An Application of Liouville–Caputo-Type Fractional Derivatives on Certain Subclasses of Bi-Univalent Functions

Abstract

In this study, we present two novel subclasses of bi-univalent functions defined in the open unit disk, utilizing Liouville–Caputo fractional derivatives. We find constraints on initial Taylor coefficients $|c_2|$, $|c_3|$ for functions in these subclasses of bi-univalent functions. Additionally, by using the values of $a_2,a_3$ we determine the Fekete–Szegö inequality results. Moreover, a few new subclasses are deduced that have not been studied in relation to Liouville–Caputo fractional derivatives so far. The implications of the results are also emphasized. Our results are concrete examples of several earlier discoveries that are not only improved but also expanded upon.

1. Introduction and Preliminary Results

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

$${\mathcal{U}}_D=\{\varsigma :\varsigma \in \mathbb{C}\mathrm{and}|\varsigma |<1\}$$

expressed as

$$\Psi \left(\varsigma \right)=\varsigma +\sum _{n=2}^{\infty }{c}_n{\varsigma }^n.$$

(1)

Additionally, $\Psi \in \mathcal{A}$ that are univalent in ${\mathcal{U}}_D$ will be indicated by $\mathcal{S}$. Some of the significant and thoroughly studied subclasses of $\mathcal{S}$ are the class of starlike functions of order $\mathcal{V}$ and the class of convex functions of order $\mathcal{V}$ in ${\mathcal{U}}_D$, as defined and denoted, respectively, below (see [1]):

$${\mathcal{S}}^{*}\left(\mathcal{V}\right):=\left\{\Psi :\Psi \in \mathcal{S}\mathrm{and}\Re \left({\displaystyle \frac{\varsigma {\Psi }^{\prime }\left(\varsigma \right)}{\Psi \left(\varsigma \right)}}\right)>\mathcal{V};\varsigma \in {\mathcal{U}}_D;0\le \mathcal{V}<1\right\}$$

(2)

and

$$\mathcal{K}\left(\mathcal{V}\right):=\left\{\Psi :\Psi \in \mathcal{S}\mathrm{and}\Re \left(1+{\displaystyle \frac{\varsigma {\Psi }^{\prime \prime }\left(\varsigma \right)}{{\Psi }^{\prime }\left(\varsigma \right)}}\right)>\mathcal{V};\varsigma \in {\mathcal{U}}_D;0\le \mathcal{V}<1\right\}.$$

(3)

The definitions of (2) and (3) easily indicate that

$$\Psi \in \mathcal{K}\left(\mathcal{V}\right)⟺\varsigma {\Psi }^{\prime }\in {\mathcal{S}}^{*}\left(\mathcal{V}\right).$$

The inverse ${\Psi }^{-1}$ for each function $\Psi \in \mathcal{S}$ is known to exist and is defined by

$${\Psi }^{-1}(\Psi \left(\varsigma \right))=\varsigma ,\varsigma \in {\mathcal{U}}_D$$

and

$$\Psi \left({\Psi }^{-1}\left(w\right)\right)=w,\left(\left|w\right|<r_0(\Psi );r_0(\Psi )\ge {\displaystyle \frac{1}{4}}\right),$$

$$\Phi \left(w\right)={\Psi }^{-1}\left(w\right)=w-c_2{w}^2+(2c_2^2-c_3)w^3-(5c_2^3-c_2{c}_3+c_4)w^4+\dots .$$

(4)

Now we denote by $\Sigma$ the class of bi-univalent functions, defined as follows:

$$\Sigma :=\left\{\Psi \in \mathcal{A}:\Psi \left(\varsigma \right)\mathrm{and}{\Psi }^{-1}\left(\varsigma \right),\mathrm{are}\mathrm{univalent}\mathrm{in}{\mathcal{U}}_D\right\}.$$

(5)

Examples of functions in the class $\Sigma$ are

$${\displaystyle \frac{\varsigma }{1-\varsigma }},-log(1-\varsigma ),{\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{1+\varsigma }{1-\varsigma }}\right)$$

and so on. The well-known Koebe function, however, is not included in $\Sigma .$ Other typical instances of functions in $\mathcal{S}$, including

$$\varsigma -{\displaystyle \frac{{\varsigma }^2}{2}}\mathrm{and}{\displaystyle \frac{\varsigma }{1-{\varsigma }^2}}$$

are also not members of $\Sigma$ (see [1,2]).

One area of study that dates back to the early days of univalent function research is the study of coefficients of the functions in specific special classes. The 1914 Gronwall’s Area Theorem, which is used to determine constraints on the coefficients of the class of meromorphic functions, is a key finding in the theory of univalent functions. In 1916, Bieberbach solved an equivalent problem for the class $\mathcal{S}$ and his famous conjecture which was only verified in 1984 sparked the development of many techniques in the geometric theory of functions of a complex variable. The first two Taylor–Maclaurin coefficients are typically estimated in the study of bi-univalent functions, just as in the case of the classes examined by Gronwall and Bieberbach.

Lewin [3] examined the bi-univalent function class $\Sigma$ in 1967 and demonstrated that $|c_2|<1.51.$ Following this, Brannan et al. [4] postulated that $|c_2|\le \sqrt{2}.$ However, Netanyahu [5] demonstrated that $\underset{\Psi \in \Sigma }{max}\left|c_2\right|={\displaystyle \frac{4}{3}}.$ The Taylor–Maclaurin coefficient problem $|c_n|$ is likely still an open problem to solve for $n\in \mathbb{N}\setminus \{1,2\};$$\mathbb{N}:=\{1,2,3,\dots \}$. According to Brannan and Taha [6] (see also [7]), the function classes ${\mathcal{S}}_{\Sigma }\left(\mathcal{V}\right)$ (and ${\mathcal{K}}_{\Sigma }\left(\mathcal{V}\right)$) of bi-starlike (and bi-convex) functions of order $\mathcal{V}$ correspond (respectively) to ${\mathcal{S}}^{*}\left(\mathcal{V}\right)$ and $\mathcal{K}\left(\mathcal{V}\right)$, provided by (2) and (3) (see [4,6,7]). These relationships highlight the intricate connections between different classes of functions and their geometric properties. Furthermore, understanding these mappings can lead to new insights in both theoretical and applied mathematics, particularly in complex analysis.

