displaystyle \frac{{\varpi }^3}{54}}-\cdots ={\displaystyle \frac{3(e^{\frac{\varpi }{3}}-1)}{e^{\frac{\varpi }{3}}}}\in \mathcal{S}$. So, we obtain that $u\left(\zeta \right)\in \sigma$ (see [46]).

Corollary 4. 

Let $\gamma \ge 1,and\lambda \ge 1$. If $\varphi \in$${\chi }_{\sigma }^{\lambda }(\gamma ,F)$, then

$$|d_2|\le {\displaystyle \frac{\left|k\right|\sqrt{\left|k\right|}}{\sqrt{|(8\gamma {\lambda }^2-7\gamma \lambda +1)k^2-4{(2\gamma \lambda -1)}^2(k^2+l)|}}},$$

$$|d_3|\le {\left({\displaystyle \frac{k}{2(2\gamma \lambda -1)}}\right)}^2+{\displaystyle \frac{\left|k\right|}{3(3\gamma \lambda -1)}},$$

and for $\mu \in \mathbb{R}$

$$\begin{array}{ccc}\hfill & |d_3-\mu d_2^2|\hfill & \hfill \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|k\right|}{3(3\gamma \lambda -1)}};\hfill & |1-\mu |\le J_2\hfill \\ {\displaystyle \frac{{2\left|k\right|}^3|1-\mu |}{|(8\gamma {\lambda }^2-7\gamma \lambda +1)k^2-4{(2\gamma \lambda -1)}^2(k^2+l)|}};\hfill & |1-\mu |\ge J_2,\hfill \end{array}\right.\end{array}$$

where

$$J_2=\left|{\displaystyle \frac{(8\gamma {\lambda }^2-7\gamma \lambda +1))k^2-4{(2\gamma \lambda -1)}^2(k^2+l)}{(3\gamma \lambda -1)k^2}}\right|.$$

Example 2. 

Letting $\gamma =1$ in the class ${\chi }_{\sigma }^{\lambda }(\gamma ,F)$, we obtain a subclass ${\chi }_{\sigma }^{\lambda }(1,F)$ of functions ϕ$\in \sigma$ satisfying

$${\displaystyle \frac{{\left[{\left(\zeta {\varphi }^{\prime }\left(\zeta \right)\right)}^{\prime }\right]}^{\lambda }}{{\varphi }^{\prime }\left(\zeta \right)}}\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},\mathrm{and}{\displaystyle \frac{{\left[{\left(\varpi {\psi }^{\prime }\left(\varpi \right)\right)}^{\prime }\right]}^{\lambda }}{{\psi }^{\prime }\left(\varpi \right)}}\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

where $F\left(\zeta \right)$ is as mentioned in (6).

Corollary 5. 

Let $\lambda \ge 1$. If $\varphi \in$${\chi }_{\sigma }^{\lambda }(1,F)$ then

$$|d_2|\le {\displaystyle \frac{\left|k\right|\sqrt{\left|k\right|}}{\sqrt{|(8{\lambda }^2-7\lambda +1)k^2-4{(2\lambda -1)}^2(k^2+l)|}}},\left|d_3\right|\le {\left({\displaystyle \frac{k}{2(2\lambda -1)}}\right)}^2+{\displaystyle \frac{\left|k\right|}{3(3\lambda -1)}},$$

and for $\mu \in \mathbb{R}$

$$\begin{array}{ccc}\hfill & |d_3-\mu d_2^2|\hfill & \hfill \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|k\right|}{3(3\lambda -1)}};\hfill & |1-\mu |\le J_3\hfill \\ {\displaystyle \frac{{2\left|k\right|}^3|1-\mu |}{|(8{\lambda }^2-7\lambda +1)k^2-4{(2\lambda -1)}^2(k^2+l)|}};\hfill & |1-\mu |\ge J_3,\hfill \end{array}\right.\end{array}$$

where

$$J_3=\left|{\displaystyle \frac{(8{\lambda }^2-7\lambda +1))k^2-4{(2\lambda -1)}^2(k^2+l)}{(3\gamma \lambda -1)k^2}}\right|.$$

Remark 4. 

