Study of quantum calculus for a new subclass of bi-univalent functions associated with the cardioid domain

Abstract

In this article, we make use of the concepts of subordination and the q-calculus theory to analyze a new class of analytic bi-univalent functions associated to the cardioid domain. Our main focus is to derive a sharp inequality for a newly defined class of analytic and bi-univalent functions in the open unit disc $\mathcal{U}$. We explore the bounds of initial coefficients, Fekete-Szegö type problems, and coefficient inequalities for newly established families. In addition, we explore some recent findings for the bi-univalent function and inverse function. Additionally, a few well-known results are mentioned to help make links between earlier and current results.

1. Introduction and definitions

Let $\mathcal{U}=\{\chi \in \mathbb{C}:\left|\chi \right|<1\}$ is the open unit disc and $\mathcal{A}$ contains a set of all analytic functions (AFs) $\beta (\chi )$, and normalized by

$$\beta \left(0\right)=0\text{ and }{\beta }^{\prime }\left(0\right)=1.$$

This series form of $\beta \in \mathcal{A}$ is given as:

$$\beta (\chi )=\chi +\sum _{n=2}^{\infty }{t}_n{\chi }^n.$$ (1)

The letter $\mathcal{S}\subset \mathcal{A}$ represents univalent functions.

In 1851, researchers began to study the functions theory formally and this area originally came to light as a promising area of research when Bieberbach [1] investigated the conjecture in 1916. Numerous eminent researchers sought to either support or contradict the Bieberbach theory between 1916 and 1985, but in 1985, De Branges [2] proved this idea. It is important to understand the theory of analytic functions (AFs) and univalent functions (UFs) and how these ideas evaluate the growth of functions within the boundaries of their designated domains. This consists the representation of Taylor coefficients, and the corresponding functional inequalities. The significant and helpful Fekete-Szegö inequality discovered by Fekete and Szegö [3] in 1933. The coefficients of UFs are the subject of a mathematical inequality known as Fekete and Szegö, which is linked to the Bieberbach conjecture. It has been shown that there are several effects of maximization of the non-linear functional $|t_3-\mu t_2^2|$. This sort of issue is referred to as a sharp Fekete-Szegö problem (FSP) (for more detail see [3]) and is presented as follows:

$$|t_3-\mu t_2^2|\le \left\{\begin{array}{ccc}\hfill 3-4\mu \hfill & \hfill \text{ if }\hfill & \hfill \mu \le 0\hfill \\ \hfill 1+2\exp \left(\frac{2\mu }{\mu -1}\right)\hfill & \hfill \text{ if }\hfill & \hfill 0\le \mu <1\hfill \\ \hfill 4\mu -3\hfill & \hfill \text{ if }\hfill & \hfill \mu \ge 1.\hfill \end{array}\right\}$$

Let two analytic functions ${\beta }_1$ and ${\beta }_2$ and there subordination form are ${\beta }_1(\chi )\prec {\beta }_2(\chi )$, $\chi \in \mathcal{U}$. Suppose a Schwarz function $u_0$ meets the following conditions $u_0(0)=0$ and $|u_0(\chi )|<1$, thus ${\beta }_1(\chi )={\beta }_2(u_0(\chi ))$, $\chi \in \mathcal{U}$. Note that, if ${\beta }_2$ is univalent in $\mathcal{U}$, then we have

$${\beta }_1(0)={\beta }_2(0)\text{and}{\beta }_1(\mathcal{U})\subset {\beta }_2(\mathcal{U}).$$

The widely used class of starlike functions, denoted as ${\mathcal{S}}^{⁎}$, is described as:

$$\beta \in {\mathcal{S}}^{⁎}\iff \mathit{Re}\left(\frac{\chi {\beta }^{{}^{\prime }}(\chi )}{\beta (\chi )}\right)>0,\chi \in \mathcal{U}$$

and it can be expressed as follows in terms of subordination:

$${\mathcal{S}}^{⁎}=\{\beta \in \mathcal{A}:\frac{\chi {\beta }^{{}^{\prime }}(\chi )}{\beta (\chi )}\prec \frac{1+\chi }{1-\chi }\}.$$

The familiar class of convex functions, denoted as $\mathcal{C}$ is described as:

$$\beta \in \mathcal{C}\iff \mathit{Re}(1+\frac{\chi {\beta }^{{}^{″}}(\chi )}{{\beta }^{{}^{\prime }}(\chi )})>0,\chi \in \mathcal{U}$$

and it can be expressed as follows in terms of subordination:

$$\mathcal{C}=\{\beta \in \mathcal{A}:1+\frac{\chi {\beta }^{{}^{″}}(\chi )}{{\beta }^{{}^{\prime }}(\chi )}\prec \frac{1+\chi }{1-\chi }\}.$$

Let $\phi \left(\chi \right)$ be the function whose real part is positive with $\phi (0)=1$, and ${\phi }^{{}^{\prime }}(0)>0$.

Ma and Minda [4] introduced a new direction by using the concepts of subordination along with the function $\phi \left(\chi \right)$ to study the class ${\mathcal{S}}^{⁎}$ and the class $\mathcal{C}$. They defined ${\mathcal{S}}^{⁎}\left(\phi \right)$ and $\mathcal{C}\left(\phi \right)$ in the following way:

$$\beta \in {\mathcal{S}}^{⁎}\left(\phi \right)⟺\frac{\chi {\beta }^{{}^{\prime }}(\chi )}{\beta (\chi )}\prec \phi \left(\chi \right)$$

and

$$\beta \in \mathcal{C}\left(\phi \right)⟺1+\frac{\chi {\beta }^{{}^{{}^{″}}}(\chi )}{{\beta }^{{}^{\prime }}(\chi )}\prec \phi \left(\chi \right).$$

Numerous new classes of analytic functions were explored in recent study as specific instances of the class ${\mathcal{S}}^{⁎}\left(\phi \right)$. As a case study, class ${\mathcal{S}}_L^{⁎}$ was examined by Sokól and Stankiewicz in [5], the class ${\mathcal{S}}^{⁎}(P,W)$ defined in [6]. The classes ${\mathcal{S}}_{\sin }^{⁎}$, ${\mathcal{S}}_{\tan }^{⁎}$, and ${\mathcal{S}}^{⁎}\left(e^{\chi }\right)$ investigated by Cho et al. in [7], [8], [9], respectively.