The well-known class of strongly bi-starlike functions ${\mathcal{S}}_{\Sigma }^{*}\left(\mathcal{V}\right)$ of order $\mathcal{V}(0<\mathcal{V}\le 1)$ fulfills the following conditions:

$$\left|arg\left({\displaystyle \frac{\varsigma {\Psi }^{\prime }\left(\varsigma \right)}{\Psi \left(\varsigma \right)}}\right)\right|<{\displaystyle \frac{\mathcal{V}\pi }{2}}$$

and

$$\left|arg\left({\displaystyle \frac{w{\Phi }^{\prime }\left(w\right)}{\Phi \left(w\right)}}\right)\right|<{\displaystyle \frac{\mathcal{V}\pi }{2}},$$

where $\varsigma ,w\in {\mathcal{U}}_D,0<\mathcal{V}\le 1,$ and $\Phi$ is as given in (4). Similarly, a function $\Psi \in \mathcal{A}$ is in the class ${\mathcal{K}}_{\Sigma }\left(\mathcal{V}\right)$ of strongly bi-convex functions of order $\mathcal{V}(0<\mathcal{V}\le 1)$ if each of the following conditions are satisfied:

$$\left|arg\left(1+{\displaystyle \frac{\varsigma {\Psi }^{\prime \prime }\left(\varsigma \right)}{{\Psi }^{\prime }\left(\varsigma \right)}}\right)\right|<{\displaystyle \frac{\mathcal{V}\pi }{2}}$$

and

$$\left|arg\left(1+{\displaystyle \frac{w{\Phi }^{\prime \prime }\left(w\right)}{{\Phi }^{\prime }\left(w\right)}}\right)\right|<{\displaystyle \frac{\mathcal{V}\pi }{2}},$$

where $\varsigma ,w\in {\mathcal{U}}_D;0<\mathcal{V}\le 1,$ and $\Phi$ is as given in (4).

For $\Psi \in {\mathcal{S}}_{\Sigma }^{*}\left(\mathcal{V}\right)$ and $\Psi \in {\mathcal{K}}_{\Sigma }\left(\mathcal{V}\right)$, Brannan and Taha [6] found non-sharp estimates on the first two Taylor–Maclaurin coefficients $|c_2|$ and $|c_3|$ (for details see [6,7]). The work of Srivastava et al. [2], which really revamped the study of bi-univalent functions, and the sources referenced therein may provide the interested reader with a brief historical review of functions in $\Psi \in \Sigma$. In a number of successors to [2], including [8,9,10,11,12], bounds were obtained for the first two Taylor–Maclaurin coefficients $|c_2|$ and $|c_3|$ of different subclasses of bi-univalent functions. Actually, the groundbreaking work of Srivastava et al. [2] seems to have successfully resurrected the study of analytic and bi-univalent functions in recent years, especially in light of the very large number of works on the topic.

Since the inception of complex function research, operators have been employed. They have made many known results simpler to use, and they may provide novel conclusions, particularly those pertaining to the convexity and starlikeness of particular functions. Most often, research involving operators leads to the introduction of new classes of analytic functions. As evidenced by the most recent results from papers [8,13], the study of bi-univalent functions using operators is also a popular approach these days. According to the most recent paper [14], there is special interest in obtaining the Fekete–Szegő functional for the newly introduced special classes.

Liouville–Caputo’s Fractional-Order Derivative (FD)

Fractional calculus (FC) is the study of non-integer-order integro-differential operators. Gottfried Wilhelm Leibniz first mentioned this topic in a letter to Guillaume de L’Hospital in 1695. W. Leibniz responded to his inquiry about what would happen if a derivative’s order was half by saying that “it will lead to a paradox, from which one day a useful consequence will be drawn” (for further information, see [15]). According to the literature, the growth of FC [16] is largely dependent on the Riemann–Liouville fractional integral and derivative. In 1984, Srivastava and Owa [17] presented the operator using the previous definitions and their well-known expansions involving fractional derivatives (FDs) and fractional integral (FIs) as

$${\mathcal{R}}^{\mathfrak{w}}:\mathcal{S}\to \mathcal{S}$$

defined by

$${\mathcal{R}}^{\mathfrak{w}}\Psi \left(\varsigma \right)=\Gamma (2-\mathfrak{w}){\varsigma }^{\mathfrak{w}}{\mathcal{I}}_{\varsigma }^{\mathfrak{w}}\Psi \left(\varsigma \right)=\varsigma +\sum _{n=2}^{\infty }\Omega (n,\mathfrak{w})c_n{\varsigma }^n,$$

where

$$\Omega (n,\mathfrak{w})={\displaystyle \frac{\Gamma (n+1)\Gamma (2-\mathfrak{w})}{\Gamma (n+1-\mathfrak{w})}}$$

and $\mathfrak{w}\in \mathbb{R};\mathfrak{w}\ne 2,3,4,\dots$.

We recall the following definitions by Srivastava et al. [17,18].

Definition 1.

In a simply connected region of the ς-plane that encompasses the origin, let $\Psi \in \mathcal{A}$ and the fractional integral (FI) of Ψ of order $\tau (\tau >0)$ be

$${\mathcal{I}}_{\varsigma }^{-\tau }\Psi \left(\varsigma \right)={\displaystyle \frac{1}{\Gamma \left(\tau \right)}}\underset{0}{\overset{\varsigma }{\int }}{\displaystyle \frac{\Psi \left(\chi \right)}{{(\varsigma -\chi )}^{1-\tau }}}d\chi ,\tau >0.$$

(6)

Also, the fractional derivatives (FDs) of Ψ of order $\tau (0\le \tau <1)$ are

$${\mathcal{I}}_{\varsigma }^{\tau }\Psi \left(\varsigma \right)={\displaystyle \frac{1}{\Gamma (1-\tau )}}{\displaystyle \frac{d}{d\varsigma }}\underset{0}{\overset{\varsigma }{\int }}{\displaystyle \frac{\Psi \left(\chi \right)}{{(\varsigma -\chi )}^{\tau }}}d\chi ,0\le \tau <1,$$

(7)

where the multiplicity of ${(\varsigma -\chi )}^{1-\tau }$ and ${(\varsigma -\chi )}^{-\tau }$ is removed by requiring $log(\varsigma -\chi )$ to be real when $\varsigma -\chi >0.$

Definition 2.