$\lambda =1$ in Corollary 4 yields Corollaries 3 and 7 of Yilmaz and Aktas [43].

Taking $\eta =1$ in the above theorem, we obtain

Corollary 6. 

Let $\gamma \ge 1,and\lambda \ge 1$. If ϕ$\in {\mathfrak{Z}}_{\sigma ,p,q}^{\lambda }(\gamma ,\varkappa )$, then

$$|d_2|\le {\displaystyle \frac{\left|k\right|\sqrt{\left|k\right|}}{\sqrt{|(P{\left[3\right]}_{p,q}+Q{\left[2\right]}_{p,q}^2)k^2-R^2{\left[2\right]}_{p,q}^2(k^2+l)|}}},\left|d_3\right|\le {\displaystyle \frac{k^2}{R^2{\left[2\right]}_{p,q}^2}}+{\displaystyle \frac{\left|k\right|}{P{\left[3\right]}_{p,q}}},$$

and for $\mu \in \mathbb{R}$

$$\begin{array}{ccc}\hfill & |d_3-\mu d_2^2|\hfill & \hfill \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|k\right|}{P{\left[3\right]}_{p,q}}};\hfill & |1-\mu |\le J_4\hfill \\ {\displaystyle \frac{{\left|k\right|}^3|1-\mu |}{|(P{\left[3\right]}_{p,q}+Q{\left[2\right]}_{p,q}^2)k^2-R^2{\left[2\right]}_{p,q}^2(k^2+l)|}};\hfill & |1-\mu |\ge J_4,\hfill \end{array}\right.\end{array}$$

where

$$J_4=\left|{\displaystyle \frac{(P{\left[3\right]}_{p,q}+Q{\left[2\right]}_{p,q}^2)k^2-R^2{\left[2\right]}^2(k^2+l)}{P{\left[3\right]}_{p,q}{k}^2}}\right|,$$

and P, Q, and R are as detailed in (11), (12), and (13), respectively.

Remark 5. 

${\mathfrak{Z}}_{\sigma ,p=1,q\to 1^{-}}^{\lambda }(\gamma ,F)\equiv {\chi }_{\sigma }^{\lambda }(\gamma ,F)$

3. Results for the Class ${\mathfrak{T}}_{\mathit{\sigma },\mathit{p},\mathit{q}}^{\mathit{\lambda },\mathit{\eta }}(\mathit{\nu },\mathit{\varkappa })$

First, for the class $\varphi \in {\mathfrak{T}}_{\sigma ,p,q}^{\lambda ,\eta }(\nu ,\varkappa )$, as defined in Definition 3, we determine the coefficient estimates.

Theorem 2. 

Let $0\le \nu \le 1,0<\eta \le 1,and\lambda \ge 1$. If ϕ$\in {\mathfrak{T}}_{\sigma ,p,q}^{\lambda ,\eta }(\nu ,\varkappa )$, then

$$|d_2|\le {\displaystyle \frac{2\eta \left|k\right|\sqrt{\left|k\right|}}{\sqrt{|(2\eta (\eta +1)(L+N)+(1-\eta )M^2)k^2-{(\eta +1)}^2{B}^2(k^2+l)|}}},$$

(31)

$$|d_3|\le {\displaystyle \frac{4{\eta }^2{k}^2}{{(\eta +1)}^2{M}^2}}+{\displaystyle \frac{2\eta \left|k\right|}{(\eta +1)L}},$$

(32)

and for $\mu \in \mathbb{R}$

$$\begin{array}{ccc}\hfill & |d_3-\mu d_2^2|\hfill & \hfill \le \left\{\begin{array}{cc}{\displaystyle \frac{2\eta \left|k\right|}{(\eta +1)L}}\hfill & ;|1-\mu |\le \mathbb{Q}\hfill \\ {\displaystyle \frac{4{\eta }^2{\left|k\right|}^3|1-\mu |}{|(2\eta (\eta +1)(L+N)+(1-\eta )M^2)k^2-{(\eta +1)}^2{B}^2(k^2+l)|}}\hfill & ;|1-\mu |\ge \mathbb{Q},\hfill \end{array}\right.\end{array}$$

