Comprehensive Subfamilies of Bi-Univalent Functions Involving a Certain Operator Subordinate to Generalized Bivariate Fibonacci Polynomials

Abstract

This paper introduces novel subfamilies of analytic and bi-univalent functions in $\mathrm{\Omega }=\left\{\varsigma \in \mathbb{C}:\left|\varsigma \right|<1\right\}$, defined by applying a linear operator associated with the Mittag–Leffler function and requiring subordination to domains related to generalized bivariate Fibonacci polynomials. The proposed framework provides a unified treatment that generalizes numerous earlier studies by incorporating parameters controlling both the operator’s fractional calculus features and the domain’s combinatorial geometry. For these subfamilies, we establish initial coefficient bounds ($\left|d_2\right|$, $\left|d_3\right|$) and solve the Fekete–Szegö problem ($\left|d_3-\xi d_2^2\right|$). The derived inequalities are interesting, and their proofs leverage the intricate interplay between the series expansions of the Mittag–Leffler function and the generating function of the Fibonacci polynomials. By specializing the parameters governing the operator and the polynomial domain, we show how our main theorems systematically recover and extend a wide range of known results from the literature, thereby demonstrating the generality and unifying power of our approach.

1. Introduction and Preliminaries

Let $\mathrm{\Omega }=\left\{\varsigma \in \mathbb{C}:\left|\varsigma \right|<1\right\}$ be an open unit disk, where $\mathbb{C}$ is the complex plane. Denote by $\mathcal{A}$ the family of all holomorphic functions $\phi$ in $\mathrm{\Omega }$ that have the next series:

$$\phi \left(\varsigma \right)=\varsigma +d_2{\varsigma }^2+d_3{\varsigma }^3+\cdots =\varsigma +\sum _{j=2}^{\infty }{d}_j{\varsigma }^j,\varsigma \in \mathrm{\Omega },$$

(1)

and let $\mathcal{S}=\{\phi \in \mathcal{A}:\phi$, is univalent in $\mathrm{\Omega }\}\subset \mathcal{A}$. For $\mathfrak{a}\left(\varsigma \right)$, $\mathfrak{b}\left(\varsigma \right)\in \mathcal{A}$, $\mathfrak{a}\left(\varsigma \right)$ is subordinate to $\mathfrak{b}\left(\varsigma \right)$, if there is a Schwarz function $\theta \left(\varsigma \right)$ that is holomorphic in $\mathrm{\Omega }$ with $\theta \left(0\right)=0$ and $\left|\theta \left(\varsigma \right)\right|<1$, such that $\mathfrak{a}\left(\varsigma \right)=\left(\mathfrak{b}\circ \theta \right)\left(\varsigma \right)=\mathfrak{b}\left(\theta \left(\varsigma \right)\right)$,$\varsigma \in \mathrm{\Omega }$. This subordination is symbalized as $\mathfrak{a}\left(\varsigma \right)\prec \mathfrak{b}\left(\varsigma \right)$, $\varsigma \in \mathrm{\Omega }$. In case, if $\mathfrak{b}\left(\varsigma \right)\in \mathcal{S}$, then

$$\mathfrak{a}\left(\varsigma \right)\prec \mathfrak{b}\left(\varsigma \right),\varsigma \in \mathrm{\Omega }\iff \mathfrak{a}\left(\mathrm{\Omega }\right)\subset \mathfrak{b}\left(\mathrm{\Omega }\right)\mathrm{with}\mathfrak{a}\left(0\right)=\mathfrak{b}\left(0\right).$$

The famous Koebe theorem ([1]) states that every univalent function $\phi \in \mathcal{S}$ of the series (1) has an inverse function ${\phi }^{-1}\left(w\right)$, where

$${\phi }^{-1}\left(w\right)=w-d_2{w}^2+(2d_2^2-d_3)w^3-(5d_2^3-5d_2{d}_3+d_4)w^4+\cdots =\chi \left(w\right)$$

(2)

satisfying $\varsigma =\chi \left(\phi \right(\varsigma \left)\right)$ and $w=\phi \left(\chi \right(w\left)\right)$,$\left|w\right|<r_0\left(\phi \right),1/4\leqq r_0\left(\phi \right),\varsigma ,w\in \mathrm{\Omega }$. A function $\phi \left(\varsigma \right)$ is said to be bi-univalent function if $\phi ,{\phi }^{-1}\in \mathcal{S}$ in $\mathrm{\Omega }$. The family $\mathrm{\Sigma }$ of bi-univalent functions was presented by Levin in his work [2]. The family $\mathrm{\Sigma }$ is not a null set where ${\displaystyle \frac{\varsigma }{1-\varsigma }}$, ${\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{1+\varsigma }{1-\varsigma }}\right)$ and $-log\left(1-\varsigma \right)$ are functions in the family $\mathrm{\Sigma }$ but the Koebe function ${\displaystyle \frac{\varsigma }{{(1-\varsigma )}^2}},$$\varsigma -{\displaystyle \frac{{\varsigma }^2}{2}}$ and ${\displaystyle \frac{\varsigma }{1-{\varsigma }^2}}$ are not elements of the family $\mathrm{\Sigma }$, although they are univalent functions in $\mathrm{\Omega }$. For a shortened analysis and to discover some of the properties of the family $\mathrm{\Sigma }$, see [3,4,5,6].

In 2010, Srivastava et al. [7] derived bounds for the coefficients $\left|d_2\right|$ and $\left|d_3\right|$ of functions in two significant subfamilies of the family $\mathrm{\Sigma }$. Motivated by this article, numerous scholars have studied new subfamilies of $\mathrm{\Sigma }$ to find new bounds for the coefficients $\left|d_2\right|$ and $\left|d_3\right|$ (see [8,9,10,11,12]). Also, for each univalent function $\phi \in \mathcal{S}$ defined by (1), the Fekete–Szegö functional $\left|d_3-\xi d_2^2\right|$, $\xi \in \mathbb{R}$, is well renowned for its rich history in the field of Geometric Function Theory (GFT). Its origin was in the refutation by Fekete and Szegö [13]. Several researchers have found coefficient estimates and Fekete–Szegö problems for subfamilies of $\mathrm{\Sigma }$ (involving different operators) that are subordinate to a known number sequence or special polynomials (refer to [14,15,16,17,18,19] for additional details). The problems of finding coefficient bounds and solving the Fekete–Szegö inequality are essentially optimization problems in mathematics. Interestingly, similar boundedness and optimization problems appear in other fields such as high-dimensional data analysis and system identification (see [20]). This shows a useful connection between geometric function theory and modern applied mathematics.

For $j\geqq 2$, generalized bivariate Fibonacci polynomials (GBFP) are defined by the recurrence relation:

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

(3)

where $k(\varkappa ,y)$ and $l(\varkappa ,y)$ are polynomials in $\varkappa$ and y, with real coefficients, ${\mathcal{G}}_0(\varkappa ,y)=0$, ${\mathcal{G}}_1(\varkappa ,y)=1$ and $k^2(\varkappa ,y)+4l(\varkappa ,y)>0$. The generating function of generalized bivariate Fibonacci polynomials is (see [21])

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

(4)

For the specific chosen of $k(\varkappa ,y)$ and $l(\varkappa ,y)$, GBFP tends to various familiar polynomials (see [22]). Readers interested in GBFP can find a brief history and more extensive information in [23] and its references. For members of certain subclasses of $\mathrm{\Sigma }$ involving GBFP, interesting properties regarding coefficient estimates and Fekete–Szegö problems have been found in [24,25].