The FD of $\Psi \in \mathcal{A}$ of order $m+\tau$ is

$${\mathcal{I}}_{\varsigma }^{m+\tau }\Psi \left(\varsigma \right)={\displaystyle \frac{d^m}{d{\varsigma }^m}}{\mathcal{I}}_{\varsigma }^{\tau }\Psi \left(\varsigma \right),0\le \tau <1;m\in {\mathbb{N}}_0.$$

(8)

Liouville–Caputo’s [19] fractional-order derivative is examined throughout this article with the assumption that

$${\mathcal{I}}^{\tau }\Psi \left(t\right)={\displaystyle \frac{1}{\Gamma (n-\tau )}}\underset{a}{\overset{t}{\int }}{\displaystyle \frac{{\Psi }^{\left(n\right)}\left(\chi \right)}{{(u-\chi )}^{\tau +1-n}}}d\chi$$

(9)

where $n-1<\Re \left(\tau \right)\le n,n\in \mathbb{N},$ and $\tau \in \mathbb{C}$. With no loss of generality, we let the base point a be located at $a=0$ and (initially) assume that the evaluation point t satisfies $t>0$ (later $\varsigma$ will be generalized to an arbitrarily placed point $\varsigma$ in the complex plane).

The modified Liouville-Caputo fractional derivative operator, which was presented and examined by Salah et al. [20], is represented as

$${\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)={\displaystyle \frac{\Gamma (2+\vartheta -\tau )}{\Gamma (\vartheta -\tau )}}{\varsigma }^{\tau -\vartheta }{\int }_0^{\varsigma }{\displaystyle \frac{{\Theta }^{\vartheta }\Psi \left(\chi \right)}{{(\varsigma -\chi )}^{\tau +1-\vartheta }}}d\chi$$

(10)

where $\vartheta \in \mathbb{R}$ and $(\vartheta -1<\tau <\vartheta <2).$ Further, Salah et al. [20] expressed the normalized (the summation form of the above operator) form of (10) in the following manner:

$$\begin{array}{ccc}\hfill {\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)& =& \varsigma +\sum _{n=2}^{\infty }{\displaystyle \frac{\Gamma (2-\tau +\vartheta )\Gamma (2-\vartheta ){(\Gamma (n+1))}^2}{\Gamma (n+1-\vartheta )\Gamma (n+\vartheta +1-\tau )}}{c}_n{\varsigma }^n,\varsigma \in {\mathcal{U}}_D\hfill \\ & =& \varsigma +\sum _{n=2}^{\infty }{\Xi }_n{c}_n{\varsigma }^n,\varsigma \in {\mathcal{U}}_D\hfill \end{array}$$

(11)

where

$$\begin{array}{ccc}\hfill {\Xi }_n& =& {\displaystyle \frac{\Gamma (2+\vartheta -\tau )\Gamma (2-\vartheta ){(\Gamma (n+1))}^2}{\Gamma (n-\vartheta +1)\Gamma (n+\vartheta -\tau +1)}}\hfill \end{array}$$

(12)

We note that

$${\mathfrak{X}}_0^0\Psi \left(\varsigma \right)=\Psi \left(\varsigma \right)and{\mathfrak{X}}_1^1\Psi \left(\varsigma \right)=\varsigma {\Psi }^{\prime }\left(\varsigma \right).$$

We express

$$\begin{array}{ccc}\hfill {\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)& =& \varsigma +{\Xi }_2{c}_2{\varsigma }^2+c_3{\Xi }_3{\varsigma }^3+\dots \varsigma \in {\mathcal{U}}_D\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill {\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)& =& w-{\Xi }_2{c}_2{w}^2-(2c_2^2-c_3){\Xi }_3{w}^3+\dots w\in {\mathcal{U}}_D.\hfill \end{array}$$

(14)

New analytic function subclasses are introduced by applying the ideas of fractional calculus or quantum calculus. As a consequence, one can look into some helpful findings like coefficient estimates, subordination properties, and the Fekete–Szegö problem. This opens up pertinent issues for researchers, like distortion theorems, closure theorems, convolution properties, and radius difficulties. Furthermore, meromorphic and multivalent functions can be included in these findings. Motivated by recent works on bi-univalent functions by Srivastava et al. [2,8], Frasin and Aouf [9], Wanas [21], and Breaz et al. [22] and employing the techniques used earlier (see [10,11,12,13,14,23,24] and also references cited therein), in this paper, we establish two new subclasses $\Sigma$ based on the modified Liouville–Caputo fractional derivative operator ${\mathfrak{X}}_{\tau }^{\vartheta }$ and find $|c_2|$ and $|c_3|$ and the Fekete–Szegö problem for functions in these new subclasses ${\mathcal{S}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\mathcal{V})$ and ${\mathcal{M}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\delta )$.

2. Coefficient Bounds for the Function Class ${\mathcal{S}}_{\Sigma }^{\mathbf{\tau },\mathbf{\vartheta }}(\mathfrak{p},\mathcal{V})$

Definition 3.

Let $\Psi \in \mathcal{A}$ be defined by (1); then $\Psi \in {\mathcal{S}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\mathcal{V})$ if the following criteria are satisfied.

$$\left|arg\left({\displaystyle \frac{\varsigma {\left({\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)\right)}^{\prime }}{(1-\mathfrak{p})\varsigma +\mathfrak{p}{\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)}}\right)\right|<{\displaystyle \frac{\mathcal{V}\pi }{2}},$$

(15)

and

$$\left|arg\left({\displaystyle \frac{w{\left({\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)\right)}^{\prime }}{(1-\mathfrak{p})w+\mathfrak{p}{\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)}}\right)\right|<{\displaystyle \frac{\mathcal{V}\pi }{2}}$$

(16)

where Φ is given by (4) and $0<\mathcal{V}\le 1,0\le \mathfrak{p}\le 1,$ and $\varsigma ,w\in {\mathcal{U}}_D$.

By fixing $\mathfrak{p}=0$ and $\mathfrak{p}=1$, we can define the following new subclasses ${\mathcal{W}}_{\Sigma }^{\tau ,\vartheta }\left(\mathcal{V}\right)$ and ${\mathcal{Y}}_{\Sigma }^{\tau ,\vartheta }\left(\mathcal{V}\right)$, which have not been studied in association with the modified Liouville-Caputo fractional derivative operator.

Remark 1.