(33)

where

$$\mathbb{Q}=\left|{\displaystyle \frac{{(2\eta (\eta +1)(L+N)+(1-\eta )M)}^2)k^2-{(\eta +1)}^2{M}^2(k^2+l)}{2\eta (1+\eta )L{k}^2}}\right|,$$

(34)

$$L=\lambda {\left[3\right]}_{p,q}-\nu ,$$

(35)

$$N={\displaystyle \frac{\lambda (\lambda -1){\left[2\right]}_{p,q}^2}{2}}-\nu \lambda {\left[2\right]}_{p,q}+{\nu }^2,$$

(36)

and

$$M=\lambda {\left[2\right]}_{p,q}-\nu .$$

(37)

Proof. 

Let $\varphi \in {\mathfrak{T}}_{\sigma ,p,q}^{\lambda ,\eta }(\nu ,\varkappa )$. Following that, due to Definition 3, we obtain

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\zeta {\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{\lambda }}{\nu \varphi \left(\zeta \right)+(1-\nu )\zeta }}+{\left({\displaystyle \frac{\zeta {\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{\lambda }}{\nu \varphi \left(\zeta \right)+(1-\nu )\zeta }}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}=F\left(\mathfrak{m}\left(\zeta \right)\right),$$

(38)

and

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\varpi {\left(D_{p,q}\psi \left(\varpi \right)\right)}^{\lambda }}{\nu \psi \left(w\right)+(1-\nu )\varpi }}+{\left({\displaystyle \frac{\varpi {\left(D_{p,q}\psi \left(\varpi \right)\right)}^{\lambda }}{\nu \psi \left(\varpi \right)+(1-\nu )\varpi }}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}=F\left(\mathfrak{n}\left(\varpi \right)\right),$$

(39)

where $\mathfrak{m}\left(\zeta \right)$ and $\mathfrak{n}\left(\varpi \right)$ are analytic functions as given in (16), with $\left|\mathfrak{m}\right(\zeta \left)\right|<1,\left|\mathfrak{n}\right(\varpi \left)\right|<1,\varsigma ,\varpi \in \mathfrak{D}$, and it is known that

$$|{\mathfrak{m}}_i|\le 1,and|{\mathfrak{n}}_i|\le 1,(i\in \mathbb{N}).$$

(40)

It follows from (38), (39) and (16) that

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\zeta {\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{\tau }}{\nu \varphi \left(\zeta \right)+(1-\nu )\zeta }}+{\left({\displaystyle \frac{\zeta {\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{\lambda }}{\nu \varphi \left(\zeta \right)+(1-\nu )\zeta }}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}=1+{\mathcal{F}}_2(\varkappa ,y)\mathfrak{m}\left(\zeta \right)+{\mathcal{F}}_3(\varkappa ,y){\mathfrak{m}}^2\left(\zeta \right)+\cdots ,$$

(41)

and

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\varpi {\left(D_{p,q}\psi \left(\varpi \right)\right)}^{\lambda }}{\nu \psi \left(\varpi \right)+(1-\nu )\varpi }}+{\left({\displaystyle \frac{w{\left(D_{p,q}\psi \left(\varpi \right)\right)}^{\lambda }}{\nu \psi \left(\varpi \right)+(1-\nu )\varpi }}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}=$$

$$1+{\mathcal{F}}_2(\varkappa ,y)\mathfrak{n}\left(\varpi \right)+{\mathcal{F}}_3(\varkappa ,y){\mathfrak{n}}^2\left(\varpi \right)+\cdots .$$

(42)

In light of (4), Equations (41) and (42) can be expressed as follows.