For the sake of brevity, we write $k(\varkappa ,y)=k$ and $l(\varkappa ,y)=l$. Therefore,

$${\mathcal{G}}_2(\varkappa ,y)=k,{\mathcal{G}}_3(\varkappa ,y)=k^2+l,\cdots$$

(5)

are evident from (3).

The Mittag–Leffler function plays a main role in a diversity of problems in mathematical analysis, fractional calculus, operator theory, fractional adaptive control systems [26], uncertainty modeling in power flow analysis [27] and other fields related to science and engineering. Furthermore, it has always been a source of inspiration for many researchers [28,29,30,31,32,33]. In 1903, Mittag–Leffler [34] introduced the function ${\mathcal{M}}_{\mu }\left(\varsigma \right)$, $\varsigma \in \mathrm{\Omega }$, defined by the next form,

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

It is convergent in the whole complex plane. For $\mu >0$, the one-parameter Mittag–Leffler function ${\mathcal{M}}_{\mu }\left(\varsigma \right)$ is an entire function of order $1/\mu$ and satisfies the recurrence relation

$${\mathcal{M}}_{\mu }\left(\varsigma \right)={\displaystyle \frac{1}{\varsigma }}\left[{\mathcal{M}}_{1-\mu }\left(\varsigma \right)-{\displaystyle \frac{1}{\mathrm{\Gamma }(1-\mu )}}\right].$$

Special values of the Mittag–Leffler function ${\mathcal{M}}_{\mu }\left(\varsigma \right)$ for integer $\mu =k$ are

$$\begin{array}{ccc}\hfill {\mathcal{M}}_0\left(\varsigma \right)& =& {\displaystyle \frac{1}{1-\varsigma }},\hfill \\ \hfill {\mathcal{M}}_1\left(\varsigma \right)& =& e^{\varsigma },\hfill \\ \hfill {\mathcal{M}}_2\left(\varsigma \right)& =& cosh\left(\sqrt{\varsigma }\right),\hfill \\ \hfill {\mathcal{M}}_2(-{\varsigma }^2)& =& cos\varsigma ,\hfill \\ \hfill {\mathcal{M}}_3\left(\varsigma \right)& =& {\displaystyle \frac{1}{3}}\left[e^{\sqrt[3]{\varsigma }}+2e^{-{\displaystyle \frac{\sqrt[3]{\varsigma }}{2}}}cos\left({\displaystyle \frac{\sqrt{3}}{2}}\sqrt[3]{\varsigma }\right)\right],\hfill \\ \hfill {\mathcal{M}}_4\left(\varsigma \right)& =& {\displaystyle \frac{1}{2}}\left[cos\left(\sqrt[4]{\varsigma }\right)+cosh\left(\sqrt[4]{\varsigma }\right)\right],\hfill \end{array}$$

and

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

where $erf\left(\varsigma \right)$ is the Gauss error function given by

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

and $erfc\left(\varsigma \right)=1-erf\left(\varsigma \right)$, where $erfc\left(\varsigma \right)$ denotes the complimentary error function. These functions are widely used in fractional dynamical systems, such as in structural control and digital signal integration (see [35]).

In [36], Srivastava and Tomovski defined the generalized Mittag–Leffler function ${\mathcal{M}}_{\mu ,\tau }^{\eta ,\delta }\left(\varsigma \right)$ as follows:

$${\mathcal{M}}_{\mu ,\tau }^{\eta ,\delta }\left(\varsigma \right)=\sum _{j=0}^{\infty }{\displaystyle \frac{{\left(\eta \right)}_{j\delta }{\varsigma }^j}{\mathrm{\Gamma }(\mu j+\tau )j!}}={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\tau \right)}}+{\displaystyle \frac{{\left(\eta \right)}_{\delta }}{\mathrm{\Gamma }(\mu +\tau )}}\varsigma +{\displaystyle \frac{{\left(\eta \right)}_{2\delta }}{2\mathrm{\Gamma }(2\mu +\tau )}}{\varsigma }^2+\cdots ,$$

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

$${\left(\eta \right)}_0=1\mathrm{and}{\left(\eta \right)}_n=\eta (\eta +1)\cdots (\eta +j-1),\left(j=1,2,3,\cdots \right).$$

If $h_1,h_2\in \mathcal{A}$ such that $h_1\left(\varsigma \right),h_2\left(\varsigma \right)$ are given by

$$h_1\left(\varsigma \right)=\varsigma +\sum _{j=2}^{\infty }{A}_j{\varsigma }^{j}and{h}_2\left(\varsigma \right)=\varsigma +\sum _{j=2}^{\infty }{B}_j{\varsigma }^j,$$

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

$$(h_1\ast h_2)\left(\varsigma \right)=h_1\left(\varsigma \right)\ast h_2\left(\varsigma \right)=\varsigma +\sum _{j=2}^{\infty }{A}_j{B}_j{\varsigma }^j=(h_2\ast h_1)\left(\varsigma \right)\in \mathcal{A},$$

which is called the convolution of $h_1$ and $h_2$ or the Hadamard product of $h_1$ and $h_2.$

In 2016, Attiya [37] defined and studied a certain linear operator ${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }:\mathcal{A}⟶\mathcal{A}$ by using convolution as follows:

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

where

$${\mathcal{E}}_{\mu ,\tau }^{\eta ,\delta }\left(\varsigma \right)={\displaystyle \frac{\mathrm{\Gamma }(\mu +\tau )}{{\left(\eta \right)}_{\delta }}}\left[{\mathcal{M}}_{\mu ,\tau }^{\eta ,\delta }\left(\varsigma \right)-{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\tau \right)}}\right]=\varsigma +{\displaystyle \frac{{(\eta +\delta )}_{\delta }}{{(\mu +\tau )}_{\mu }2!}}{\varsigma }^2+{\displaystyle \frac{{(\eta +\delta )}_{2\delta }}{{(\mu +\tau )}_{2\mu }3!}}{\varsigma }^3+\cdots .$$

Hence, we get

$${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)=\varsigma +\sum _{j=2}^{\infty }{\displaystyle \frac{{(\eta +\delta )}_{\left(j-1\right)\delta }}{{(\mu +\tau )}_{\left(j-1\right)\mu }j!}}{d}_j{\varsigma }^j.$$

Remark 1.