Taking $\mathfrak{p}=0$ in the class ${\mathcal{S}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\mathcal{V}),$ we have ${\mathcal{S}}_{\Sigma }^{\tau ,\vartheta }(0,\mathcal{V})\equiv {\mathcal{W}}_{\Sigma }^{\tau ,\vartheta }\left(\mathcal{V}\right)$ and $\Psi \in {\mathcal{W}}_{\Sigma }^{\tau ,\vartheta }\left(\mathcal{V}\right)$ if the following conditions are satisfied:

$$\left|arg\left({\left({\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)\right)}^{\prime }\right)\right|<{\displaystyle \frac{\mathcal{V}\pi }{2}},$$

(17)

and

$$\left|arg\left({\left({\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)\right)}^{\prime }\right)\right|<{\displaystyle \frac{\mathcal{V}\pi }{2}},$$

(18)

where the function Φ is given by (4) and $0<\mathcal{V}\le 1,0\le \mathfrak{p}\le 1,$ and $\varsigma ,w\in {\mathcal{U}}_D$.

Remark 2.

Taking $\mathfrak{p}=1$ in the class ${\mathcal{S}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\mathcal{V}),$ we have ${\mathcal{S}}_{\Sigma }^{\tau ,\vartheta }(1,\mathcal{V})\equiv {\mathcal{Y}}_{\Sigma }^{\tau ,\vartheta }\left(\mathcal{V}\right)$ and $\Psi \in {\mathcal{Y}}_{\Sigma }^{\tau ,\vartheta }\left(\mathcal{V}\right)$ if the following conditions are satisfied:

$$\left|arg\left({\displaystyle \frac{\varsigma {\left({\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)\right)}^{\prime }}{{\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)}}\right)\right|<{\displaystyle \frac{\mathcal{V}\pi }{2}},$$

(19)

and

$$\left|arg\left({\displaystyle \frac{w{\left({\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)\right)}^{\prime }}{{\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)}}\right)\right|<{\displaystyle \frac{\mathcal{V}\pi }{2}},$$

(20)

where the function Φ is given by (4) and $0<\mathcal{V}\le 1,0\le \mathfrak{p}\le 1,$ and $\varsigma ,w\in {\mathcal{U}}_D.$

We note that for $\vartheta =\tau =\mathfrak{p}=0,$ the class ${\mathcal{S}}_{\Sigma }^{k,\mathfrak{p}}\left(\alpha \right)\equiv {\mathcal{H}}_{\Sigma }^{\alpha }$, introduced and studied by Srivastava et al. [2]. Putting $\vartheta =\tau =0$ and $\mathfrak{p}=1$, the class ${\mathcal{S}}_{\Sigma }^{0,1}\left(\mathcal{V}\right)\equiv {\mathcal{S}}_{\Sigma }^{*}\left(\mathcal{V}\right)$. When $\vartheta =\tau =\mathfrak{p}=1$, the class ${\mathcal{S}}_{\Sigma }^{1,1}\left(\mathcal{V}\right)\equiv {\mathcal{K}}_{\Sigma }\left(\mathcal{V}\right)$. For $\vartheta =\tau =0$, the class was introduced and studied in [23].

Now we find $|c_2|$ and $|c_3|$ for $\Psi \in {\mathcal{S}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\mathcal{V})$; we recall the following result.

Lemma 1

([25]). If ${\rm Y}\in \wp$ expressed as

$${\rm Y}\left(\varsigma \right)=1+{\upsilon }_1\varsigma +{\upsilon }_2{\varsigma }^2+\dots \varsigma \in {\mathcal{U}}_D$$

then

$$|{\upsilon }_n|\le 2;(n\ge 1)$$

where ℘ is the family of all analytic functions in ${\mathcal{U}}_D$ for which $\Re \{{\rm Y}(\varsigma \left)\right\}>0.$

Theorem 1.

For $0<\mathcal{V}\le 1,0\le \mathfrak{p}\le 1,$ let Ψ be given by (1), and if $\Psi \in {\mathcal{S}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\mathcal{V}),$ then

$$|c_2|\le {\displaystyle \frac{2\mathcal{V}}{\sqrt{\left[4(1-\mathcal{V})+\mathfrak{p}(\mathfrak{p}-4)\right]{\Xi }_2^2+2\mathcal{V}(3-\mathfrak{p}){\Xi }_3}}},$$

(21)

and

$$|c_3|\le {\displaystyle \frac{2\mathcal{V}}{(3-\mathfrak{p}){\Xi }_3}}+{\displaystyle \frac{4{\mathcal{V}}^2}{\left[4(1-\mathcal{V})+\mathfrak{p}(\mathfrak{p}-4)\right]{\Xi }_2^2+2\mathcal{V}(3-\mathfrak{p}){\Xi }_3}}.$$

(22)

where ${\Xi }_n(n=2,3)$ are given by (12).

Proof. 

It is derived from (15) and (16) that

$${\displaystyle \frac{\varsigma {\left({\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)\right)}^{\prime }}{(1-\mathfrak{p})\varsigma +\mathfrak{p}{\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)}}={\left[p\left(\varsigma \right)\right]}^{\mathcal{V}}$$

(23)

and

$${\displaystyle \frac{w{\left({\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)\right)}^{\prime }}{(1-\mathfrak{p})w+\mathfrak{p}{\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)}}.={\left[q\left(w\right)\right]}^{\mathcal{V}}$$

(24)

$p,q\in \wp$ and are assumed as

$$p\left(\varsigma \right)=1+p_1\varsigma +p_2{\varsigma }^2+\dots$$

(25)

and

$$q\left(w\right)=1+q_{1}w+q_2{w}^2+\dots .$$

(26)

Thus,

$${\left[p\left(\varsigma \right)\right]}^{\mathcal{V}}=1+\mathcal{V}{p}_1\varsigma +{\displaystyle \frac{1}{2}}\left[\mathcal{V}(\mathcal{V}-1)p_1^2+2\mathcal{V}{p}_2\right]{\varsigma }^2+\dots$$

and

$${\left[q\left(w\right)\right]}^{\mathcal{V}}=1+\mathcal{V}{q}_{1}w+{\displaystyle \frac{1}{2}}\left[\mathcal{V}(\mathcal{V}-1)q_1^2+2\mathcal{V}{q}_2\right]w^2+\dots$$