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\zeta {\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{\lambda }}{\nu \varphi \left(\zeta \right)+(1-\nu )\zeta }}+{\left({\displaystyle \frac{\zeta {\left(D_{p,q}\varphi \left(\zeta \right)\right)}^{\lambda }}{\nu \varphi \left(\zeta \right)+(1-\nu )\zeta }}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}=$$

$$1+{\mathcal{F}}_2(\varkappa ,y){\mathfrak{m}}_1\zeta +[{\mathcal{F}}_2(\varkappa ,y){\mathfrak{m}}_2+{\mathcal{F}}_3(\varkappa ,y){\mathfrak{m}}_1^2]{\zeta }^2+\cdots ,$$

(43)

and

$${\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\varpi {\left(D_{p,q}\psi \left(\varpi \right)\right)}^{\lambda }}{\nu \psi \left(\varpi \right)+(1-\nu )\varpi }}+{\left({\displaystyle \frac{w{\left(D_{p,q}\psi \left(\varpi \right)\right)}^{\lambda }}{\nu \psi \left(\varpi \right)+(1-\nu )\varpi }}\right)}^{{\displaystyle \frac{1}{\eta }}}\right\}=$$

$$1+{\mathcal{F}}_2(\varkappa ,y){\mathfrak{n}}_1\varpi +[{\mathcal{F}}_2(\varkappa ,y){\mathfrak{n}}_2+{\mathcal{F}}_3(\varkappa ,y){\mathfrak{n}}_1^2]{\varpi }^2+\cdots .$$

(44)

Comparing (43) and (44), we have

$${\displaystyle \frac{(\eta +1)M}{2\eta }}{d}_2={\mathcal{F}}_2(\varkappa ,y){\mathfrak{m}}_1,$$

(45)

$$\left({\displaystyle \frac{\eta +1}{2\eta }}\right)\left(L{d}_3+N{d}_2^2\right)+\left({\displaystyle \frac{1-\eta }{4{\eta }^2}}\right)M^2{d}_2^2={\mathcal{F}}_2(\varkappa ,y){\mathfrak{m}}_2+{\mathcal{F}}_3(\varkappa ,y){\mathfrak{m}}_1^2,$$

(46)

$$-{\displaystyle \frac{(\eta +1)M}{2\eta }}{d}_2={\mathcal{F}}_2(\varkappa ,y){\mathfrak{n}}_1,$$

(47)

and

$$\left({\displaystyle \frac{\eta +1}{2\eta }}\right)\left(L(2d_2^2-d_3)+N{d}_2^2\right)+\left({\displaystyle \frac{1-\eta }{4{\eta }^2}}\right)M^2{d}_2^2={\mathcal{F}}_2(\varkappa ,y){\mathfrak{n}}_2+{\mathcal{F}}_3(\varkappa ,y){\mathfrak{n}}_1^2,$$

(48)

where L, N, and M are as in (35), (36) and (37), respectively. From (45) and (47), we easily obtain

$${\mathfrak{m}}_1=-{\mathfrak{n}}_1$$

(49)

and also

$${\displaystyle \frac{{(\eta +1)}^2{M}^2}{2{\eta }^2}}{d}_2^2=({\mathfrak{m}}_1^2+{\mathfrak{n}}_1^2){\left({\mathcal{H}}_2\left(\varkappa \right)\right)}^2.$$

(50)

The bound on $|d_2|$, is obtained by adding (46) and (48).

$$\left(\left({\displaystyle \frac{\eta +1}{\eta }}\right)(L+N)+\left({\displaystyle \frac{1-\eta }{2{\eta }^2}}\right)M^2\right)d_2^2={\mathcal{F}}_2(\varkappa ,y)({\mathfrak{m}}_2+{\mathfrak{n}}_2)+{\mathcal{F}}_3(\varkappa ,y)({\mathfrak{m}}_1^2+{\mathfrak{n}}_1^2).$$

(51)

Substituting the value of ${\mathfrak{m}}_1^2+{\mathfrak{n}}_1^2$ from (50) in (51), we obtain

$$d_2^2={\displaystyle \frac{2{\eta }^2{\mathcal{F}}_2^3(\varkappa ,y)\left(\varkappa \right)({\mathfrak{m}}_2+{\mathfrak{n}}_2)}{(2\eta (\eta +1)(L+N)+(1-\eta )M^2){\mathcal{F}}_2^2(\varkappa ,y)-{(\eta +1)}^2{M}^2{\mathcal{F}}_3(\varkappa ,y)}}.$$

(52)

Applying (40) for the coefficients ${\mathfrak{m}}_2$ and ${\mathfrak{n}}_2$, we obtain (31).