Setting the parameters as $\eta =2$, $\delta =1$, and $\mu =0$ reduces the operator ${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)$ to ${\mathcal{H}}_{0,\tau }^{2,1}\phi \left(\varsigma \right)={\displaystyle \frac{{\left[\varsigma {\phi }^{\prime }\left(\varsigma \right)\right]}^{\prime }}{2}}$, which coincides with the operator introduced by Livingston [38]. Similarly, choosing $\eta =0$, $\delta =1$, and $\mu =0$ yields the integral form ${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)={\int }_0^{\varsigma }{\displaystyle \frac{\phi \left(s\right)}{s}}ds$, known as the Alexander integral operator [39,40]. These concrete reductions demonstrate how the Mittag–Leffler framework systematically encapsulates both classical differential and integral operators within a unified analytic structure.

Remark 2.

For $\eta =\delta =1$ and $\mu =0$, we have ${\mathcal{H}}_{0,\tau }^{1,1}\phi \left(\varsigma \right)=\phi \left(\varsigma \right)$.

This paper introduces and investigates new subfamilies of analytic and bi-univalent functions by applying a linear operator ${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)$ based on the generalized Mittag–Leffler function and imposing subordination to domains associated with the generalized bivariate Fibonacci polynomials. The Mittag–Leffler function provides a flexible fractional calculus operator, while the Fibonacci polynomials define a diverse family of geometric shapes. By unifying these two parameterized frameworks, we establish bounds for the initial Taylor coefficients $\left|d_2\right|$ and $\left|d_3\right|,$ and solve the corresponding Fekete–Szegö problem for the functional $\left|d_3-\xi d_2^2\right|$.

2. Main Results

Using the operator ${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)$ and the generalized bivariate Fibonacci polynomials $\mathcal{G}(\varkappa ,y,z)$, we define the following families.

Definition 1.

Let $0\leqq \lambda \leqq 1$. If $\phi \in \mathrm{\Sigma }$ satisfies

$$\lambda \left(\frac{{\varsigma }^2{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{″}\right]}^2}{{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }}+2\varsigma {\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{″}+{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }\right)+(1-\lambda )\frac{\varsigma {\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }\right]}^2}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)}\prec F\left(\varsigma \right),\varsigma \in \mathrm{\Omega },$$

(6)

and

$$\lambda \left(\frac{w^2{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{″}\right]}^2}{{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }}+2w{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{″}+{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right)+(1-\lambda )\frac{w{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right]}^2}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)}\prec F\left(w\right),w\in \mathrm{\Omega },$$

(7)

then we say that $\phi \in {\mathfrak{J}}_{\mu ,\tau }^{\eta ,\delta }(\lambda ,F)$, where $\chi \left(w\right)={\phi }^{-1}\left(w\right)$ and

$$F\left(\varsigma \right)={\displaystyle \frac{\mathcal{G}(\varkappa ,y,\varsigma )}{\varsigma }}={\displaystyle \frac{1}{1-k\varsigma -l{\varsigma }^2}},k\ne 0,k^2+4l>0.$$

(8)

Remark 3.

The subfamilies are derived for particular choices of λ in the family ${\mathfrak{J}}_{\mu ,\tau }^{\eta ,\delta }(\lambda ,F)$:

1. ${\mathfrak{R}}_{\mu ,\tau }^{\eta ,\delta }\left(F\right)\equiv {\mathfrak{J}}_{\mu ,\tau }^{\eta ,\delta }(0,F)$ is a subfamily of functions such that $\phi \in \mathrm{\Sigma }$ fulfills

$${\displaystyle \frac{\varsigma {\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }\right]}^2}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)}}\prec F\left(\varsigma \right),\varsigma \in \mathrm{\Omega },$$

(9)

and

$${\displaystyle \frac{w{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right]}^2}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)}}\prec F\left(w\right),w\in \mathrm{\Omega },$$

(10)

where $F\left(\varsigma \right)$ is as mentioned in (8).

2. ${\mathfrak{Y}}_{\mu ,\tau }^{\eta ,\delta }\left(F\right)\equiv {\mathfrak{J}}_{\mu ,\tau }^{\eta ,\delta }(1,F)$ represents a set of functions $\phi \in \mathrm{\Sigma }$ that meet

$$\frac{{\varsigma }^2{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{″}\right]}^2}{{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }}+2\varsigma {\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{″}+{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }\prec F\left(\varsigma \right),\varsigma \in \mathrm{\Omega },$$

(11)

and

$$\frac{w^2{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{″}\right]}^2}{{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }}+2w{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{″}+{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\prec F\left(w\right),w\in \mathrm{\Omega },$$

(12)

where $\chi \left(w\right)={\phi }^{-1}\left(w\right)$ and $F\left(\varsigma \right)$ is as stated in (8).

Definition 2.

Let $0\leqq \lambda \leqq 1$. If $\phi \in \mathrm{\Sigma }$ satisfies

$$\lambda \left(\frac{w^2{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{″}\right]}^2}{{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }}+2w{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{″}+{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }\right)+(1-\lambda ){\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }\right]}^2\prec F\left(\varsigma \right),\varsigma \in \mathrm{\Omega },$$

(13)

and

$$\lambda \left(\frac{w^2{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{″}\right]}^2}{{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }}+2w{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{″}+{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right)+(1-\lambda ){\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right]}^2\prec F\left(w\right),w\in \mathrm{\Omega },$$

(14)

then we say that $\phi \in {\mathfrak{E}}_{\mu ,\tau }^{\eta ,\delta }(\lambda ,F)$, where $\chi \left(w\right)={\phi }^{-1}\left(w\right)$ and $F\left(z\right)$ is as mentioned in (8).

Remark 4.

The subfamilies are derived for particular choices of λ in the family ${\mathfrak{E}}_{\mathrm{\Sigma }}(\lambda ,F)$:

1. ${\mathfrak{M}}_{\mu ,\tau }^{\eta ,\delta }\left(F\right)\equiv {\mathfrak{E}}_{\mu ,\tau }^{\eta ,\delta }(0,F)$, is the collection of functions $\phi \in \mathrm{\Sigma }$ that meet

$${\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right]}^2\prec F\left(\varsigma \right),\varsigma \in \mathrm{\Omega },$$

(15)

and

$${\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right]}^2\prec F\left(w\right),w\in \mathrm{\Omega },$$

(16)

where $F\left(z\right)$ is as mentioned in (8).

2. ${\mathfrak{E}}_{\mu ,\tau }^{\eta ,\delta }(1,F)\equiv {\mathfrak{Y}}_{\mu ,\tau }^{\eta ,\delta }\left(F\right)$.

Throughout our following discussion, we will assume, unless otherwise stated, that $\mu ,\tau ,\eta ,\delta \in \mathbb{C}$, k and l are polynomials in $\varkappa$ and y, with real coefficients such that $k^2+4l>0$ and all the powers are considered the principal ones. In the first theorem, for functions $\phi$ in the newly introduced subfamily ${\mathfrak{J}}_{\mu ,\tau }^{\eta ,\delta }(\lambda ,F)$, we establish the upper bounds for the initial Maclaurin coefficients $\left|d_2\right|$ and $\left|d_3\right|$, and solve the Fekete–Szegö inequality $\left|d_3-\xi d_2^2\right|$.

Theorem 1.