Since $\Psi \in \mathcal{A}$, as given in (1), we get

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\varsigma {\left({\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)\right)}^{\prime }}{(1-\mathfrak{p})\varsigma +\mathfrak{p}{\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)}}& =& 1+(2-\mathfrak{p}){\Xi }_2{c}_2\varsigma +[\mathfrak{p}(\mathfrak{p}-2){\Xi }_2^2{a}_2^2+(3-\mathfrak{p}){\Xi }_3{c}_3]{\varsigma }^2+\dots ,\hfill \end{array}$$

(27)

Similarly by using $\Phi$ as given by (4), we get

$$\begin{array}{ccc}\hfill {\displaystyle \frac{w{\left({\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)\right)}^{\prime }}{(1-\mathfrak{p})w+\mathfrak{p}{\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)}}& =& 1-(2-\mathfrak{p}){\Xi }_2{c}_{2}w\hfill \\ & +& [[({\mathfrak{p}}^2-2\mathfrak{p}){\Xi }_2^2+(6-2\mathfrak{p}){\Xi }_3]c_2^2-(3-\mathfrak{p}){\Xi }_3{c}_3]w^2+\dots \hfill \end{array}$$

(28)

Substituting the above equations in (23) and (24) and equating the corresponding coefficients of $\varsigma ,w,{\varsigma }^2,w^2,$ we obtain

$$(2-\mathfrak{p}){\Xi }_2{c}_2=\mathcal{V}{p}_1$$

(29)

$$({\mathfrak{p}}^2-2\mathfrak{p}){\Xi }_2^2{c}_2^2+(3-\mathfrak{p}){\Xi }_3{c}_3={\displaystyle \frac{1}{2}}\left[\mathcal{V}(\mathcal{V}-1)p_1^2+2\mathcal{V}{p}_2\right]$$

(30)

$$-(2-\mathfrak{p}){\Xi }_2{c}_2=\mathcal{V}{q}_1$$

(31)

and

$$[({\mathfrak{p}}^2-2\mathfrak{p}){\Xi }_2^2+(6-2\mathfrak{p}){\Xi }_3]c_2^2-(3-\mathfrak{p}){\Xi }_3{c}_3={\displaystyle \frac{1}{2}}\left[\mathcal{V}(\mathcal{V}-1)q_1^2+2\mathcal{V}{q}_2\right].$$

(32)

From (29) and (31), we have

$$p_1=-q_1$$

(33)

and

$$2{(2-\mathfrak{p})}^2{\Xi }_2^2{c}_2^2={\mathcal{V}}^2(p_1^2+q_1^2).$$

(34)

From (30), (32), and (34), we obtain

$$c_2^2={\displaystyle \frac{{\mathcal{V}}^2(p_2+q_2)}{\left[\mathcal{V}({\mathfrak{p}}^2-4\mathfrak{p})-(\mathcal{V}-1){(2-\mathfrak{p})}^2\right]{\Xi }_2^2+\mathcal{V}(6-2\mathfrak{p}){\Xi }_3}}.$$

(35)

By Lemma 1, we have $|p_2|\le 2$ and $|q_2|\le 2;$ thus we immediately get

$$|c_2|\le {\displaystyle \frac{2\mathcal{V}}{\sqrt{\left[4(1-\mathcal{V})+\mathfrak{p}(\mathfrak{p}-4)\right]{\Xi }_2^2+2\mathcal{V}(3-\mathfrak{p}){\Xi }_3}}}.$$

This yields $|c_2|$, as asserted in (21).

Next, by subtracting (32) from (30), we get

$$(6-2\mathfrak{p}){\Xi }_3{c}_3+2(3-\mathfrak{p}){\Xi }_3{c}_2^2=\mathcal{V}(p_2-q_2)+{\displaystyle \frac{\mathcal{V}(\mathcal{V}-1)}{2}}(p_1^2-q_1^2).$$

(36)

It follows from (33), (34), and (36) that

$$\begin{array}{ccc}\hfill c_3& =& {\displaystyle \frac{\mathcal{V}(p_2-q_2)}{{\Xi }_3(6-2\mathfrak{p})}}+c_2^2\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill c_3& =& {\displaystyle \frac{\mathcal{V}(p_2-q_2)}{{\Xi }_3(6-2\mathfrak{p})}}+{\displaystyle \frac{{\mathcal{V}}^2(p_2+q_2)}{\left[\mathcal{V}({\mathfrak{p}}^2-4\mathfrak{p})-(\mathcal{V}-1){(2-\mathfrak{p})}^2\right]{\Xi }_2^2+\mathcal{V}(6-2\mathfrak{p}){\Xi }_3}}.\hfill \end{array}$$

(38)

By applying Lemma 1 for the coefficients $p_1,$$p_2,$$q_1$, and $q_2,$ we get

$$\begin{array}{ccc}\hfill |c_3|& \le & {\displaystyle \frac{2\mathcal{V}}{(3-\mathfrak{p}){\Xi }_3}}+{\displaystyle \frac{4{\mathcal{V}}^2}{\left[4(1-\mathcal{V})+\mathfrak{p}(\mathfrak{p}-4)\right]{\Xi }_2^2+\mathcal{V}(6-2\mathfrak{p}){\Xi }_3}}\hfill \\ & =& {\displaystyle \frac{2\mathcal{V}}{(3-\mathfrak{p}){\Xi }_3}}+{\displaystyle \frac{4{\mathcal{V}}^2}{\left[4(1-\mathcal{V})+\mathfrak{p}(\mathfrak{p}-4)\right]{\Xi }_2^2+2\mathcal{V}(3-\mathfrak{p}){\Xi }_3}}.\hfill \end{array}$$

This completes the proof of Theorem 1. □

3. Initial Bounds for the Function Class ${\mathcal{M}}_{\Sigma }^{\mathbf{\tau },\mathbf{\vartheta }}(\mathfrak{p},\mathbf{\delta })$

Definition 4.

Let $\Psi \in \mathcal{A}$ be given by (1); then $\Psi \in {\mathcal{M}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\delta )$ if the following criteria are satisfied:

$$\Re \left({\displaystyle \frac{\varsigma {\left({\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)\right)}^{\prime }}{(1-\mathfrak{p})\varsigma +\mathfrak{p}{\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)}}\right)>\delta ,$$

(39)

and

$$\Re \left({\displaystyle \frac{w{\left({\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)\right)}^{\prime }}{(1-\mathfrak{p})w+\mathfrak{p}{\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)}}\right)>\delta ,$$

(40)

where Φ is given by (4) and $0\le \mathfrak{p}\le 1,0\le \delta <1,$ and $\varsigma \in {\mathcal{U}}_D$.