Lewin [10] looked into the (Σ) class of bi-univalent functions and established that $\left|t_2\right|<1.5$. Lewin finding was generalized by Brannan and Clunie [11] to $\left|t_2\right|\le \sqrt{2}$, and subsequently, Netanyahu [12] demonstrated that $\left|t_2\right|\le \frac{4}{3}$. Bi-convex and bi-starlike functions were actually suggested by Brannan and Taha [13], where as Tan [14] discovered some preliminary coefficient estimates for a class. The study of subfamilies Σ has been a popular research topic for the past ten years. Finding the initial coefficient bounds for particular subfamilies generally attracted attention of Σ. Srivastava et al. [15] new revival of the study of coefficient problems regarding bi-univalent functions is remarkable. Two fascinating subfamilies of the function family Σ were introduced in 2010 by Srivastava et al. [15]. For functions in these subfamilies, he discovered bounds $\left|t_2\right|$ and $\left|t_3\right|$. Frasin and Aouf [16] started to found the $\left|t_2\right|$ and $\left|t_3\right|$ of the functions that are members of two new subclasses of the function class Σ. The study of functions from the class that are related to certain polynomials, such as the Lucas, Horadam, Fibonacci, Legendrae, Chebyshev and Gegenbauer polynomials, is the main focus of current research. Serivastava et al. [17] proposed a new subclass of bi-univalent functions by making use of the Horadam polynomials, while in [18], Altınkaya and Yalçin examined the Chebyshev polynomial coefficient issue of a few subclasses of UFs. See the following articles, for several special subfamilies connected with any of the aforementioned polynomials, as shown in [19], [20], [21], [22].

For each $\beta \in \mathcal{S}$, its inverse is described as:

$${\beta }^{-1}(\beta (\chi ))=\chi ,\chi \in \mathcal{U}$$

and

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

The ${\beta }^{-1}$ have the following series form:

$${\beta }^{-1}(w)=w+D_2{w}^2+D_3{w}^3+D_4{w}^4...,$$ (2)

where

$$D_2=-t_2,\text{and}{D}_3=2t_2^2-t_3.$$ (3)

The functions ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ are the examples of the family Σ which are given as follows:

$${\beta }_1\left(\chi \right)=\chi {(1-\chi )}^{-1},{\beta }_2\left(\chi \right)=\frac{-1}{\chi }\log (1-\chi )\text{ and }{\beta }_3\left(\chi \right)=-\log (1-\chi ).$$

The inverse functions of ${\beta }_1$, and ${\beta }_2$, that correspond

$${\beta }_2^{-1}\left(w\right)=\frac{e^{2w}-1}{e^{2w}+1},{\beta }_1^{-1}\left(w\right)=\frac{w}{1+w}\text{, and }{\beta }_3^{-1}\left(w\right)=1-e^{-w}.$$

Several analytic function subclasses have been created employing the idea of subordination, including the different types of domains; some of them have been studied in the following and references [23], [24], [25], [26], [27].

The shell-like curves shape depends on the function

$$p(\chi )=\frac{1+{\tau }^2{\chi }^2}{1-\tau \chi -{\tau }^2{\chi }^2},\tau =\frac{1-\sqrt{5}}{2}$$

and the series form of $p(\chi )$ is:

$$p(\chi )=1+\sum _{n=1}^{\infty }(F_{n-1}+F_{n+1}){\tau }^n{\chi }^n,$$

where

$$F_n=\frac{{(1-\tau )}^n-{\tau }^n}{\sqrt{5}}$$

produce a sequence of Fibonacci numbers coefficients. The image of the unit circle under the function given in (4) results in the Maclaurin conchoid.

$$p\left(e^{i\phi }\right)=i\frac{\sin \phi (4\cos \phi -1)}{2(3-2\cos \phi )(1+\cos \phi )}+\frac{\sqrt{5}}{2(3-2\cos \phi )},0\le \phi <2\pi .$$ (4)

Jacek Dziok, in his work [28], introduces a class of starlike functions that are link with a shell-like curve and Fibonacci numbers. In addition, following on the work of [28], Malik et al. have recently introduced a novel class of analytic functions known as $\mathcal{CP}[P,W]$, which are closely linked to cardioid-like curve functions $\tilde{p}(\chi ,P,W)$.

Definition 1

[29]. Let $-1<W<P\le 1$, $\tau =\frac{1-\sqrt{5}}{2}$, $\chi \in \mathcal{U}$. If $p\in \mathcal{CP}[P,W]$ and

$$p(\chi )\prec \tilde{p}(\chi ,P,W),$$

where $\tilde{p}(\chi ,P,W)$ defined by

$$\tilde{p}(\chi ,P,W)=\frac{2P{\tau }^2{\chi }^2+(P-1)\tau \chi +2}{2W{\tau }^2{\chi }^2+(W-1)\tau \chi +2}.$$ (5)

Geometrical interpretations: The examining the class $\mathcal{CP}[P,W]$ in depth may benefit from a geometric description of $\tilde{p}(\chi ,P,W)$. If we let

$$W_{\tilde{p}}(e^{i\theta },P,W)=u\text{ and }{I}_{\tilde{p}}(e^{i\theta },P,W)=v$$

then a cardioid-like curve represents the image of the unit circle at $\tilde{p}(e^{i\theta },P,W)$ and u and v defined by

$$u=\frac{4+(P-1)(W-1){\tau }^2+4PW{\tau }^4+2\sigma \cos \theta +4(P+W){\tau }^2\cos 2\theta }{4+{(W-1)}^2{\tau }^2+4W^2{\tau }^4+4(W-1)(\tau +W{\tau }^3)\cos \theta +8W{\tau }^2\cos 2\theta },v=\frac{(P-W)(\tau -{\tau }^3)\sin \theta +2{\tau }^2\sin 2\theta }{4+{(W-1)}^2{\tau }^2+4W^2{\tau }^4+4(W-1)(\tau +W{\tau }^3)\cos \theta +8W{\tau }^2\cos 2\theta },$$ (6)

where

$$\sigma =(P+W-2)\tau +(2PW-P-W){\tau }^3,$$

and

$$\tilde{p}(0,P,W)=1\text{ and }\tilde{p}(1,P,W)=\frac{PW+9(P+W)+1+4(W-P)\sqrt{5}}{W^2+18W+1}.$$

The formula for the cusp of the cardioid-like curve described by (6) is

$$\gamma (P,W)=\tilde{p}(e^{\pm i\arccos \left(\frac{1}{4}\right)},P,W)=\frac{2PW-3(P+W)+2+(P-W)\sqrt{5}}{2(W^2-3W+1)}.$$

To learn more about geometric principles, see [29]. The Fig. 1, Fig. 2 and Fig. 3 show that $\tilde{p}(\chi ,P,W)$ maps $\mathcal{U}$ onto cardioid region.