To obtain the bound on $|d_3|$, we deduct (48) from (46):

$$d_3=d_2^2+{\displaystyle \frac{{\mathcal{F}}_2(\varkappa ,y)({\mathfrak{m}}_2-{\mathfrak{n}}_2)}{\left(\frac{\eta +1}{\eta }\right)L}}.$$

(53)

Then, in view of (49) and (50), (53) becomes

$$d_3={\displaystyle \frac{2{\eta }^2{\mathcal{F}}_2^2(\varkappa ,y)({\mathfrak{m}}_1^2+{\mathfrak{n}}_1^2)}{{(\eta +1)}^2{M}^2}}+{\displaystyle \frac{\eta {\mathcal{F}}_2(\varkappa ,y)({\mathfrak{m}}_2-{\mathfrak{n}}_2)}{(\eta +1)L}},$$

and applying (40) for the coefficients ${\mathfrak{m}}_1$, ${\mathfrak{m}}_2$, ${\mathfrak{n}}_1$ and ${\mathfrak{n}}_2$ we obtain (32).

From (52) and (53), for $\mu \in \mathbb{R}$, we obtain

$$|d_3-\mu d_2^2|=|{\mathcal{F}}_2(\varkappa ,y)|\left|\left({\mathfrak{B}}_{\mathfrak{1}}(\mu ,\varkappa )+{\displaystyle \frac{\eta }{(\eta +1)L}}\right){\mathfrak{m}}_2+\left({\mathfrak{B}}_{\mathfrak{1}}(\mu ,\varkappa )-{\displaystyle \frac{\eta }{(\delta +1)L}}\right){\mathfrak{n}}_2\right|,$$

where

$${\mathfrak{B}}_{\mathfrak{1}}(\mu ,\varkappa )={\displaystyle \frac{2{\eta }^2(1-\mu ){\mathcal{F}}_2^2(\varkappa ,y)}{(2\eta (\eta +1)(L+N)+(1-\eta )M^2){\mathcal{F}}_2^2(\varkappa ,y)-{(\eta +1)}^2{M}^2{\mathcal{F}}_3(\varkappa ,y)}}.$$

Clearly

$$|d_3-\mu d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{2\eta |{\mathcal{F}}_2(\varkappa ,y)|}{(\eta +1)L}}\hfill & ;0\le |{\mathfrak{B}}_{\mathfrak{1}}(\mu ,\varkappa )|\le {\displaystyle \frac{\eta }{(\eta +1)L}}\hfill \\ 2|{\mathcal{F}}_2(\varkappa ,y)\left|\right|{\mathfrak{B}}_{\mathfrak{1}}(\mu ,\varkappa )|\hfill & ;|{\mathfrak{B}}_{\mathfrak{1}}(\mu ,\varkappa )|\ge {\displaystyle \frac{\eta }{(\eta +1)L}},\hfill \end{array}\right.$$

from which we conclude (33) with $\mathbb{Q}$ as in (34). □

Corollary 7. 

Let $\nu =0$ in the above theorem. Then, for any function $\varphi \in {\mathfrak{C}}_{\sigma ,p,q}^{\lambda ,\eta }\left(F\right)$ the upper bounds of $|d_2|,|d_3|$, and $|d_3-\mu d_2^2|,\mu \in \Re ,$ are given by (31), (32) and (33), respectively, with $L=L_1=\lambda {\left[3\right]}_{p,q},N=N_1={\displaystyle \frac{\lambda (\lambda -1)}{2}}{\left[2\right]}_{p,q}^2$ and $M=M_1=\lambda {\left[2\right]}_{p,q}$. $L_1,N_1,and{M}_1$ are to be used in place of $L,N,andM$ for $\mathbb{Q}$ in (34).

Corollary 8. 

Let $\nu =1$ in the above theorem. Then, for any function $\varphi \in {\mathfrak{D}}_{\sigma ,p,q}^{\lambda ,\eta }\left(F\right)$ the upper bounds of $|d_2|,|d_3|$, and $|d_3-\mu d_2^2|,\mu \in \Re ,$ are given by (31), (32) and (33), respectively, with $L=L_2=\lambda {\left[3\right]}_{p,q}-1,N=N_2={\displaystyle \frac{\lambda (\lambda -1)}{2}}{\left[2\right]}_{p,q}^2-\lambda {\left[2\right]}_{p,q}+1andM=M_2=\lambda {\left[2\right]}_{p,q}-1$. $L_2,N_2,and{M}_2$ are to be used in place of $L,N,andM$ for $\mathbb{Q}$ in (34).