By fixing $\mathfrak{p}=0$ and $\mathfrak{p}=1$, we can define the following new subclasses ${\mathcal{M}}_{\Sigma }^{\tau }\left(\delta \right)$ and ${\mathcal{M}}_{\Sigma }^{\tau ,*}\left(\delta \right)$, which have not been studied in association with the modified Liouville-Caputo fractional derivative operator

Remark 3.

By fixing $\mathfrak{p}=0$ in the class ${\mathcal{M}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\delta ),$ we have ${\mathcal{M}}_{\Sigma }^{\tau ,\vartheta }(0,\delta )\equiv {\mathcal{M}}_{\Sigma }^{\tau }\left(\delta \right)$ and $\Psi \in {\mathcal{M}}_{\Sigma }^{\tau }\left(\delta \right)$ if the following criteria hold:

$$\Re \left({\left({\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)\right)}^{\prime }\right)>\delta ,and\Re \left({\left({\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)\right)}^{\prime }\right)>\delta ,$$

(41)

where $0\le \delta <1,\varsigma ,w\in {\mathcal{U}}_D$, and Φ is given by (4).

Remark 4.

Taking $\mathfrak{p}=1$ in the class ${\mathcal{M}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\delta ),$ we have ${\mathcal{M}}_{\Sigma }^{\tau ,\vartheta }(1,\delta )\equiv {\mathcal{M}}_{\Sigma }^{\tau ,*}\left(\delta \right)$ and $\Psi \in {\mathcal{M}}_{\Sigma }^{\tau ,*}\left(\delta \right)$ if the conditions below hold:

$$\Re \left({\displaystyle \frac{\varsigma {\left({\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)\right)}^{\prime }}{{\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)}}\right)>\delta and\Re \left({\displaystyle \frac{w{\left({\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)\right)}^{\prime }}{{\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)}}\right)>\delta$$

(42)

where Φ is given by (4) and $0\le \delta <1$ and $\varsigma ,w\in {\mathcal{U}}_D$.

In the following theorem we obtain the estimates on the coefficients $|c_2|$ and $|c_3|$ for $\Psi \in {\mathcal{M}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\delta ).$

Theorem 2.

If Ψ is as in (1) and $\Psi \in {\mathcal{M}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\delta ),$ then

$$|c_2|\le \sqrt{{\displaystyle \frac{2(1-\delta )}{({\mathfrak{p}}^2-2){\Xi }_2^2+(3-\mathfrak{p}){\Xi }_3}}},$$

(43)

and

$$|c_3|\le {\displaystyle \frac{2(1-\delta )}{(3-\mathfrak{p}){\Xi }_3}}+{\displaystyle \frac{2(1-\delta )}{({\mathfrak{p}}^2-2){\Xi }_2^2+(3-\mathfrak{p}){\Xi }_3}},$$

(44)

where ${\Xi }_n(n=2,3)$ are given by (12).

Proof. 

From (39) and (40) and for $p,q\in \wp$ we have

$${\displaystyle \frac{\varsigma {\left({\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)\right)}^{\prime }}{(1-\mathfrak{p})\varsigma +\mathfrak{p}{\mathfrak{X}}_{\tau }^{\vartheta }\Psi \left(\varsigma \right)}}=\delta +(1-\delta )p\left(\varsigma \right)$$

(45)

and

$${\displaystyle \frac{w{\left({\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)\right)}^{\prime }}{(1-\mathfrak{p})w+\mathfrak{p}{\mathfrak{X}}_{\tau }^{\vartheta }\Phi \left(w\right)}}=\delta +(1-\delta )q\left(w\right),$$

(46)

where $p\left(z\right)$ and $q\left(w\right)$ have the forms (25) and (26), respectively. Now using (25)–(28) and equating coefficients in (45) and (46), we get

$$(2-\mathfrak{p}){\Xi }_2{c}_2=(1-\delta )p_1$$

(47)

$$({\mathfrak{p}}^2-2\mathfrak{p}){\Xi }_2^2{c}_2^2+(3-\mathfrak{p}){\Xi }_3{c}_3=(1-\delta )p_2$$

(48)

$$-(2-\mathfrak{p}){\Xi }_2{c}_2=(1-\delta )q_1$$

(49)

and

$$[({\mathfrak{p}}^2-2\mathfrak{p}){\Xi }_2^2+(6-2\mathfrak{p}){\Xi }_3]c_2^2-(3-\mathfrak{p}){\Xi }_3{c}_3=(1-\delta )q_2.$$

(50)

From (47) and (49), we get

$$p_1=-q_1.$$

Also, adding (48) and (50), we obtain

$$c_2^2={\displaystyle \frac{(1-\delta )(p_2+q_2)}{2\left[({\mathfrak{p}}^2-2){\Xi }_2^2+(3-\mathfrak{p}){\Xi }_3\right]}}.$$

(51)

When we apply Lemma 1 to the coefficients $p_2$ and $q_2$, we get

$$|c_2|\le \sqrt{{\displaystyle \frac{2(1-\delta )}{({\mathfrak{p}}^2-2){\Xi }_2^2+(3-\mathfrak{p}){\Xi }_3}}}.$$

As stated in (43), this provides the bound on $|c_2|$. The bound on $|c_3|$ may then be found by subtracting (50) from (48).

$$(6-2\mathfrak{p}){\Xi }_3{c}_3-(6-2\mathfrak{p}){\Xi }_3{c}_2^2=(1-\delta )(p_2-q_2).$$

(52)

$$c_3=c_2^2+{\displaystyle \frac{(1-\delta )(p_2-q_2)}{(6-2\mathfrak{p}){\Xi }_3}}.$$

(53)

Lemma 1 is applied again for the coefficients $p_1,$$p_2,$$q_1$, and $q_2,$ and we easily obtain

$$|c_3|\le {\displaystyle \frac{2(1-\delta )}{(3-\mathfrak{p}){\Xi }_3}}+{\displaystyle \frac{2(1-\delta )}{({\mathfrak{p}}^2-2){\Xi }_2^2+(3-\mathfrak{p}){\Xi }_3}}$$

This concludes the proof of Theorem 2. □

4. Fekete–Szegö Inequality

In this section, we apply the techniques presented in [10,11,22] to prove Fekete–Szegö inequalities for $\Psi \in {\mathcal{S}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\mathcal{V})$ and $\Psi \in {\mathcal{M}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\delta )$. We use the following lemmas proved by Zaprawa in [26].