Figure 1.

The curve (6) with P = 0.8; W = 0.6 and the curve (6) with P = 0.5; W=-0.5.

Figure 2.

The curve (6) with P = 0.6; W = 0.8 and the curve (6) with P =-0.5; W = 0.5.

Figure 3.

The Fig. 3 shows the images of certain concentric circles of cardioid domain.

The quantum calculus, also known as q-calculus, is a significant mathematical instrument used for the examination of various families of AFs. Its applications in mathematics and allied fields have sparked inspiration among academics. Within the field of engineering science and technology, engineers and scientists frequently utilize the q-calculus in the areas of mass and heat transfer, nonlinear differential equations and fuzzy differential equations. The q-difference operator (${\partial }_q)$ and the q-calculus operator were first formulated as pioneering contributions to the subject. Jackson [30], [31] provided the definition of this operator. Ismail et al. [32] were the pioneers in using the q-difference operator to expand the class of ${\mathcal{S}}^{⁎}$ in quantum calculus. In addition, Srivastava (see [33]) was the pioneer in using the fundamental (or q-) hypergeometric functions in Geometric Functions Theory (GFT). Subsequently, several mathematicians carried out noteworthy research, which has significantly influenced the field of GFT. Attiya et al. [34] conducted research on the new uses of this operator in relation to the q-raina function. Raza et al. [35] then developed and explained a group of star-shaped functions that are associated with the symmetric booth lemniscate. See the works [36], [37], [38] for additional details on q-calculus operator theory in GFT. Based on earlier work in the field of GFT that used the q-calculus, we use the technique of subordination to create two new subclasses of $\mathcal{K}(P,W,q,\xi )$ and ${\mathcal{K}}_{\mathrm{\Sigma }}(P,W,q,\xi )$. In detail, we look into the FSP and sharp initial coefficient bounds for the subclass of q-starlike functions. We also look into a new result for the class of bi-univalent functions. Furthermore, our findings yield numerous known results that serve as corollaries of the principal outcomes.

The following is the way Jackson [30] introduced the q-difference operator for analytic functions as follows:

$${\partial }_q\beta (\chi )=\frac{\beta (q\chi )-\beta (\chi )}{\chi (q-1)},\chi \in \mathcal{U}=1+\sum _{n=1}^{\infty }{[n]}_q{t}_n{\chi }^{n-1},$$

where

$${[n]}_q=\frac{1-q^n}{1-q},n\in \mathbb{N}.$$

We now consider the notions of subordinations and the aforementioned q-difference operator (${\partial }_q)$, and we give new subclasses associated with the cardioid domain.

Definition 2

The class $\mathcal{K}(P,W,q,\xi )$, contains a function β of the form (1), if

$$\frac{\chi {\partial }_q\beta (\chi )}{\beta (\chi )}+\xi \frac{{\chi }^2{\partial }_q({\partial }_q\beta (\chi ))}{\beta (\chi )}\prec \tilde{p}(\chi ,P,W),$$

where $\xi \ge 0$, $0<q<1$ and $\tilde{p}(\chi ,P,W)$ is given by (5).

Special cases:

(1): $\mathcal{K}(P,W,q\to 1-,0)={\mathcal{S}}^{⁎}(P,W)$, studied in [39].

(2): $\mathcal{K}(1,-1,q\to 1-,0)=SL$, studied by Sokół in [28].

Definition 3

Let $\xi \ge 0$, the class ${\mathcal{K}}_{\mathrm{\Sigma }}(P,W,q,\xi )$ of bi-univalent functions consisting of the functions in $\mathcal{A}$, such that

$$\beta \in \mathcal{K}(P,W,q,\xi )\text{and }{\beta }^{-1}\in \mathcal{K}(P,W,q,\xi ).$$

2. A set of lemmas

To demonstrate our findings, we shall employ the subsequent lemmas.

Lemma 1

[29] . Suppose that $\tilde{p}(\chi ,P,W)$ , given by (5) and if $p(\chi )\prec \tilde{p}(\chi ,P,W)$ . Then

$$\mathit{Re}(p(\chi ))>\alpha ,$$

where

$$\alpha =\frac{2(P+W-2)\tau +2(2PW-P-W){\tau }^3+16(P+W){\tau }^2\eta }{4(W-1)(\tau +W{\tau }^3)+32W{\tau }^2\eta },\eta =\frac{4+{\tau }^2-W^2{\tau }^2-4W^2{\tau }^4-(1-W{\tau }^2)T(W)}{4\tau (1+W^2{\tau }^2)},T(W)=\sqrt{5(2W{\tau }^2-(W-1)\tau +2)(2W{\tau }^2+(W-1)\tau +2)}$$

and

$$-1<W<P\le 1,\mathit{\text{and}}\tau =\frac{1-\sqrt{5}}{2}.$$

Note that $\tilde{p}(\chi ,P,W)$ is univalent within $\left|\chi \right|<{\tau }^2$.

Lemma 2

[29] . Let $\tilde{p}(\chi ,P,W)=1+\sum _{n=1}^{\infty }{\tilde{p}}_n{\chi }^n$ , where $\tilde{p}(\chi ,P,W)$ , given by (5) . Then

$${\tilde{p}}_n=\{\begin{array}{cc}\hfill (P-W)\frac{\tau }{2}\hfill & \hfill \mathit{\text{for}}n=1,\hfill \\ \hfill (P-W)(5-W)\frac{{\tau }^2}{2^2}\hfill & \hfill \mathit{\text{for}}n=2,\hfill \\ \hfill \frac{1-W}{2}\tau p_{n-1}-W{\tau }^2{p}_{n-2}\hfill & \hfill \mathit{\text{for}}n\ge 3,\hfill \end{array}$$ (7)

where $-1<W<P\le 1$ .

