A Class of Bi-Bazilevič Mappings Generated via Miller-Ross Type Poisson Distribution Subordinate to Chebyshev Polynomials

Abstract

Bazilevič mappings are considered very important in the theory of geometric mappings because they provide a way to generalize and study the properties of important classes of univalent mappings. Their importance is not only in the deepening of the theory, but also in the practical means of modeling phenomena in applied science and engineering, physics, and differential equations. This paper, in this sense, provides a new subclass of bi-Bazilevič mappings with the use of advanced analytical methods, Chebyshev polynomials on one side, and a Miller–Ross-type Poisson distribution on the other side. The Poisson distribution is considered one of the most important models of probability distributions with a large scope of application in the various sciences. The main components of this study are the definition and the study of this new class of functions, in which the initial Taylor–Maclaurin coefficients, in particular, $q_2$ and $q_3$, are determined and estimated for mappings in this subclass. Also, the classical Fekete–Szegö problem is solved and the first-order limits of this important functional are obtained with respect to the newly introduced bi-Bazilevič mappings. The outcomes contribute to expanding both the theoretical and practical aspects of this type of mapping.

1. Introduction

The last few decades have seen a significant increase in studies addressing the distribution of random variables. This is largely due to the importance of the probability density mapping of a distribution, which is pivotal for statistical inference. This importance has warranted further research into the characteristics and utility of the diverse distribution mappings. Many of these distributions derive their importance from their ability to describe phenomena observed in real-world contexts; prominent examples include the binomial being used for binary experiments, hyper-geometric distributions being used for sampling without replacement and the Poisson distribution for modeling rare events. A random variable x is formally within mathematical statistics, defined as following a Poisson distribution if its probability density mapping is specified as follows:

$$\hslash \left(x\right)={\displaystyle \frac{e^{-p}}{x!}}{p}^x,x\ge 0,$$

(1)

where p represents the parameter governing the distributions.

Let $\Lambda$ consist of all analytic mappings $\hslash \left(\xi \right)$ that satisfy the normalization condition expressed as

$$\hslash \left(\xi \right)=\xi +\sum _{i=2}^{\infty }{q}_i{\xi }^i,(\xi \in \mathbb{U})$$

(2)

where $\mathbb{U}=\{\xi \in \mathbb{C}:\mid \xi \mid <1\}$.

Additionally, we denote, using $\Omega$, the subclass of mappings $\hslash \left(\xi \right)\in \Lambda$ that are univalent in $\mathbb{U}$.

For two analytic mappings, ℏ and ģ in $\mathbb{U}$, it is said that the mapping ℏ is subordinate to the mapping ģ denoted by $\hslash \prec ģ$ if there exists an analytic Schwarz mapping d in $\mathbb{U}$ with $d\left(0\right)=0$ and $\mid d\left(\xi \right)\mid <1,(\xi \in \mathbb{U})$ such that $\hslash \left(\xi \right)=ģ\left(d\right(\xi \left)\right)$; that is, the mapping ℏ is obtained by composing ģ with a Schwarz mapping.

Moreover, when the mapping ģ be univalent in $\mathbb{U}$, we have the following equivalence: $\hslash \left(\xi \right)\prec ģ\left(\xi \right)$ if and only if $\hslash \left(0\right)=ģ\left(0\right)$ and $\hslash \left(\mathbb{U}\right)\subset ģ\left(\mathbb{U}\right)$.

It is widely recognized that each mapping $\hslash \in \Omega$ has an inverse ${\hslash }^{-1}$, given by the series expansion

$${\hslash }^{-1}(\hslash \left(\xi \right))=\xi$$

and

$${\hslash }^{-1}(\hslash \left(d\right))=d,$$

$$\mid d \mid <r_0(\hslash );r_0(\hslash )\ge {\displaystyle \frac{1}{4}},$$

where

$${\hslash }^{-1}\left(d\right)=d-q_2{d}^2+(2q_2^2-q_3)d^3-(5a_2^3-5q_2{q}_3+q_4)d^4+\dots .$$

(3)

A mapping $\hslash \left(\xi \right)$ is called bi-univalent if both $\hslash \left(\xi \right)$ and its inverse ${\hslash }^{-1}\left(\xi \right)$ are univalent in $\mathbb{U}$.