Lemma 2

([26]). Let $k,l\in \mathbb{R}$ and ${\varsigma }_1,{\varsigma }_2\in \mathbb{C}.$ If $\left|{\varsigma }_1\right|<R$ and $\left|{\varsigma }_2\right|<R$, then

$$\left|(k+l){\varsigma }_1+(k-l){\varsigma }_2\right|\le \left\{\begin{array}{cc}2\left|k\right|R,& \left|k\right|\ge \left|l\right|,\\ 2\left|l\right|R,& \left|k\right|\le \left|l\right|.\end{array}\right.$$

Theorem 3.

Let Ψ be given by (1). If $\Psi \in {\mathcal{S}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\mathcal{V}),$$0<\mathcal{V}\le 1,0\le \mathfrak{p}\le 1$, and $\varrho \in \mathbb{R},$ then

$$|a_3-\varrho a_2^2\mid \le \left\{\begin{array}{cc}{\displaystyle \frac{2\mathcal{V}}{(3-\mathfrak{p}){\Xi }_3}},\hfill & 0\le \mid \varphi \left(\varrho \right)\mid \le {\displaystyle \frac{\mathcal{V}}{2(3-\mathfrak{p}){\Xi }_3}}\hfill \\ 4\left|\varphi \right(\varrho \left)\right|,\hfill & \left|\varphi \left(\varrho \right)\right|\ge {\displaystyle \frac{\mathcal{V}}{2(3-\mathfrak{p}){\Xi }_3}}\hfill \end{array}\right.$$

where

$$\varphi \left(\varrho \right)={\displaystyle \frac{(1-\varrho ){\mathcal{V}}^2}{\left[4(1-\mathcal{V})+\mathfrak{p}(\mathfrak{p}-4)\right]{\Xi }_2^2+2\mathcal{V}(3-\mathfrak{p}){\Xi }_3}}.$$

Proof. 

From (35) and (37)

$$\begin{array}{ccc}\hfill a_3-\varrho a_2^2& =& {\displaystyle \frac{\mathcal{V}(p_2-q_2)}{{\Xi }_3(6-2\mathfrak{p})}}+{\displaystyle \frac{(1-\varrho ){\mathcal{V}}^2(p_2+q_2)}{\left[4(1-\mathcal{V})+\mathfrak{p}(\mathfrak{p}-4)\right]{\Xi }_2^2+\mathcal{V}(6-2\mathfrak{p}){\Xi }_3}}\hfill \\ & =& \left[\varphi \left(\varrho \right)+{\displaystyle \frac{\mathcal{V}}{2(3-\mathfrak{p}){\Xi }_3}}\right]p_2+\left[\varphi \left(\varrho \right)-{\displaystyle \frac{\mathcal{V}}{2(3-\mathfrak{p}){\Xi }_3}}\right]q_2\hfill \end{array}$$

where

$$\varphi \left(\varrho \right)={\displaystyle \frac{(1-\varrho ){\mathcal{V}}^2}{\left[4(1-\mathcal{V})+\mathfrak{p}(\mathfrak{p}-4)\right]{\Xi }_2^2+2\mathcal{V}(3-\mathfrak{p}){\Xi }_3}}.$$

(54)

Thus by applying Lemmas 1 and 2, we get

$$|a_3-\varrho a_2^2\mid \le \left\{\begin{array}{cc}{\displaystyle \frac{2\mathcal{V}}{(3-\mathfrak{p}){\Xi }_3}},\hfill & 0\le \mid \varphi \left(\varrho \right)\mid \le {\displaystyle \frac{\mathcal{V}}{2(3-\mathfrak{p}){\Xi }_3}}\hfill \\ 4\left|\varphi \right(\varrho \left)\right|,\hfill & \left|\varphi \left(\varrho \right)\right|\ge {\displaystyle \frac{\mathcal{V}}{2(3-\mathfrak{p}){\Xi }_3}}.\hfill \end{array}\right.$$

In particular by fixing $\varrho =1,$ we get

$$|a_3-a_2^2\mid \le {\displaystyle \frac{2\mathcal{V}}{(3-\mathfrak{p}){\Xi }_3}}.$$

□

Theorem 4.

Let Ψ be given by (1). If $\Psi \in {\mathcal{M}}_{\Sigma }^{\tau ,\vartheta }(\mathfrak{p},\delta )$ and $\aleph \in \mathbb{R}$, then

$$|a_3-\aleph a_2^2\mid \le \left\{\begin{array}{cc}{\displaystyle \frac{1-\delta }{(3-\mathfrak{p}){\Xi }_3}},\hfill & 0\le \mid \varphi (\aleph )\mid \le {\displaystyle \frac{1-\delta }{(3-\mathfrak{p}){\Xi }_3}}\hfill \\ 4\left|\varphi \right(\aleph \left)\right|,\hfill & \left|\varphi (\aleph )\right|\ge {\displaystyle \frac{1-\delta }{(3-\mathfrak{p}){\Xi }_3}}\hfill \end{array}\right.$$

where

$$\varphi (\aleph )={\displaystyle \frac{(1-\aleph )(1-\delta )}{2\left[({\mathfrak{p}}^2-2){\Xi }_2^2+(3-\mathfrak{p}){\Xi }_3\right]}}.$$

Proof. 