Lemma 3

[29] . Let $p(\chi )=1+\sum _{n=1}^{\infty }{p}_n{\chi }^n\prec \tilde{p}(\chi ,P,W)=1+\sum _{n=1}^{\infty }{\tilde{p}}_n{\chi }^n$ , where $\tilde{p}(\chi ,P,W)$ , given by (5) . Then

$$|p_2-v{p}_1^2|\le \frac{(P-W)\left|\tau \right|}{4}\max \{2,\left|\tau (v(P-W)+W-5)\right|\},v\in \mathbb{C}.$$

Lemma 4

[12] . Let $p(\chi )=1+\sum _{n=1}^{\infty }{c}_n{\chi }^n$ and $p\in \mathcal{P}$ , then

$$|c_2-\frac{v}{2}{c}_1^2|\le \max \{2,2|v-1|\}=\left\{\begin{array}{cc}\hfill 2|v-1|\hfill & \hfill v\in \mathbb{C},\hfill \\ \hfill 2,\hfill & \hfill \mathit{\text{if}}0\le v\le 2,\hfill \end{array}\right\}$$

and

$$\left|c_n\right|\le 2,\mathit{\text{for}}n\ge 1.$$

Lemma 5

[40] . Let $f(\chi )=\sum _{n=1}^{\infty }{t}_n{\chi }^n$ be analytic in $\mathcal{U}$ and $\beta (\chi )=\sum _{n=1}^{\infty }{b}_n{\chi }^n$ be convex in $\mathcal{U}$ . If $f(\chi )\prec \beta (\chi )$ , then

$$\left|t_n\right|<\left|b_1\right|,n=1,2,3....$$

The order of this section is as follows. We begin by estimating the coefficients for $\beta \in \mathcal{K}(P,W,q,\xi )$ and $\beta \in {\mathcal{K}}_{\mathrm{\Sigma }}(q,\xi ,P,W)$. Theorem 1, the first of our primary findings, establishes initial coefficients of functions $\beta \in \mathcal{K}(P,W,q,\xi )$. The Fekete-Szego problems for the class $\mathcal{K}(P,W,q,\xi )$ is solved by Theorem 2, the second of our key findings, while Theorem 3 provides bounds on coefficients for $\beta \in \mathcal{K}(P,W,q,\xi )$. After that, we look at the upper bounds for initial coefficients and the Fekete-Szego problem involving inverse functions (${\beta }^{-1}$) for the class $\mathcal{K}(P,W,q,\xi )$ in Theorem 4 and Theorem 5. Finally, in Theorem 6, we explore an innovative consequence using bi-univalent functions of the class ${\mathcal{K}}_{\mathrm{\Sigma }}(P,W,q,\xi )$.

3. Main results

Sharp coefficient estimates for the functions $\beta \in \mathcal{K}(P,W,q,\xi )$:

Theorem 1

Let $\beta \in \mathcal{K}(P,W,q,\xi )$ be given by (1) , $-1\le W<P\le 1$ . Then

$$\left|t_2\right|\le \frac{(P-W)\left|\tau \right|}{2((1+\xi ){\left[2\right]}_q-1)},\left|t_3\right|\le \frac{(P-W){\left|\tau \right|}^2}{4({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)}\{5-W+\frac{(P-W)}{(1+\xi ){\left[2\right]}_q-1}\}.$$

The results are sharp.

Proof

Let $\beta \in \mathcal{K}(P,W,q,\xi )$. Then

$$\frac{\chi {\partial }_q\beta (\chi )}{\beta (\chi )}+\xi \frac{{\chi }^2{\partial }_q({\partial }_q\beta (\chi ))}{\beta (\chi )}\prec \tilde{p}(\chi ,P,W),$$

where

$$\tilde{p}(\chi ,P,W)=\frac{2P{\tau }^2{\chi }^2+(P-1)\tau \chi +2}{2W{\tau }^2{\chi }^2+(W-1)\tau \chi +2}.$$

We have a function $u_0$ with a subordination principle and

$$u_0(0)=0\text{ and }|u_0(\chi )|<1$$

thus

$$\frac{\chi {\partial }_q\beta (\chi )}{\beta (\chi )}+\xi \frac{{\chi }^2{\partial }_q({\partial }_q\beta (\chi ))}{\beta (\chi )}=\tilde{p}(P,W;u(\chi )).$$

Let

$$u_0\left(\chi \right)=\frac{p(\chi )-1}{p(\chi )+1}=\frac{1}{2}{c}_1\chi +\frac{1}{2}(c_2-\frac{1}{2}{c}_1^2){\chi }^2+\frac{1}{2}(c_3-c_1{c}_2+\frac{1}{4}{c}_1^3){\chi }^3+\cdots .$$

Since $\tilde{p}(\chi ,P,W)=1+\sum _{n=1}^{\infty }{\tilde{p}}_n{\chi }^n$, then

$$\tilde{p}(u_0(\chi ),P,W)=1+\frac{{\tilde{p}}_1{c}_1}{2}\chi +(\frac{1}{2}(c_2-\frac{1}{2}{c}_1^2){\tilde{p}}_1+\frac{{\tilde{p}}_2{c}_1^2}{4}){\chi }^2+....$$ (8)

Also consider the function

$$\tilde{p}(\chi ,P,W)=\frac{2P{\tau }^2{\chi }^2+(P-1)\tau \chi +2}{2W{\tau }^2{\chi }^2+(W-1)\tau \chi +2}.$$

Suppose $\tau \chi ={\mathrm{\Psi }}_0$. Then

$$\tilde{p}(\chi ,P,W)=(P{\mathrm{\Psi }}_0^2+\frac{(P-1)}{2}{\mathrm{\Psi }}_0+1)[1+\frac{1}{2}(1-W){\mathrm{\Psi }}_0+\left(\frac{W^2-6W+1}{4}\right){\mathrm{\Psi }}_0^2+...]=1+\frac{1}{2}(P-W){\mathrm{\Psi }}_0+\frac{1}{4}(P-W)(5-W){\mathrm{\Psi }}_0^2+....$$

This indicates that

$$\tilde{p}(\chi ,P,W)=1+\frac{1}{2}(P-W)\tau \chi +\frac{1}{4}(P-W)(5-W){\tau }^2{\chi }^2+....$$ (9)

It is evident from (8) that

$$\tilde{p}(u(\chi ),P,W)=1+\frac{1}{4}(P-W)\tau c_1\chi +(\frac{1}{4}(P-W)\tau (c_2-\frac{1}{2}{c}_1^2)+\frac{(P-W)(5-W){\tau }^2{c}_1^2}{16}){\chi }^2+....$$ (10)

Since $\beta \in \mathcal{K}(P,W,q,\xi )$, then

$$\frac{\chi {\partial }_q\beta (\chi )}{\beta (\chi )}+\xi \frac{{\chi }^2{\partial }_q({\partial }_q\beta (\chi ))}{\beta (\chi )}=1+((1+\xi ){\left[2\right]}_q-1)t_2\chi +\{({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)t_3-((1+\xi ){\left[2\right]}_q-1)t_2^2\}{\chi }^2+....$$ (11)

Compare the coefficients from (10) and (11), we have

$$t_2=\frac{(P-W)\tau c_1}{4((1+\xi ){\left[2\right]}_q-1)}.$$ (12)