Let $\xi \in \mathbb{R}$, $0\leqq \lambda \leqq 1$ and F be in (8). If $\phi \in \mathrm{\Sigma }$ is assigned to the family ${\mathfrak{J}}_{\mu ,\tau }^{\eta ,\delta }(\lambda ,F)$, then

$$\left|d_2\right|\leqq \sqrt{{\displaystyle \frac{2{\left|k\right|}^3}{\left|\left\{\frac{(3\lambda +1){\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}+\frac{5(2\lambda +1){(\eta +\delta )}_{2\delta }}{3{(\mu +\tau )}_{2\mu }}\right\}{k}^2-\frac{9{(\lambda +1)}^2{\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}\left(k^2+l\right)\right|}}},$$

(17)

$$\left|d_3\right|\leqq {\displaystyle \frac{4{\left|{(\mu +\tau )}_{\mu }\right|}^2{\left|k\right|}^2}{9{(\lambda +1)}^2{\left|{(\eta +\delta )}_{\delta }\right|}^2}}+{\displaystyle \frac{6\left|{(\mu +\tau )}_{2\mu }\right|\left|k\right|}{5(2\lambda +1)\left|{(\eta +\delta )}_{2\delta }\right|}},$$

(18)

and

$$\left|d_3-\xi d_2^2\right|\leqq \left\{\begin{array}{cc}{\displaystyle \frac{6\left|{(\mu +\tau )}_{2\mu }\right|\left|k\right|}{5(2\lambda +1)\left|{\left(\eta +\delta \right)}_{2\delta }\right|}}\hfill & ;\left|1-\xi \right|\leqq \daleth \hfill \\ {\displaystyle \frac{2{\left|k\right|}^3\left|1-\xi \right|}{\left|\left\{\frac{(3\lambda +1){\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}+\frac{5(2\lambda +1){(\eta +\delta )}_{2\delta }}{3{(\mu +\tau )}_{2\mu }}\right\}{k}^2-\frac{9{(\lambda +1)}^2{\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}\left(k^2+l\right)\right|}}\hfill & ;\left|1-\xi \right|\geqq \daleth .\hfill \end{array}\right.$$

(19)

where

$$\daleth =\left|1+{\displaystyle \frac{3{(\mu +\tau )}_{2\mu }{\left[{(\eta +\delta )}_{\delta }\right]}^2\left[\left(3\lambda +1\right)k^2-9{(\lambda +1)}^2\left(k^2+l\right)\right]}{10(2\lambda +1){\left(\eta +\delta \right)}_{2\delta }{\left[{(\mu +\tau )}_{\mu }\right]}^2{k}^2}}\right|.$$

(20)

Proof. 

Let $\phi \in {\mathfrak{J}}_{\mu ,\tau }^{\eta ,\delta }(\lambda ,F)$. Then, from conditions (6) and (7), we can find that

$$\lambda \left(\frac{{\varsigma }^2{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{″}\right]}^2}{{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }}+2\varsigma {\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{″}+{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }\right)+(1-\lambda )\frac{\varsigma {\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }\right]}^2}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)}=F\left(\mathfrak{u}\left(\varsigma \right)\right),\varsigma \in \mathrm{\Omega }$$

(21)

and

$$\lambda \left(\frac{w^2{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{″}\right]}^2}{{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }}+2w{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{″}+{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right)+(1-\lambda )\frac{w{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right]}^2}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)}=F\left(\mathfrak{v}\left(w\right)\right),w\in \mathrm{\Omega },$$

(22)

such that $\mathfrak{u}\left(\varsigma \right)={\displaystyle \sum _{j=1}^{\infty }}{\mathfrak{u}}_j{\varsigma }^j,and\mathfrak{v}\left(w\right)={\displaystyle \sum _{j=1}^{\infty }}{\mathfrak{v}}_j{w}^j,\varsigma ,w\in \mathrm{\Omega }$ are Schwarz functions with the property (See [1])

$$\left|{\mathfrak{u}}_j\right|\leqq 1,and\left|{\mathfrak{v}}_j\right|\leqq 1(j\in \mathbb{N}).$$

(23)

Using simple mathematical techniques, we can find from (21) and (22) that

$$\lambda \left(\frac{{\varsigma }^2{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{″}\right]}^2}{{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }}+2\varsigma {\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{″}+{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }\right)+(1-\lambda )\frac{\varsigma {\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }\right]}^2}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)}$$

$$=1+\frac{3(\lambda +1){(\eta +\delta )}_{\delta }}{2{(\mu +\tau )}_{\mu }}{d}_2\varsigma +\left[\frac{5(2\lambda +1){(\eta +\delta )}_{2\delta }}{6{(\mu +\tau )}_{2\mu }}{d}_3+\frac{(3\lambda +1){\left[{(\eta +\delta )}_{\delta }\right]}^2}{4{\left[{(\mu +\tau )}_{\mu }\right]}^2}{d}_2^2\right]{\varsigma }^2+\cdots ,$$

(24)

$$F\left(\mathfrak{u}\left(\varsigma \right)\right)=1+{\mathcal{G}}_2(\varkappa ,y){\mathfrak{u}}_1\varsigma +\left[{\mathcal{G}}_2(\varkappa ,y){\mathfrak{u}}_2+{\mathcal{G}}_3(\varkappa ,y){\mathfrak{u}}_1^2\right]{\varsigma }^2+\cdots$$

(25)

and

$$\lambda \left(\frac{w^2{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{″}\right]}^2}{{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }}+2w{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{″}+{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right)+(1-\lambda )\frac{w{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right]}^2}{{\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)}$$

$$=1-\frac{3(\lambda +1){(\eta +\delta )}_{\delta }}{2{(\mu +\tau )}_{\mu }}{d}_{2}w++\left[\frac{5(2\lambda +1){(\eta +\delta )}_{2\delta }}{6{(\mu +\tau )}_{2\mu }}\left(2d_2^2-d_3\right)+\frac{(3\lambda +1){\left[{(\eta +\delta )}_{\delta }\right]}^2}{4{\left[{(\mu +\tau )}_{\mu }\right]}^2}{d}_2^2\right]w^2+\cdots ,$$

(26)

$$F\left(\mathfrak{v}\left(w\right)\right)=1+{\mathcal{G}}_2(\varkappa ,y){\mathfrak{v}}_{1}w+\left[{\mathcal{G}}_2(\varkappa ,y){\mathfrak{v}}_2+{\mathcal{G}}_3(\varkappa ,y){\mathfrak{v}}_1^2\right]w^2+\cdots .$$

(27)

From the comparison of Equations (24) and (25), we conclude due to equality (21) that

$${\displaystyle \frac{3(\lambda +1){(\eta +\delta )}_{\delta }}{2{(\mu +\tau )}_{\mu }}}{d}_2={\mathcal{G}}_2(\varkappa ,y){\mathfrak{u}}_1$$

(28)

and

$${\displaystyle \frac{5(2\lambda +1){(\eta +\delta )}_{2\delta }}{6{(\mu +\tau )}_{2\mu }}}{d}_3+{\displaystyle \frac{(3\lambda +1){\left[{(\eta +\delta )}_{\delta }\right]}^2}{4{\left[{(\mu +\tau )}_{\mu }\right]}^2}}{d}_2^2={\mathcal{G}}_2(\varkappa ,y){\mathfrak{u}}_2+{\mathcal{G}}_3(\varkappa ,y){\mathfrak{u}}_1^2.$$