From Theorem 2, taking $p=1$ and $q\to 1^{-}$, we obtain

Corollary 9. 

Let $0\le \nu \le 1,0<\eta \le 1,and\lambda \ge 1$. If $\varphi \in$${\Gamma }_{\sigma }^{\lambda ,\eta }(\nu ,F)$ then

$$|d_2|\le$$

$${\displaystyle \frac{2\eta \left|k\right|\sqrt{\left|k\right|}}{\sqrt{|(2\eta (\eta +1)((2\lambda +1)(\lambda -\nu )+{\nu }^2)+(1-\eta ){(2\lambda -\nu )}^2)k^2-{(\eta +1)}^2{(2\lambda -\nu )}^2(k^2+l)|}}},$$

$$|d_3|\le {\left({\displaystyle \frac{2\eta k}{(\eta +1)(2\lambda -\nu )}}\right)}^2+{\displaystyle \frac{2\eta \left|k\right|}{(\eta +1)(3\lambda -\nu )}},$$

and for $\mu \in \mathbb{R}$

$$\begin{array}{ccc}\hfill & |d_3-\mu d_2^2|\hfill & \hfill \le \left\{\begin{array}{cc}{\displaystyle \frac{2\eta \left|k\right|}{(\eta +1)(3\lambda -\nu )}};\hfill & |1-\mu |\le {\mathbb{Q}}_1\hfill \\ {\displaystyle \frac{4{\eta }^2{\left|k\right|}^3|1-\mu |}{|(2\eta (\eta +1)((2\lambda +1)(\lambda -\nu )+{\nu }^2)+(1-\eta ){(2\lambda -\nu )}^2)k^2-{(\eta +1)}^2{(2\lambda -\nu )}^2(k^2+l)|}};\hfill & |1-\mu |\ge {\mathbb{Q}}_1,\hfill \end{array}\right.\end{array}$$

where

$${\mathbb{Q}}_1=$$

$$\left|{\displaystyle \frac{(2\eta (\eta +1)((2\lambda +1)(\lambda -\nu )+{\nu }^2)+(1-\eta ){(2\lambda -\nu )}^2)k^2-{(\eta +1)}^2{(2\lambda -\nu )}^2(k^2+l)}{2\eta (\eta +1)(3\lambda -\nu )k^2}}\right|.$$

Here are a few examples of the above corollary’s special cases.

Example 3. 

Letting $\eta =1$ in the class ${\Gamma }_{\sigma }^{\lambda ,\eta }(\nu ,F)$, we obtain a subclass ${\Psi }_{\sigma }^{\lambda }(\nu ,F)\equiv {\Gamma }_{\sigma }^{\lambda ,1}(\nu ,F)$ of functions ϕ$\in \sigma$ satisfying

$${\displaystyle \frac{\zeta {\left({\varphi }^{\prime }\left(\zeta \right)\right)}^{\lambda }}{\nu \varphi \left(\zeta \right)+(1-\nu )\zeta }}\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},$$

and

$${\displaystyle \frac{\varpi {\left({\psi }^{\prime }\left(\varpi \right)\right)}^{\lambda }}{\nu \psi \left(\varpi \right)+(1-\nu )\varpi }}\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

where $F\left(\zeta \right)$ is as mentioned in (6), $0\le \nu \le 1,and\lambda \ge 1$.

Corollary 10. 