Applying modulus, we have

$$\left|t_2\right|\le \frac{(P-W)\left|\tau \right|}{2((1+\xi ){\left[2\right]}_q-1)}.$$

When we compare the coefficients from (10) and (11), we have

$$t_3=\frac{(P-W)\tau }{4({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)}\{c_2-\frac{v}{2}{c}_1^2\},$$ (13)

where

$$v=1-\frac{(5-W)\tau }{2}-\frac{(P-W)\tau }{2((1+\xi ){\left[2\right]}_q-1)},$$

which demonstrating that $v>2$ for the connection $P>W$. So, by using Lemma 4, the desired outcome is achieved. For sharpness, let ${\beta }_n:\mathcal{U}\to \mathbb{C}$ defined by

$${\beta }_{⁎}(\chi )=\chi +\frac{\tau (P-W)}{2((1+\xi ){\left[2\right]}_q-1)}{\chi }^2+\frac{{\tau }^2(P-W)}{4({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)}(5-W+\frac{P-W}{(1+\xi ){\left[2\right]}_q-1}){\chi }^3+....$$ (14)

Then it is clear that

$${\beta }_{⁎}(0)=0\text{ and }{\beta }_{⁎}^{{}^{\prime }}(0)=1.$$

It is readily demonstrable that

$$\frac{\chi {\partial }_q{\beta }_{⁎}(\chi )}{{\beta }_{⁎}(\chi )}+\xi \frac{{\chi }^2{\partial }_q({\partial }_q{\beta }_{⁎}(\chi ))}{{\beta }_{⁎}(\chi )}=\tilde{p}(\chi ,P,W),$$

where $\tilde{p}(\chi ,P,W)$ is provided in (9). This show that ${\beta }_{⁎}\in \mathcal{K}(P,W,q,\xi )$. Therefore result is sharp for ${\beta }_{⁎}$ stated in (14).

The following known result was demonstrated in [41], when $\xi =0$ and $q\to 1-$ in Theorem 1.

Corollary 1

[41] . For $-1\le W<P\le 1$ . Let $\beta \in {\mathcal{S}}^{⁎}(P,W)$ defined in (1) . Then

$$\left|t_2\right|\le \frac{(P-W)\left|\tau \right|}{2},\left|t_3\right|\le \frac{(P-W){\left|\tau \right|}^2}{8}(P-2W+5).$$

Theorem 2

Let $\beta \in \mathcal{K}(P,W,q,\xi )$ and of the form (1) . Then

$$|t_3-\mu t_2^2|\le \frac{(P-W)\left|\tau \right|}{4({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)}\max \{2,|W-5+\tau \left(\frac{P-W}{(1+\xi ){\left[2\right]}_q-1}(\frac{({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)\mu }{(1+\xi ){\left[2\right]}_q-1}-1)\right)|\}.$$

This result is sharp.

Proof

Since $\beta \in \mathcal{K}(P,W,q,\xi )$, then by Schwarz function, such that

$$\frac{\chi {\partial }_q\beta (\chi )}{\beta (\chi )}+\xi \frac{{\chi }^2{\partial }_q({\partial }_q\beta (\chi ))}{\beta (\chi )}=\tilde{p}(u(\chi ),P,W),\chi \in \mathcal{U}.$$

Therefore

$$\chi +(1+\xi ){\left[2\right]}_q{t}_2{\chi }^2+{\left[3\right]}_q(1+\xi {\left[2\right]}_q)t_3{\chi }^3+...=\{\chi +t_2{\chi }^2+t_3{\chi }^3+...\}\{1+p_1\chi +p_2{\chi }^2+...\}.$$

When we evaluate the coefficients on either side, we obtain

$$t_2=\frac{p_1}{(1+\xi ){\left[2\right]}_q-1},\text{and}({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)t_3=p_1{t}_2+p_2.$$

This implies that

$$|t_3-\mu t_2^2|=\frac{1}{{\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1}|p_2-v{p}_1^2|,$$

where

$$v=\frac{1}{(1+\xi ){\left[2\right]}_q-1}(\mu \frac{((1+\xi {\left[2\right]}_q){\left[3\right]}_q-1)}{(1+\xi ){\left[2\right]}_q-1}-1).$$

Apply the Lemma 3 for v, We've achieved the desired result. The equality

$$|t_3-\mu t_2^2|=\frac{(P-W){\left|\tau \right|}^2}{4({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)}\{|W-5+\frac{P-W}{(1+\xi ){\left[2\right]}_q-1}(\frac{({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)\mu }{(1+\xi ){\left[2\right]}_q-1}-1)|\}$$

is hold for ${\beta }_{⁎}$ given in (14). Suppose that ${\beta }_0:\mathcal{U}\to \mathbb{C}$ be given as:

$${\beta }_0(\chi )=\chi +\frac{\tau (P-W)}{4((1+\xi ){\left[2\right]}_q-1)}{\chi }^3+....$$ (15)

Hence, clear that

$${\beta }_0(0)=0\text{and}{\beta }_0^{{}^{\prime }}(0)=1$$

and

$$\frac{\chi {\partial }_q\beta (\chi )}{\beta (\chi )}+\xi \frac{{\chi }^2{\partial }_q({\partial }_q\beta (\chi ))}{\beta (\chi )}=\tilde{p}({\chi }^2,P,W).$$

This show that ${\beta }_0\in \mathcal{K}(P,W,q,\xi )$. Hence the equality $|t_3-\mu t_2^2|=\frac{(P-W)\left|\tau \right|}{2((1+\xi ){\left[2\right]}_q-1)}$ hold for the function ${\beta }_0$ defined in (15).

We get the known result for taking $q\to 1-$ and $\xi =0$ in Theorem 2.

Corollary 2

[41] . Let β defined in (1) and belong to class ${\mathcal{S}}^{⁎}(P,W)$ . Then

$$|t_3-\mu t_2^2|\le \frac{(P-W)\left|\tau \right|}{8}\max \{2,\left|\tau (-(P-2W+5)+2(P-W)\mu )\right|\}.$$

Theorem 3

Let $\beta \in \mathcal{A}$ , be defined in (1) . If $\beta \in \mathcal{K}(P,W,q,\xi )$ . Then

$$\left|t_2\right|\le \frac{\left|{\tilde{p}}_1\right|}{{\left[2\right]}_q(1+\xi )-1}$$

and

$$\left|t_n\right|\le \frac{\left|{\tilde{p}}_1\right|}{{\left[n\right]}_q(1+\xi {[n-1]}_q)-1}\prod _{k=2}^{n-1}(1+\frac{\left|{\tilde{p}}_1\right|}{{\left[k\right]}_q(1+\xi {[k-1]}_q)-1}),\mathit{\text{for}}n\ge 3,$$

where ${\tilde{p}}_1$ is demonstrated in (7) .