(29)

Similarly, consequent to (22), we find by comparing the terms of the same degree in (26) and (27) that

$$-{\displaystyle \frac{3(\lambda +1){(\eta +\delta )}_{\delta }}{2{(\mu +\tau )}_{\mu }}}{d}_2={\mathcal{G}}_2(\varkappa ,y){\mathfrak{v}}_1$$

(30)

and

$$\frac{5(2\lambda +1){(\eta +\delta )}_{2\delta }}{6{(\mu +\tau )}_{2\mu }}\left(2d_2^2-d_3\right)+\frac{(3\lambda +1){\left[{(\eta +\delta )}_{\delta }\right]}^2}{4{\left[{(\mu +\tau )}_{\mu }\right]}^2}{d}_2^2={\mathcal{G}}_2(\varkappa ,y){\mathfrak{v}}_2+{\mathcal{G}}_3(\varkappa ,y){\mathfrak{v}}_1^2.$$

(31)

Equating (28) and (30) gives us

$${\mathfrak{u}}_1=-{\mathfrak{v}}_1.$$

(32)

Also, by squaring (28) and (30) and adding them together, we get

$${\displaystyle \frac{9{(\lambda +1)}^2{\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}}{d}_2^2=({\mathfrak{u}}_1^2+{\mathfrak{v}}_1^2){\mathcal{G}}_2^2(\varkappa ,y).$$

(33)

Addition of (29) and (31) yields

$$\left[\frac{(3\lambda +1){\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}+\frac{5(2\lambda +1){(\eta +\delta )}_{2\delta }}{3{(\mu +\tau )}_{2\mu }}\right]d_2^2={\mathcal{G}}_2(\varkappa ,y)({\mathfrak{u}}_2+{\mathfrak{v}}_2)+{\mathcal{G}}_3(\varkappa ,y)({\mathfrak{u}}_1^2+{\mathfrak{v}}_1^2).$$

(34)

Replacing ${\mathfrak{u}}_1^2+{\mathfrak{v}}_1^2$ from (33) in (34), we get

$$d_2^2={\displaystyle \frac{{\mathcal{G}}_2^3(\varkappa ,y)({\mathfrak{u}}_2+{\mathfrak{v}}_2)}{\left\{\frac{(3\lambda +1){\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}+\frac{5(2\lambda +1){(\eta +\delta )}_{2\delta }}{3{(\mu +\tau )}_{2\mu }}\right\}{\mathcal{G}}_2^2(\varkappa ,y)-\frac{9{(\lambda +1)}^2{\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}{\mathcal{G}}_3(\varkappa ,y)}}.$$

(35)

Utilizing (5) for ${\mathcal{G}}_2(\varkappa ,y)$ and ${\mathcal{G}}_3(\varkappa ,y)$, and applying (23) to ${\mathfrak{u}}_2$,${\mathfrak{v}}_2$ produces (17). Now, we subtract (31) from (29) to find the bound on $\left|d_3\right|$:

$$d_3=d_2^2+{\displaystyle \frac{3{(\mu +\tau )}_{2\mu }{\mathcal{G}}_2(\varkappa ,y)\left({\mathfrak{u}}_2-{\mathfrak{v}}_2\right)}{5(2\lambda +1){(\eta +\delta )}_{2\delta }}}.$$

(36)

If we replace $d_2^2$ from (33) in (36), we get

$$d_3={\displaystyle \frac{2{\left[{(\mu +\tau )}_{\mu }\right]}^2{\mathcal{G}}_2^2(\varkappa ,y)\left({\mathfrak{u}}_1^2+{\mathfrak{v}}_1^2\right)}{9{(\lambda +1)}^2{\left[{(\eta +\delta )}_{\delta }\right]}^2}}+{\displaystyle \frac{3{(\mu +\tau )}_{2\mu }{\mathcal{G}}_2(\varkappa ,y)\left({\mathfrak{u}}_2-{\mathfrak{v}}_2\right)}{5(2\lambda +1){(\eta +\delta )}_{2\delta }}}.$$

(37)

We derive (18) from (37) by using (5) and (23). Finally, we calculate the estimate on $\left|d_3-\xi d_2^2\right|$ using the values of $d_2^2$ and $d_3$ from (35) and (36), respectively. Consequently, we have

$$\begin{array}{cc}\hfill \left|d_3-\xi d_2^2\right|& =\left|\begin{array}{c}{\displaystyle \frac{3{(\mu +\tau )}_{2\mu }{\mathcal{G}}_2(\varkappa ,y)\left({\mathfrak{u}}_2-{\mathfrak{v}}_2\right)}{5(2\lambda +1){(\eta +\delta )}_{2\delta }}}\\ \\ +{\displaystyle \frac{\left(1-\xi \right){\mathcal{G}}_2^3(\varkappa ,y)({\mathfrak{u}}_2+{\mathfrak{v}}_2)}{\left\{\frac{(3\lambda +1){\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}+\frac{5(2\lambda +1){(\eta +\delta )}_{2\delta }}{3{(\mu +\tau )}_{2\mu }}\right\}{\mathcal{G}}_2^2(\varkappa ,y)-\frac{9{(\lambda +1)}^2{\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}{\mathcal{G}}_3(\varkappa ,y)}}\end{array}\right|\hfill \\ \hfill & =\left|{\mathcal{G}}_2(\varkappa ,y)\right|\left|\left({\displaystyle \frac{3{(\mu +\tau )}_{2\mu }}{5(2\lambda +1){\left(\eta +\delta \right)}_{2\delta }}}+\mathcal{B}\left(\xi ,F\right)\right){\mathfrak{u}}_2\right.\hfill \\ \hfill & \left.-\left({\displaystyle \frac{3{(\mu +\tau )}_{2\mu }}{5(2\lambda +1){(\eta +\delta )}_{2\delta }}}-\mathcal{B}\left(\xi ,F\right)\right){\mathfrak{v}}_2\right|,\hfill \end{array}$$

where

$$\mathcal{B}\left(\xi ,F\right)={\displaystyle \frac{\left(1-\xi \right){\mathcal{G}}_2^2(\varkappa ,y)}{\left\{\frac{(3\lambda +1){\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}+\frac{5(2\lambda +1){(\eta +\delta )}_{2\delta }}{3{(\mu +\tau )}_{2\mu }}\right\}{\mathcal{G}}_2^2(\varkappa ,y)-\frac{9{(\lambda +1)}^2{\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}{\mathcal{G}}_3(\varkappa ,y)}}.$$

Clearly, we have

$$\left|d_3-\xi d_2^2\right|\leqq \left\{\begin{array}{cc}{\displaystyle \frac{6\left|{(\mu +\tau )}_{2\mu }\right|\left|{\mathcal{G}}_2(\varkappa ,y)\right|}{5(2\lambda +1)\left|{\left(\eta +\delta \right)}_{2\delta }\right|}}\hfill & ;\left|\mathcal{B}(\xi ,F)\right|\leqq {\displaystyle \frac{3\left|{(\mu +\tau )}_{2\mu }\right|}{5(2\lambda +1)\left|{\left(\eta +\delta \right)}_{2\delta }\right|}}\hfill \\ 2\left|{\mathcal{G}}_2(\varkappa ,y)\right|\left|\mathcal{B}(\xi ,F)\right|\hfill & ;\left|\mathcal{B}(\xi ,F)\right|\geqq {\displaystyle \frac{3\left|{(\mu +\tau )}_{2\mu }\right|}{5(2\lambda +1)\left|{\left(\eta +\delta \right)}_{2\delta }\right|}}.\hfill \end{array}\right.$$

(38)

We derive (19) from (38), where ℸ is the same as in (20). □

Next, we provide the initial upper bounds for functions in the subfamily ${\mathfrak{E}}_{\mu ,\tau }^{\eta ,\delta }(\lambda ,F)$.