From (51) and (53)

$$\begin{array}{ccc}\hfill a_3-\aleph a_2^2& =& {\displaystyle \frac{(1-\delta )(p_2-q_2)}{(6-2\mathfrak{p}){\Xi }_3}}+{\displaystyle \frac{(1-\aleph )(1-\delta )(p_2+q_2)}{2\left[({\mathfrak{p}}^2-2){\Xi }_2^2+(3-\mathfrak{p}){\Xi }_3\right]}}\hfill \\ & =& \left[\varphi (\aleph )+{\displaystyle \frac{1-\delta }{2(3-\mathfrak{p}){\Xi }_3}}\right]p_2+\left[\varphi (\aleph )-{\displaystyle \frac{1-\delta }{2(3-\mathfrak{p}){\Xi }_3}}\right]q_2\hfill \end{array}$$

where

$$\varphi (\aleph )={\displaystyle \frac{(1-\aleph )(1-\delta )}{2\left[({\mathfrak{p}}^2-2){\Xi }_2^2+(3-\mathfrak{p}){\Xi }_3\right]}}.$$

Thus by applying Lemmas 1 and 2, we get

$$|a_3-\aleph a_2^2\mid \le \left\{\begin{array}{cc}{\displaystyle \frac{1-\delta }{(3-\mathfrak{p}){\Xi }_3}},\hfill & 0\le \mid \varphi (\aleph )\mid \le {\displaystyle \frac{1-\delta }{(3-\mathfrak{p}){\Xi }_3}}\hfill \\ 4\left|\varphi \right(\aleph \left)\right|,\hfill & \left|\varphi (\aleph )\right|\ge {\displaystyle \frac{1-\delta }{(3-\mathfrak{p}){\Xi }_3}}.\hfill \end{array}\right.$$

In particular by fixing $\aleph =1,$ we get

$$\mid a_3-a_2^2\mid \le {\displaystyle \frac{1-\delta }{(3-\mathfrak{p}){\Xi }_3}}.$$

□

5. Corollaries and Consequences of the Estimates

According to Theorem 1, if we fix $\mathfrak{p}=0$ and $\mathfrak{p}=1$ we derive the subsequent corollaries, which give the initial Taylor coefficient estimates for the function classes stated in Remarks 1 to 4.

Corollary 1

([2]). If $\Psi \in \mathcal{A}$ is as in (1) and $\Psi \in {\mathcal{W}}_{\Sigma }^{\tau ,\vartheta }\left(\mathcal{V}\right)$, then

$$|c_2|\le {\displaystyle \frac{2\mathcal{V}}{\sqrt{6\mathcal{V}{\Xi }_3-4(\mathcal{V}-1){\Xi }_2^2}}},$$

and

$$|c_3|\le {\displaystyle \frac{2\mathcal{V}}{3{\Xi }_3}}+{\displaystyle \frac{4{\mathcal{V}}^2}{\left[6\mathcal{V}{\Xi }_3-4(1-\mathcal{V})\right]{\Xi }_2^2}},$$

where $0<\mathcal{V}\le 1,$ and ${\Xi }_n(n=2,3)$ are given by (12).

Fixing $\mathfrak{p}=1$ in Theorem 1, we state the following.

Corollary 2.

If $\Psi \in \mathcal{A}$ is as in (1) and $\Psi \in {\mathcal{Y}}_{\Sigma }^{\tau ,\vartheta }\left(\mathcal{V}\right)$, then

$$|c_2|\le {\displaystyle \frac{2\mathcal{V}}{\sqrt{4\mathcal{V}{\Xi }_3+\left[1-4\mathcal{V}\right]{\Xi }_2^2}}},$$

and

$$|c_3|\le {\displaystyle \frac{2\mathcal{V}}{2{\Xi }_3}}+{\displaystyle \frac{4{\mathcal{V}}^2}{\left[1-4\mathcal{V}\right]{\Xi }_2^2+4\mathcal{V}{\Xi }_3}},$$

where $0<\mathcal{V}\le 1,$ and ${\Xi }_n(n=2,3)$ are given by (12).

Fixing $\mathfrak{p}=0$ and $\mathfrak{p}=1$ in Theorem 2, we state the following corollaries, respectively.

Corollary 3.

Let Ψ be given by (1), and if $\Psi \in {\mathcal{M}}_{\Sigma }^{\tau ,\vartheta }\left(\delta \right),$ then

$$|c_2|\le \sqrt{{\displaystyle \frac{2(1-\delta )}{3{\Xi }_3-2{\Xi }_2^2}}},$$

and

$$|c_3|\le {\displaystyle \frac{2(1-\delta )}{3{\Xi }_3}}+{\displaystyle \frac{2(1-\delta )}{3{\Xi }_3-2{\Xi }_2^2}},$$

where $0\le \delta <1$ and ${\Xi }_n(n=2,3)$ are given by (12).

Corollary 4.

Let Ψ be given by (1), and if $\Psi \in {\mathcal{M}}_{\Sigma }^{\tau ,*}\left(\delta \right),$ then

$$|c_2|\le \sqrt{{\displaystyle \frac{2(1-\delta )}{2{\Xi }_3-{\Xi }_2^2}}},$$

and

$$|c_3|\le {\displaystyle \frac{1-\delta }{{\Xi }_3}}+{\displaystyle \frac{2(1-\delta )}{2{\Xi }_3-{\Xi }_2^2}},$$

where $0\le \delta <1$ and ${\Xi }_n(n=2,3)$ are given by (12).

Remark 5.

Fixing $\mathfrak{p}=0$ and $\mathfrak{p}=1$ in Theorems 3 and 4, one can easily deduce the Fekete–Szegö inequality for the function classes defined in Remarks 1 to 4. Additionally ${\mathfrak{X}}_0^0\Psi \left(\varsigma \right)=\Psi \left(\varsigma \right)$ and ${\mathfrak{X}}_1^1\Psi \left(\varsigma \right)=\varsigma {\Psi }^{\prime }\left(\varsigma \right)$ by suitably assuming the above results confirm the results studied earlier in the literature (see [4,6,7]). We let interested readers perform this exercise.

6. Conclusions

For each of these bi-univalent function classes given in Definitions 3 and 4 and the subclasses stated in Remarks 1–4, we have calculated the initial Taylor–Maclaurin coefficients $|c_2|$ and $|c_3|$ as well as estimates for the Fekete–Szegö functional. The operator utilized to define the new classes for which coefficient estimates are derived illustrates another facet of the originality of the findings in this study. In a future study, the exploration of upper bounds for the Zaclman conjecture and the investigation of Hankel determinants of orders two and three within the aforementioned subclasses indicate potential for new paths of research and exploration by subordinating with the Limaçon domain and leaf-like domain [14,24]. This resurgence has led to a deeper understanding of their properties and applications, inspiring further research into related function classes. As a result, mathematicians are now exploring new avenues to refine existing bounds and establish novel results that could enhance the field even further.