Proof

Let $\beta \in \mathcal{K}(P,W,q,\xi )$ then

$$K(\chi )\prec \tilde{p}(\chi ,P,W),$$

where, $\tilde{p}(\chi ,P,W)$ is defined by (5), and suppose that

$$K(\chi )=\frac{\chi {\partial }_q\beta (\chi )}{\beta (\chi )}+\xi \frac{{\chi }^2{\partial }_q({\partial }_q\beta (\chi ))}{\beta (\chi )}.$$ (16)

Lemma 5 is used to obtain

$$\left|\frac{K^{(j)}(0)}{j!}\right|=\left|c_j\right|\le \left|{\tilde{p}}_1\right|,j\in \mathbb{N},$$ (17)

where

$$K(\chi )=1+c_1\chi +c_2{\chi }^2+....$$

Considering (16), we have

$$({\left[n\right]}_q(1+\xi {[n-1]}_q)-1)t_n=\{c_{n-1}+c_{n-2}{t}_2+...+c_1{t}_{n-1}\}=\sum _{i=1}^{n-1}{c}_i{t}_{n-i},t_1=1.$$ (18)

Solving (17) and (18), we get

$$({\left[n\right]}_q(1+\xi {[n-1]}_q)-1)t_n\le \left|{\tilde{p}}_1\right|\sum _{i=1}^{n-1}\left|t_{n-i}\right|.$$

Using $n=2,3,4$, we obtain

$$\left|t_2\right|\le \frac{\left|{\tilde{p}}_1\right|}{{\left[2\right]}_q(1+\xi )-1},\left|t_3\right|\le \frac{\left|{\tilde{p}}_1\right|}{{\left[3\right]}_q(1+2\xi )-1}(1+\frac{\left|{\tilde{p}}_1\right|}{{\left[2\right]}_q(1+\xi )-1})$$

and

$$\left|t_4\right|\le \frac{\left|{\tilde{p}}_1\right|}{{\left[4\right]}_q(1+3\xi )-1}(1+\left|t_2\right|+\left|t_3\right|)=\frac{\left|{\tilde{p}}_1\right|}{{\left[4\right]}_q(1+3\xi )-1}(1+\frac{\left|{\tilde{p}}_1\right|}{{\left[3\right]}_q(1+2\xi )-1})(1+\frac{\left|{\tilde{p}}_1\right|}{{\left[2\right]}_q(1+\xi )-1}).$$

We get by using mathematical induction

$$\left|t_n\right|\le \frac{\left|{\tilde{p}}_1\right|}{{\left[n\right]}_q(1+\xi {[n-1]}_q)-1}\prod _{k=2}^{n-1}(1+\frac{\left|{\tilde{p}}_1\right|}{{\left[k\right]}_q(1+\xi {[k-1]}_q)-1}),\text{ for }n\ge 3.$$

The proof of Theorem 3 is now complete.

This follows from the following consequence when we assume that $\xi =0$ in Theorem 3.

Corollary 3

Let for β defined by (1) and $\beta \in {\mathcal{S}}^{⁎}(P,W,q)$ . Then

$$\left|t_2\right|\le \frac{\left|{\tilde{p}}_1\right|}{{\left[2\right]}_q-1}$$

and

$$\left|t_n\right|\le \frac{\left|{\tilde{p}}_1\right|}{{\left[n\right]}_q-1}\prod _{k=2}^{n-1}(1+\frac{\left|{\tilde{p}}_1\right|}{{\left[k\right]}_q-1}),\mathit{\text{for}}n\ge 3,$$

where ${\tilde{p}}_1$ is given by (7) .

We have following unknown result by taking $\xi =0$ and $q\to 1-$ in Theorem 3.

Corollary 4

Let for β defined by (1) and $\beta \in {\mathcal{S}}^{⁎}(P,W)$ . Then

$$\left|t_2\right|\le \left|{\tilde{p}}_1\right|$$

and

$$\left|t_n\right|\le \frac{\left|{\tilde{p}}_1\right|}{n-1}\prod _{k=2}^{n-1}(1+\frac{\left|{\tilde{p}}_1\right|}{k-1}),\mathit{\text{for}}n\ge 3,$$

where ${\tilde{p}}_1$ is given by (7) .

Inverse coefficients

Theorem 4

Let $\beta \in \mathcal{K}(P,W,q,\xi )$ be given by (1) , and ${\beta }^{-1}$ of the form (2) , $-1\le W<P\le 1$ . Then

$$\left|D_2\right|\le \frac{(P-W)\left|\tau \right|}{2({\left[2\right]}_q(1+\xi )-1)}$$ (19)

and

$$\begin{gathered} \left|D_3\right|\le \frac{(P-W)\left|\tau \right|}{4({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)}\max \{2,|\tau ((5-W)+\frac{(P-W)}{{\left[2\right]}_q(1+\xi )-1} \\ (1-\frac{2({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)}{{\left[2\right]}_q(1+\xi )-1}))|\}. \end{gathered}$$ (20)

Also, bounds $\left|D_2\right|$ and $\left|D_3\right|$ are sharp.

Proof

Let $\beta \in \mathcal{K}(P,W,q,\xi )$, then using (12) and (13), we have

$$t_2=\frac{(P-W)\tau c_1}{4({\left[2\right]}_q(1+\xi )-1)}$$ (21)

and

$$t_3=\frac{(P-W)\tau }{4({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)}(c_2-\frac{c_1^2}{2}+\frac{(5-W)\tau }{4}{c}_1^2+\frac{(P-W)\tau }{4((1+\xi ){\left[2\right]}_q-1)}{c}_1^2).$$ (22)

We know that $\beta ({\beta }^{-1})(w)=w$. So from (2), we have

$$D_2=-t_2.$$ (23)

As a result of resolving (21) and (23), we have

$$\left|D_2\right|\le \frac{(P-W)\left|\tau \right|}{2({\left[2\right]}_q(1+\xi )-1)}.$$

From (3), we have

$$D_3=2t_2^2-t_3.$$ (24)

Putting (21) and (22), in (24), we get

$$\left|D_3\right|=\frac{(P-W)\left|\tau \right|}{4({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)}|c_2-\frac{1}{2}V{c}_1^2|,$$

where

$$V=1-\frac{\tau }{2}((5-W)+\frac{(P-W)}{{\left[2\right]}_q(1+\xi )-1}(1-\frac{2({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)}{{\left[2\right]}_q(1+\xi )-1})).$$

Hence by applications of the Lemma 4, we have

$$\begin{gathered} \left|D_3\right|\le \frac{(P-W)\left|\tau \right|}{4({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)}\times \max \{2,|\tau (5-W+\frac{(P-W)}{{\left[2\right]}_q(1+\xi )-1} \\ (1-\frac{2({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)}{{\left[2\right]}_q(1+\xi )-1}))|\}. \end{gathered}$$

Thus, the desired result is shown. The results (19) and (20) are sharp for ${\beta }_{⁎}$ demonstrated in (14). The result

$$\left|D_3\right|\le \frac{(P-W)\left|\tau \right|}{4({\left[2\right]}_q(1+\xi )-1)}$$

is sharp for the function ${\beta }_0$ given in (15).