Theorem 2.

Let $\xi \in \mathbb{R},0\leqq \lambda \leqq 1$, and F be as in (8). If $\phi \in \mathrm{\Sigma }$ is assigned to the family ${\mathfrak{E}}_{\mu ,\tau }^{\eta ,\delta }(\lambda ,F),$ then

$$\left|d_2\right|\leqq \sqrt{{\displaystyle \frac{2{\left|k\right|}^3}{\left|\left\{\frac{(3\lambda +2){(\eta +\delta )}_{2\delta }}{{(\mu +\tau )}_{2\mu }}+\frac{2{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}\right\}{k}^2-\frac{2{\left(\lambda +2\right)}^2{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}(k^2+l)\right|}}},$$

(39)

$$\left|d_3\right|\leqq {\displaystyle \frac{{\left|{(\mu +\tau )}_{\mu }\right|}^2{\left|k\right|}^2}{{\left(\lambda +2\right)}^2{\left|{(\eta +\delta )}_{\delta }\right|}^2}}+{\displaystyle \frac{2\left|{(\mu +\tau )}_{2\mu }\right|\left|k\right|}{\left(3\lambda +2\right)\left|{(\eta +\delta )}_{2\delta }\right|}}$$

(40)

and

$$|d_3-\xi d_2^2|\leqq \left\{\begin{array}{cc}{\displaystyle \frac{2\left|{(\mu +\tau )}_{2\mu }\right|\left|k\right|}{\left(3\lambda +2\right)\left|{\left(\eta +\delta \right)}_{2\delta }\right|}}\hfill & ;\left|1-\xi \right|\leqq \mathcal{Z}\hfill \\ {\displaystyle \frac{2{\left|k\right|}^3\left|1-\xi \right|}{\left|\left\{\frac{(3\lambda +2){(\eta +\delta )}_{2\delta }}{{(\mu +\tau )}_{2\mu }}+\frac{2{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}\right\}{k}^2-\frac{2{\left(\lambda +2\right)}^2{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}\left(k^2+l\right)\right|}}\hfill & ;\left|1-\xi \right|\geqq \mathcal{Z},\hfill \end{array}\right.$$

(41)

where

$$\mathcal{Z}=\left|1+{\displaystyle \frac{2{(\mu +\tau )}_{2\mu }{\left[{(\eta +\delta )}_{\delta }\right]}^2\left[k^2-{\left(\lambda +2\right)}^2\left(k^2+l\right)\right]}{(3\lambda +2)\left|{\left(\eta +\delta \right)}_{2\delta }\right|{\left[{(\mu +\tau )}_{\mu }\right]}^2{k}^2}}\right|.$$

Proof. 

Let $\phi \in {\mathfrak{E}}_{\mathrm{\Sigma }}(\lambda ,F)$. Then, from (13) and (14), we get

$$\lambda \left(\frac{{\varsigma }^2{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{″}\right]}^2}{{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }}+2\varsigma {\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{″}+{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }\right)+(1-\lambda ){\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\phi \left(\varsigma \right)\right)}^{\prime }\right]}^2=F\left(\mathfrak{g}\left(\varsigma \right)\right),\varsigma \in \mathrm{\Omega }$$

(42)

and

$$\lambda \left(\frac{w^2{\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{″}\right]}^2}{{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }}+2w{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{″}+{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right)+(1-\lambda ){\left[{\left({\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }\chi \left(w\right)\right)}^{\prime }\right]}^2=F\left(\mathfrak{h}\left(w\right)\right),w\in \mathrm{\Omega }.$$

(43)

where

$$\mathfrak{g}\left(\varsigma \right)=\sum _{j=1}^{\infty }{\mathfrak{g}}_j{\varsigma }^j={\mathfrak{g}}_1\varsigma +{\mathfrak{g}}_2{\varsigma }^2+{\mathfrak{g}}_3{\varsigma }^3+\cdots ,\varsigma \in \mathrm{\Omega }$$

(44)

and

$$\mathfrak{h}\left(w\right)=\sum _{j=1}^{\infty }{\mathfrak{h}}_j{w}^j={\mathfrak{h}}_{1}w+{\mathfrak{h}}_2{w}^2+{\mathfrak{h}}_3{w}^3+\cdots ,w\in \mathrm{\Omega }$$

(45)

are Schwarz functions with the following properties:

$$\left|{\mathfrak{g}}_j\right|\leqq 1,and\left|{\mathfrak{h}}_j\right|\leqq 1(j\in \mathbb{N}).$$

(46)

Utilizing the series expansions of $\phi \left(\varsigma \right)$ and $\mathfrak{g}\left(\varsigma \right)$ given by (1) and (44), respectively, we obtain

$$1+{\displaystyle \frac{\left(\lambda +2\right){(\eta +\delta )}_{\delta }}{{(\mu +\tau )}_{\mu }}}{d}_2\varsigma +\left[{\displaystyle \frac{\left(3\lambda +2\right){(\eta +\delta )}_{2\delta }}{2{(\mu +\tau )}_{2\mu }}}{d}_3+{\displaystyle \frac{{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}}{d}_2^2\right]{\varsigma }^2+\cdots$$

$$={\mathcal{G}}_1(\varkappa ,y)+{\mathcal{G}}_2(\varkappa ,y){\mathfrak{g}}_1\varsigma +\left[{\mathcal{G}}_2(\varkappa ,y){\mathfrak{g}}_2+{\mathcal{G}}_3(\varkappa ,y){\mathfrak{g}}_1^2\right]{\varsigma }^2+\cdots .$$

(47)

Moreover, by utilizing the series expansion of $\chi \left(w\right)$ and $\mathfrak{h}\left(w\right)$ given by (1) and (45), respectively, we obtain

$$1-{\displaystyle \frac{2(\lambda +2){(\eta +\delta )}_{\delta }}{{(\mu +\tau )}_{\mu }2!}}{d}_{2}w+\left[{\displaystyle \frac{(3\lambda +2){(\eta +\delta )}_{2\delta }}{2{(\mu +\tau )}_{2\mu }}}\left(2d_2^2-d_3\right)+{\displaystyle \frac{{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}}{d}_2^2\right]w^2-\cdots$$

$$={\mathcal{G}}_1(\varkappa ,y)+{\mathcal{G}}_2(\varkappa ,y){\mathfrak{h}}_{1}w+\left[{\mathcal{G}}_2(\varkappa ,y){\mathfrak{h}}_2+{\mathcal{G}}_3(\varkappa ,y){\mathfrak{h}}_1^2\right]w^2+\cdots .$$