Let $0\le \nu \le 1,and\lambda \ge 1$. If $\varphi \in$ ${\Psi }_{\sigma }^{\lambda }(\nu ,F)$ then

$$|d_2|\le {\displaystyle \frac{\left|k\right|\sqrt{\left|k\right|}}{\sqrt{|((2\lambda +1)(\lambda -\nu )+{\nu }^2)k^2-{(2\lambda -\nu )}^2(k^2+l)|}}},$$

$$|d_3|\le {\left({\displaystyle \frac{k}{2\lambda -\nu }}\right)}^2+{\displaystyle \frac{\left|k\right|}{3\lambda -\nu }},$$

and for $\mu \in \mathbb{R}$

$$\begin{array}{cc}\hfill |d_3-\mu d_2^2|& \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|k\right|}{3\lambda -\nu }};\hfill & |1-\mu |\le {\mathbb{Q}}_2\hfill \\ {\displaystyle \frac{{\left|k\right|}^3|1-\mu |}{|((2\lambda +1)(\lambda -\nu )+{\nu }^2)k^2-{(2\lambda -\nu )}^2(k^2+l)|}};\hfill & |1-\mu |\ge {\mathbb{Q}}_2,\hfill \end{array}\right.\hfill \end{array}$$

where

$${\mathbb{Q}}_2=\left|{\displaystyle \frac{((2\lambda +1)(\lambda -\nu )+{\nu }^2)k^2-{(2\lambda -\nu )}^2(k^2+l)}{(3\lambda -\nu )k^2}}\right|.$$

Example 4. 

Letting $\nu =1$, then ${\Psi }_{\sigma }^{\lambda }(1,F)$ is the subclass of functions ϕ$\in \sigma$ satisfying

$${\displaystyle \frac{\zeta {\left({\varphi }^{\prime }\left(\zeta \right)\right)}^{\lambda }}{\varphi \left(\zeta \right)}}\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},\mathrm{and}{\displaystyle \frac{\varpi {\left({\psi }^{\prime }\left(\varpi \right)\right)}^{\lambda }}{\psi \left(\varpi \right)}}\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

where $F\left(\zeta \right)$ is as mentioned in (6).

Corollary 11. 

Let $\lambda \ge 1$. If $\varphi \in$${\Psi }_{\sigma }^{\lambda }(1,F)$ then

$$|d_2|\le {\displaystyle \frac{\left|k\right|\sqrt{\left|k\right|}}{\sqrt{|\lambda (2\lambda -1)k^2-{(2\lambda -1)}^2(k^2+l)|}}},\left|d_3\right|\le {\left({\displaystyle \frac{k}{2\lambda -1}}\right)}^2+{\displaystyle \frac{\left|k\right|}{3\lambda -1}},$$

and for $\mu \in \mathbb{R}$

$$\begin{array}{cc}\hfill |d_3-\mu d_2^2|& \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|k\right|}{3\lambda -1}};\hfill & |1-\mu |\le \left|{\displaystyle \frac{\lambda (2\lambda -1)k^2-{(2\lambda -1)}^2(k^2+l)}{(3\lambda -1)k^2}}\right|\hfill \\ {\displaystyle \frac{{\left|k\right|}^3|1-\mu |}{|\lambda (2\lambda -1)k^2-{(2\lambda -1)}^2(k^2+l)|}};\hfill & |1-\mu |\ge \left|{\displaystyle \frac{\lambda (2\lambda -1)k^2-{(2\lambda -1)}^2(k^2+l)}{(3\lambda -1)k^2}}\right|.\hfill \end{array}\right.\hfill \end{array}$$

Remark 6. 

$\lambda =1$ in Corollary 11 yields Corollaries 2 and 6 of Yilmaz and Aktas [43].

Example 5. 

Letting $\nu =0$, then ${\Psi }_{\sigma }^{\lambda }(0,F)$ is the subclass of functions ϕ$\in \sigma$ satisfying

$${\left({\varphi }^{\prime }\left(\zeta \right)\right)}^{\lambda }\prec F\left(\zeta \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\zeta )}{\zeta }},\mathrm{and}{\left({\psi }^{\prime }\left(\varpi \right)\right)}^{\lambda }\prec F\left(\varpi \right)={\displaystyle \frac{\mathcal{F}(\varkappa ,y,\varpi )}{\varpi }},$$

where $F\left(\zeta \right)$ is as mentioned in $\left(\right)$.