The following consequence can be obtained when we assume that $\xi =0$ and $q\to 1-$ in Theorem 4.

Corollary 5

[41] . Let $-1\le W<P\le 1$ and $\beta \in {\mathcal{S}}^{⁎}(P,W)$ be given by (1) , and ${\beta }^{-1}$ defined in (2) . Then

$$\left|D_2\right|\le \frac{(P-W)\left|\tau \right|}{2}$$

and

$$\left|D_3\right|\le \frac{(P-W)\left|\tau \right|}{8}\max \{2,\tau |3P-2W-5|\}.$$

Also, bounds $\left|D_2\right|$ and $\left|D_3\right|$ are sharp.

Theorem 5

Let $-1\le W<P\le 1$ . Let $\beta \in \mathcal{K}(P,W,q,\xi )$ , where β and ${\beta }^{-1}$ given in (1) and (2) . Then

$$\begin{gathered} |D_3-\mu D_2^2|\le \frac{(P-W)\left|\tau \right|}{4({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)}\times \max \{2,|\tau (\frac{1}{{\left[2\right]}_q(1+\xi )-1}\left(\frac{(2-\mu )({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)}{{\left[2\right]}_q(1+\xi )-1}(P-W)\right) \\ -\frac{1}{{\left[2\right]}_q(1+\xi )-1}(P-W)+W-5)|\}. \end{gathered}$$

The result is sharp.

Proof

Since

$$D_2=-t_2$$

and

$$D_3=2t_2^2-t_3.$$

Therefore by using $t_2=\frac{p_1}{{\left[2\right]}_q(1+\xi )-1}$ and $t_3=\frac{1}{{\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1}(p_1{t}_2+p_2)$, one can we write

$$|D_3-\mu D_2^2|=\frac{1}{({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)}|p_2-v{p}_1^2|,$$

where

$$v=\frac{1}{{\left[2\right]}_q(1+\xi )-1}(\frac{(2-\mu )({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)}{{\left[2\right]}_q(1+\xi )-1}-1).$$

Hence, by using the Lemma 3, we obtain

$$\begin{gathered} |D_3-\mu D_2^2|\le \frac{(P-W)\left|\tau \right|}{4({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)} \\ \max \{2,|\tau (\frac{1}{{\left[2\right]}_q(1+\xi )-1}\left(\frac{(2-\mu )({\left[3\right]}_q(1+\xi ){\left[2\right]}_q-1)}{{\left[2\right]}_q(1+\xi )-1}(P-W)\right) \\ -\frac{1}{{\left[2\right]}_q(1+\xi )-1}(P-W)+W-5)|\}. \end{gathered}$$

The equality is held for ${\beta }_{⁎}$ and ${\beta }_0$, as demonstrated in (14) and (15).

The following consequence can be obtained when we assume that $\xi =0$ and $q\to 1-$ in Theorem 5.

Corollary 6

[41] . Let $\mu \in \mathbb{C}$ , $\left|\chi \right|<{\tau }^2$ , and $-1\le W<P\le 1$ , let $\beta \in {\mathcal{S}}^{⁎}(P,W)$ , where β and ${\beta }^{-1}$ of the form (1) and (2) . Then

$$|D_3-\mu D_2^2|\le \frac{(P-W)\left|\tau \right|}{8}\max \{2,\left|\tau (3P-2W-5-2\mu (P-W))\right|\}.$$

The result is sharp.

Finally, we fined a new result for bi-univalent function $\beta \in {\mathcal{K}}_{\mathrm{\Sigma }}(P,W,q,\xi )$.

Theorem 6

Let $\beta \in {\mathcal{K}}_{\mathrm{\Sigma }}(P,W,q,\xi )$ . Then

$$\left|t_2\right|\le \frac{{\tilde{p}}_1\sqrt{{\tilde{p}}_1}}{\sqrt{({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-(1+\xi ){\left[2\right]}_q)p_1^2+({\tilde{p}}_1-{\tilde{p}}_2){((1+\xi ){\left[2\right]}_q-1)}^2}}$$

and

$$\left|t_3\right|\le \frac{\left|{\tilde{p}}_1\right|T_1(\xi ,q)+\left|{\tilde{p}}_1^3\right|\left(((1+\xi {\left[2\right]}_q){\left[3\right]}_q-1)\right)}{((1+\xi {\left[2\right]}_q){\left[3\right]}_q-1)T_1(\xi ,q)},$$

where ${\tilde{p}}_1$ and ${\tilde{p}}_2$ is given by (7) .

Proof

If $\beta \in {\mathcal{K}}_{\mathrm{\Sigma }}(P,W,q,\xi )$, then β∈ $\mathcal{K}(P,W,q,\xi )$ and $G={\beta }^{-1}\in$ $\mathcal{K}(P,W,q,\xi )$. Hence

$$M(\chi )=\frac{\chi {\partial }_q\beta (\chi )}{\beta (\chi )}+\xi \frac{{\chi }^2{\partial }_q({\partial }_q\beta (\chi ))}{\beta (\chi )}=\tilde{p}(u_0(\chi ),P,W)$$ (25)

and

$$L(w)=\frac{\chi {\partial }_{q}G(w)}{G(w)}+\xi \frac{{\chi }^2{\partial }_q({\partial }_{q}G(w))}{G(w)}=\tilde{p}(v_0(w),P,W).$$ (26)

The series form of $\tilde{p}(\chi ,P,W)$ is

$$\tilde{p}(\chi ,P,W)=1+\sum _{n=1}^{\infty }{\tilde{p}}_n{\chi }^n,$$

where ${\tilde{p}}_n$ is given by (7). Let we have function

$$p_1(\chi )=\frac{1+u_0(\chi )}{1-u_0(\chi )}=1+c_1\chi +c_2{\chi }^2+...,$$

thus

$$u_0(\chi )=\frac{1}{2}{c}_1\chi +\frac{1}{2}(c_2-\frac{1}{2}{c}_1^2){\chi }^2...\text{ .}$$