(48)

Therefore, by comparing the corresponding coefficients in (47) and (48), it is possible to get

$${\displaystyle \frac{\left(\lambda +2\right){(\eta +\delta )}_{\delta }}{{(\mu +\tau )}_{\mu }}}{d}_2={\mathcal{G}}_2(\varkappa ,y){\mathfrak{g}}_1,$$

(49)

$${\displaystyle \frac{\left(3\lambda +2\right){(\eta +\delta )}_{2\delta }}{2{(\mu +\tau )}_{2\mu }}}{d}_3+{\displaystyle \frac{{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}}{d}_2^2={\mathcal{G}}_2(\varkappa ,y){\mathfrak{g}}_2+{\mathcal{G}}_3(\varkappa ,y){\mathfrak{g}}_1^2,$$

(50)

$$-{\displaystyle \frac{\left(\lambda +2\right){(\eta +\delta )}_{\delta }}{{(\mu +\tau )}_{\mu }}}{d}_2={\mathcal{G}}_2(\varkappa ,y){\mathfrak{h}}_1$$

(51)

and

$${\displaystyle \frac{(3\lambda +2){(\eta +\delta )}_{2\delta }}{2{(\mu +\tau )}_{2\mu }}}\left(2d_2^2-d_3\right)+{\displaystyle \frac{{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}}{d}_2^2={\mathcal{G}}_2(\varkappa ,y){\mathfrak{h}}_2+{\mathcal{G}}_3(\varkappa ,y){\mathfrak{h}}_1^2.$$

(52)

Now, the results (39)–(41) follow from (49)–(52), using the same method as used in Theorem 1 in relation to (28) to (31). □

Remark 5.

The coefficient bounds derived here are analogous to the stability bounds in interval power flow models and adaptive system identification [41].

3. Special Cases

In this section, we present some special cases for our results with specific parameters, such as $\lambda$.

When $\lambda =0$, Theorem 1 would yield the next corollary.

Corollary 1.

Let $\xi \in \mathbb{R}$ and F be as in (8). If $\phi \in {\mathfrak{R}}_{\mu ,\tau }^{\eta ,\delta }\left(F\right),$ then

$$\left|d_2\right|\leqq \sqrt{{\displaystyle \frac{2{\left|k\right|}^3}{\left|\left\{\frac{5{(\eta +\delta )}_{2\delta }}{3{(\mu +\tau )}_{2\mu }}+\frac{{\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}\right\}{k}^2-\frac{9{\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}\left(k^2+l\right)\right|}}},$$

$$\left|d_3\right|\leqq {\displaystyle \frac{4{\left|{(\mu +\tau )}_{\mu }\right|}^2{\left|k\right|}^2}{9{\left|{(\eta +\delta )}_{\delta }\right|}^2}}+{\displaystyle \frac{6\left|{(\mu +\tau )}_{2\mu }\right|\left|k\right|}{5\left|{(\eta +\delta )}_{2\delta }\right|}}$$

and

$$\left|d_3-\xi d_2^2\right|\leqq \left\{\begin{array}{cc}{\displaystyle \frac{6\left|{(\mu +\tau )}_{2\mu }\right|\left|k\right|}{5\left|{\left(\eta +\delta \right)}_{2\delta }\right|}}\hfill & ;\left|1-\xi \right|\leqq {\daleth }_1\hfill \\ {\displaystyle \frac{2{\left|k\right|}^3\left|1-\xi \right|}{\left|\left\{\frac{5{(\eta +\delta )}_{2\delta }}{3{(\mu +\tau )}_{2\mu }}+\frac{{\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}\right\}{k}^2-\frac{9{\left[{(\eta +\delta )}_{\delta }\right]}^2}{2{\left[{(\mu +\tau )}_{\mu }\right]}^2}\left(k^2+l\right)\right|}}\hfill & ;\left|1-\xi \right|\geqq {\daleth }_1.\hfill \end{array}\right.$$

where

$$\daleth =\left|1-{\displaystyle \frac{3{(\mu +\tau )}_{2\mu }{\left[{(\eta +\delta )}_{\delta }\right]}^2\left(8k^2+9l\right)}{10{\left(\eta +\delta \right)}_{2\delta }{\left[{(\mu +\tau )}_{\mu }\right]}^2{k}^2}}\right|.$$

When $\lambda =1$, Theorem 1 would yield the following corollary.

Corollary 2.

Let $\xi \in \mathbb{R}$ and F be as in (8). If $\phi \in {\mathfrak{Y}}_{\mu ,\tau }^{\eta ,\delta }\left(F\right),$ then

$$\left|d_2\right|\leqq \sqrt{{\displaystyle \frac{2{\left|k\right|}^3}{\left|\left\{\frac{2{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}+\frac{5{(\eta +\delta )}_{2\delta }}{{(\mu +\tau )}_{2\mu }}\right\}{k}^2-\frac{18{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}\left(k^2+l\right)\right|}}},$$

$$\left|d_3\right|\leqq {\displaystyle \frac{{\left|{(\mu +\tau )}_{\mu }\right|}^2{\left|k\right|}^2}{9{\left|{(\eta +\delta )}_{\delta }\right|}^2}}+{\displaystyle \frac{2\left|{(\mu +\tau )}_{2\mu }\right|\left|k\right|}{5\left|{(\eta +\delta )}_{2\delta }\right|}}$$

and

$$\left|d_3-\xi d_2^2\right|\leqq \left\{\begin{array}{cc}{\displaystyle \frac{2\left|{(\mu +\tau )}_{2\mu }\right|\left|k\right|}{5\left|{\left(\eta +\delta \right)}_{2\delta }\right|}}\hfill & ;\left|1-\xi \right|\leqq {\daleth }_2\hfill \\ {\displaystyle \frac{2{\left|k\right|}^3\left|1-\xi \right|}{\left|\left\{\frac{2{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}+\frac{5{(\eta +\delta )}_{2\delta }}{{(\mu +\tau )}_{2\mu }}\right\}{k}^2-\frac{18{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}\left(k^2+l\right)\right|}}\hfill & ;\left|1-\xi \right|\geqq {\daleth }_2,\hfill \end{array}\right.$$

where

$$\daleth =\left|1-{\displaystyle \frac{2{(\mu +\tau )}_{2\mu }{\left[{(\eta +\delta )}_{\delta }\right]}^2\left(8k^2+9l\right)}{5{\left(\eta +\delta \right)}_{2\delta }{\left[{(\mu +\tau )}_{\mu }\right]}^2{k}^2}}\right|.$$

Taking $\eta =\delta =1$ and $\mu =0$ in the above corollary, we obtain the next example.

Example 1.