The class ${\Psi }_{\sigma }^{\lambda }(0,F)$ is not an empty set, which is an important point to note. Consider the following example, where $\lambda =1,k=1andl=0$. The function $u\left(\zeta \right)$=$={\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{1+\zeta }{1-\zeta }}\right)\in \mathcal{S}$ is in the class ${\Psi }_{\sigma }^{\lambda }(0,F)$. Because, its inverse $v\left(\varpi \right)={\displaystyle \frac{e^{2\varpi }-1}{e^{2\varpi }+1}}\in \mathcal{S}$. So, we get that $u\left(\zeta \right)\in \sigma$ (see [46]).

Corollary 12. 

Let $\lambda \ge 1$. If $\varphi \in$${\Psi }_{\sigma }^{\lambda }(0,F)$, then

$$|d_2|\le {\displaystyle \frac{\left|k\right|\sqrt{\left|k\right|}}{\sqrt{\left|(2\lambda +1)\lambda \right)k^2-4{\lambda }^2(k^2+l)|}}},\left|d_3\right|\le {\left({\displaystyle \frac{k}{2\lambda }}\right)}^2+{\displaystyle \frac{\left|k\right|}{3\lambda }},$$

and for $\mu \in \mathbb{R}$

$$\begin{array}{cc}\hfill |d_3-\mu d_2^2|& \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|k\right|}{3\lambda }};\hfill & |1-\mu |\le \left|{\displaystyle \frac{(2\lambda +1)\lambda )k^2-4{\lambda }^2(k^2+l)}{3\lambda k^2}}\right|\hfill \\ {\displaystyle \frac{{\left|k\right|}^3|1-\mu |}{\left|(2\lambda +1)\lambda \right)k^2-4{\lambda }^2(k^2+l)|}};\hfill & |1-\mu |\ge \left|{\displaystyle \frac{(2\lambda +1)\lambda )k^2-4{\lambda }^2(k^2+l)}{3\lambda k^2}}\right|.\hfill \end{array}\right.\hfill \end{array}$$

Corollary 13. 

Let $0\le \nu \le 1,and\lambda \ge 1$. If ϕ$\in {\mathfrak{O}}_{\sigma ,p,q}^{\lambda }(\nu ,F)$, then

$$|d_2|\le {\displaystyle \frac{\left|k\right|\sqrt{\left|k\right|}}{\sqrt{|(L+N)k^2-M^2(k^2+l)|}}},\left|d_3\right|\le {\displaystyle \frac{k^2}{M^2}}+{\displaystyle \frac{\left|k\right|}{L}},$$

and for $\mu \in \mathbb{R}$

$$\begin{array}{cc}\hfill |d_3-\mu d_2^2|& \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|k\right|}{L}}\hfill & ;|1-\mu |\le \left|{\displaystyle \frac{(L+N)k^2-M^2(k^2+l)}{Lk2}}\right|\hfill \\ {\displaystyle \frac{{\left|k\right|}^3|1-\mu |}{|(L+N)k^2-M^2(k^2+l)|}}\hfill & ;|1-\mu |\ge \left|{\displaystyle \frac{(L+N)k^2-M^2(k^2+l)}{L{k}^2}}\right|,\hfill \end{array}\right.\hfill \end{array}$$

where L, N, and M are as mentioned in (31), (32), and (33), respectively.

Remark 7. 

${\mathfrak{O}}_{\sigma ,p=1,q\to 1^{-}}^{\lambda }(\nu ,F)\equiv {\Psi }_{\sigma }^{\lambda }(\nu ,F).$

4. Conclusions

The current work defines two subfamilies of $\sigma$ associated with GBFP and derives upper bounds of $|d_2|$, $|d_3|$ and the Fekete–Szegö functional $|d_3-\mu d_2^2|,\mu \in \mathbb{R}$ for functions in these subfamilies. We have been able to draw attention to several implications by altering the parameters in Theorem 1 and Theorem 2. Relevant connections to the current research are also discovered. The subfamilies this paper studies may encourage researchers to concentrate on the (p, q) -derivative operator. Future studies might look into extending obtained results to fractional derivatives, higher-order Hankel determinants, or Toeplitz determinants. The findings presented here give these advancements a strong foundation and emphasize the importance of geometric factors in the study of analytic function theory.