Let we have function

$$p_2(w)=\frac{1+v_0(w)}{1-v_0(w)}=1+d_{1}w+d_2{w}^2+...,$$

thus

$$v_0(\chi )=\frac{1}{2}{d}_{1}w+\frac{1}{2}(d_2-\frac{1}{2}{d}_1^2)w^2...\text{ .}$$

Therefore we have

$$\tilde{p}(u_0(\chi ),P,W)=1+\frac{{\tilde{p}}_1{c}_1}{2}\chi +(\frac{1}{2}(c_2-\frac{1}{2}{c}_1^2){\tilde{p}}_1+\frac{{\tilde{p}}_2{c}_1^2}{4}){\chi }^2+....$$ (27)

and

$$\tilde{p}(v_0(w),P,W)=1+\frac{{\tilde{p}}_1{d}_1}{2}w+(\frac{1}{2}(d_2-\frac{1}{2}{d}_1^2){\tilde{p}}_1+\frac{{\tilde{p}}_2{d}_1^2}{4})w^2+....$$ (28)

$$M(\chi )=1+((1+\xi ){\left[2\right]}_q-1)t_2\chi +\{({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)t_3-((1+\xi ){\left[2\right]}_q-1)t_2^2\}{\chi }^2+....$$ (29)

The series form of $L(w)$ is

$$L(w)=1-((1+\xi ){\left[2\right]}_q-1)t_{2}w+\{({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)(2t_2^2-t_3)-((1+\xi ){\left[2\right]}_q-1)t_2^2\}{w}^2+....$$ (30)

Using (29) and (27) in (25) and then equating the coefficients, we have

$$((1+\xi ){\left[2\right]}_q-1)t_2=\frac{{\tilde{p}}_1{c}_1}{2},$$ (31)

$$({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)t_3-((1+\xi ){\left[2\right]}_q-1)t_2^2=\frac{1}{2}(c_2-\frac{1}{2}{c}_1^2){\tilde{p}}_1+\frac{{\tilde{p}}_2{c}_1^2}{4}.$$ (32)

Again using (30) and (28) in (26) and then equating the coefficients, we have

$$-((1+\xi ){\left[2\right]}_q-1)t_2=\frac{{\tilde{p}}_1{d}_1}{2},$$ (33)

$$({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)(2t_2^2-t_3)-((1+\xi ){\left[2\right]}_q-1)t_2^2=\frac{1}{2}(d_2-\frac{1}{2}{d}_1^2){\tilde{p}}_1+\frac{{\tilde{p}}_2{d}_1^2}{4},$$ (34)

where ${\tilde{p}}_1$ and ${\tilde{p}}_2$ is given by (7). From (31) and (33), we obtain

$$-d_1=c_1$$

and

$${((1+\xi ){\left[2\right]}_q-1)}^2{t}_2^2=\frac{{\tilde{p}}_1^2(c_1^2+d_1^2)}{4}.$$

Adding, (32) and (34), and then using (33), we obtain

$$t_2^2=\frac{{\tilde{p}}_1^3(c_2+d_2)}{4\{({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-(1+\xi ){\left[2\right]}_q)p_1^2+({\tilde{p}}_1-{\tilde{p}}_2){((1+\xi ){\left[2\right]}_q-1)}^2\}}.$$

Applying the caratheodory lemma to the modulus yields

$$\left|t_2\right|\le \frac{{\tilde{p}}_1\sqrt{{\tilde{p}}_1}}{\sqrt{({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-(1+\xi ){\left[2\right]}_q)p_1^2+({\tilde{p}}_1-{\tilde{p}}_2){((1+\xi ){\left[2\right]}_q-1)}^2}}.$$ (35)

We now subtract from (32) and (34) for the result $\left|t_3\right|$, and we obtain

$$2\left(({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)\right)t_3=\frac{1}{2}{\tilde{p}}_1(c_2-d_2)+2\left(({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)\right)t_2^2.$$

Taking the modulus, we have

$$2\left(({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)\right)\left|t_3\right|\le 2{\tilde{p}}_1+2\left(({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)\right){\left|t_2\right|}^2.$$

Then, in view of (35), we obtain

$$\left|t_3\right|\le \frac{\left|{\tilde{p}}_1\right|T_1(\xi ,q)+\left|{\tilde{p}}_1^3\right|({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)}{({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-1)T_1(\xi ,q)},$$

where

$$T_1(\xi ,q)=\{({\left[3\right]}_q(1+\xi {\left[2\right]}_q)-(1+\xi ){\left[2\right]}_q)\left|p_1^2\right|+|{\tilde{p}}_1-{\tilde{p}}_2|{((1+\xi ){\left[2\right]}_q-1)}^2\}.$$

Hence we completed our result.

4. Conclusions

There are three parts of this article. The introduction and typical terminology are included in Section 1. Additionally, a novel classes of analytical functions was established in this part, which is related to the operator theory of the q-calculus and the cardioid domain, and numerous common lemmas were presented in Section 2. Interesting topics we examined for functions belonging to the classes $\mathcal{K}(P,W,q,\xi )$ and ${\mathcal{K}}_{\mathrm{\Sigma }}(P,W,q,\xi )$ in Section 3 included the first two initial coefficients bounds, estimates for the Fekete-Szegö type functional, and other useful findings. In this article, it has been shown that all of the bounds are sharp. The inverse functions were additionally investigated at with similar sharp outcomes. Some of the main consequences that are currently recognized as existing are also highlighted in our research.

Additional research proposals can explore the application of the concept of subordination and the q-calculus theory to generate results pertaining to the newly defined classes. Moreover, this study's approach has the potential to establish multiple novel subclasses of meromorphic, multivalent, and harmonic functions, enabling the examination of their characteristics. The researchers themselves, inspired by the findings presented in this paper, will be the sole individuals to pioneer studies that utilize these classes.

Funding

This work was funded through Arab Open University research fund no.(AOUKSA-524008).

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

CRediT authorship contribution statement

Khaled Matarneh: Writing – review & editing, Writing – original draft. Ahmad A. Abubakar: Writing – review & editing. Mohammad Faisal Khan: Writing – review & editing. Suha B. Al-Shaikh: Writing – review & editing, Supervision, Funding acquisition. Mustafa Kamal: Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

No data was used for the research described in the article.