If the functions $\phi ,\chi \in \mathrm{\Sigma }$ satisfy the next conditions:

$${\displaystyle \frac{\varsigma {\phi }^{\prime }\left(\varsigma \right)}{\phi \left(\varsigma \right)}}{\phi }^{\prime }\left(\varsigma \right)\prec {\displaystyle \frac{1}{1-k\varsigma -l{\varsigma }^2}},\varsigma \in \mathrm{\Omega }$$

and

$${\displaystyle \frac{w{\chi }^{\prime }\left(w\right)}{\chi \left(w\right)}}{\chi }^{\prime }\left(w\right)\prec {\displaystyle \frac{1}{1-kw-l{w}^2}},w\in \mathrm{\Omega },$$

then

$$\left|d_2\right|\leqq \sqrt{{\displaystyle \frac{{\left|k\right|}^3}{3\left|k^2+3l\right|}}},\left|d_3\right|\leqq {\displaystyle \frac{{\left|k\right|}^2}{9}}+{\displaystyle \frac{\left|k\right|}{5}},$$

and, for $\xi \in \mathbb{R}$, we have

$$\left|d_3-\xi d_2^2\right|\leqq \left\{\begin{array}{cc}{\displaystyle \frac{\left|k\right|}{5}}\hfill & ;\left|1-\xi \right|\leqq {\displaystyle \frac{3}{5}}\left|{\displaystyle \frac{k^2+3l}{k^2}}\right|\hfill \\ {\displaystyle \frac{{\left|k\right|}^3\left|1-\xi \right|}{3\left|k^2+9l\right|}}\hfill & ;\left|1-\xi \right|\geqq {\displaystyle \frac{3}{5}}\left|{\displaystyle \frac{k^2+3l}{k^2}}\right|.\hfill \end{array}\right.$$

When $\lambda =0$, Theorem 2 reduce to the next corollary.

Corollary 3.

Let $\xi \in \mathbb{R}$ and F be as in (8). If $\phi \in {\mathfrak{M}}_{\mu ,\tau }^{\eta ,\delta }\left(F\right),$ then

$$\left|d_2\right|\leqq \sqrt{{\displaystyle \frac{{\left|k\right|}^3}{\left|\left\{\frac{{(\eta +\delta )}_{2\delta }}{{(\mu +\tau )}_{2\mu }}+\frac{{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}\right\}{k}^2-\frac{4{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}(k^2+l)\right|}}},$$

$$\left|d_3\right|\leqq {\displaystyle \frac{{\left|{(\mu +\tau )}_{\mu }\right|}^2{\left|k\right|}^2}{4{\left|{(\eta +\delta )}_{\delta }\right|}^2}}+{\displaystyle \frac{\left|{(\mu +\tau )}_{2\mu }\right|\left|k\right|}{\left|{(\eta +\delta )}_{2\delta }\right|}}$$

and

$$\left|d_3-\xi d_2^2\right|\leqq \left\{\begin{array}{cc}{\displaystyle \frac{\left|{(\mu +\tau )}_{2\mu }\right|\left|k\right|}{\left|{\left(\eta +\delta \right)}_{2\delta }\right|}}\hfill & ;\left|1-\xi \right|\leqq {\mathcal{Z}}_1\hfill \\ {\displaystyle \frac{{\left|k\right|}^3\left|1-\xi \right|}{\left|\left\{\frac{{(\eta +\delta )}_{2\delta }}{{(\mu +\tau )}_{2\mu }}+\frac{{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}\right\}{k}^2-\frac{4{\left[{(\eta +\delta )}_{\delta }\right]}^2}{{\left[{(\mu +\tau )}_{\mu }\right]}^2}\left(k^2+l\right)\right|}}\hfill & ;\left|1-\xi \right|\geqq {\mathcal{Z}}_1,\hfill \end{array}\right.$$

where

$${\mathcal{Z}}_1=\left|1-{\displaystyle \frac{{(\mu +\tau )}_{2\mu }{\left[{(\eta +\delta )}_{\delta }\right]}^2\left(3k^2+4l\right)}{{\left(\eta +\delta \right)}_{2\delta }{\left[{(\mu +\tau )}_{\mu }\right]}^2{k}^2}}\right|.$$

Taking $\eta =\delta =1$ and $\mu =0$ in the above corollary, we obtain the following example.

Example 2.

If the functions $\phi ,\chi \in \mathrm{\Sigma }$ satisfy the next conditions

$${\left[{\phi }^{\prime }\left(\varsigma \right)\right]}^2\prec {\displaystyle \frac{1}{1-k\varsigma -l{\varsigma }^2}},\varsigma \in \mathrm{\Omega }$$

and

$${\left[{\chi }^{\prime }\left(w\right)\right]}^2\prec {\displaystyle \frac{1}{1-kw-l{w}^2}},w\in \mathrm{\Omega },$$

then

$$\left|d_2\right|\leqq \sqrt{{\displaystyle \frac{{\left|k\right|}^3}{2\left|3k^2+8l\right|}}},\left|d_3\right|\leqq {\displaystyle \frac{{\left|k\right|}^2}{16}}+{\displaystyle \frac{\left|k\right|}{6}},$$

and for $\xi \in \mathbb{R}$, we have

$$\left|d_3-\xi d_2^2\right|\leqq \left\{\begin{array}{cc}{\displaystyle \frac{\left|k\right|}{6}}\hfill & ;\left|1-\xi \right|\leqq \left|{\displaystyle \frac{3k^2+8l}{3k^2}}\right|,\hfill \\ {\displaystyle \frac{{\left|k\right|}^3\left|1-\xi \right|}{2\left|3k^2+8l\right|}}\hfill & ;\left|1-\xi \right|\geqq \left|{\displaystyle \frac{3k^2+8l}{3k^2}}\right|.\hfill \end{array}\right.$$

4. Conclusions

Many mathematicians have recently conducted research on special functions due to their use in various mathematical and scientific fields. This study introduces new subfamilies of holomorphic functions defined using convolution operator ${\mathcal{H}}_{\mu ,\tau }^{\eta ,\delta }$ based on the generalized Mittag–Leffler function and generalized bivariate Fibonacci polynomials. The key findings are given below:

• Two new subfamilies ${\mathfrak{J}}_{\mu ,\tau }^{\eta ,\delta }(\lambda ,F)$ and ${\mathfrak{E}}_{\mu ,\tau }^{\eta ,\delta }(\lambda ,F)$ of biunivalent functions are established and analyzed.
• The initial bounds of the Maclaurin coefficients $\left|d_2\right|and\left|d_3\right|$ inequality are derived.
• The Fekete–Szegö inequality $|d_3-\xi d_2^2|$ is solved for a real parameter $\xi$.
• Specializing the parameters yields new results for certain subfamilies of biunivalent functions that had not been explored before in geometric function theory.

Open Problems. The following list of the related open problems will presumably of interest to the readers:

• Determine the upper bounds for higher-order coefficients $|d_j|$ ($j\geqq 4$).
• Extend the framework to other special function operators (e.g., Wright or Fox–H functions).
• Investigate coefficient-related problems such as Hankel determinants, Bohr radii and inclusion relations for these geometrically defined subfamilies.