One may introduce $\Sigma$ as the collective designation for every bi-univalent mappings within $\mathbb{U}$ that follows the representation (2).

In recent times, Orthogonal polynomials have attracted extensive research attention across multiple fields, driven by their relevance to physics and mathematical statistics, engineering and probability theory. From a mathematical perspective, orthogonal polynomials frequently emerge as solutions to ordinary differential equations when specific conditions dictated by a particular model are applied. The classical orthogonal polynomials are those most commonly found in applications, including Chebyshev, Horadam, Gegenbauer, Jacobi and Fibonacci polynomials. For references focusing on the theory of geometric mapping with orthogonal polynomials, refer to [1,2,3,4].

Considerable attention is given in the literature to Chebyshev orthogonal polynomials, where, in recent years, numerous writers have explored the use of Chebyshev polynomials in the examination of analytic and bi-univalent mappings. For example, Altinkaya and Yalcin [5] determined coefficient limits for specific subclasses of bi-univalent mappings by employing Chebyshev polynomials and subsequently investigated the Chebyshev polynomial coefficient issue for univalent mappings in [6]. Bulut, Magesh, and Abirami [7] presented an extensive category of analytic bi-univalent mappings utilizing Chebyshev polynomials, whereas Bulut, Magesh, and Balaji [8] obtained preliminary coefficient estimates for these mappings. Dziok, Raina, and Sokól [9] employed Chebyshev polynomials to establish new categories of analytic functions and derived precise limits. Moreover, Güney [10] offered preliminary estimates for the bounds of Chebyshev polynomial coefficients in bi-univalent mappings.

The Chebyshev polynomials of the first and second kinds are defined as follows:

$$H_i\left(s\right)=cos\left(i\alpha \right),$$

(4)

$$T_i\left(s\right)={\displaystyle \frac{sin(i+1)\alpha }{sin\alpha }}.$$

(5)

respectively, with $s\in [-1,1]$, where $i=$ indicates the polynomial degree and $s=cos\alpha$. The polynomials given in (4) and (5) are related by the following expressions:

$${\displaystyle \frac{d{T}_i\left(s\right)}{ds}}=i{H}_{i-1}\left(s\right),T_i\left(s\right)=H_i\left(s\right)-s{H}_{i-1}\left(s\right),$$

(6)

$$2T_i\left(s\right)=H_i\left(s\right)-H_{i-2}\left(s\right).$$

(7)

Note that if $s=cos\alpha ,\alpha \in (-\pi /3,\pi /3)$, then

$$Q(\xi ,s)={\displaystyle \frac{1}{1-2s\xi +{\xi }^2}}$$

(8)

$$={\displaystyle \frac{1}{1-2cos\alpha \xi +{\xi }^2}}$$

$$=1+\sum _{i=0}^{\infty }{\displaystyle \frac{sin(i+1)\alpha }{sin\alpha }}{\xi }^i.$$

Thus,

$$Q(\xi ,s)=1+H_1\left(s\right)\xi +H_2\left(s\right){\xi }^2+\dots (s\in (-1,1),\xi \in \mathbb{U}),$$

$$1+2cos\alpha \xi +(3co{s}^2\alpha -si{n}^2\alpha ){\xi }^2+\dots ,$$

where

$$H_{i-1}\left(s\right)={\displaystyle \frac{sin(iarccoss)}{\sqrt{1-s^2}}},i\ge 1.$$

This represents the second type of Chebyshev polynomial. It is acknowledged that

$$H_i\left(s\right)=2s{H}_{i-1}\left(s\right)-H_{i-2}\left(s\right),$$

(9)

and

$$H_1\left(s\right)=2s$$

(10)

$$H_2\left(s\right)=4s^2-1$$

(11)

$$H_3\left(s\right)=8s^3-4s.$$

$$⋮$$

The standard generating mapping for Chebyshev polynomials $T_i\left(s\right),s\in [-1,1]$ of the first type takes the following shape.

$$\sum _{i=0}^{\infty }{T}_i\left(s\right){\xi }^i={\displaystyle \frac{1-s\xi }{1-2s\xi +{\xi }^2}}.$$

Assume that ${\Psi }_{\delta ,c}\left(\xi \right)$ is called the Miller–Ross mapping [11] which is defined by

$${\Psi }_{\delta ,c}\left(\xi \right)={\xi }^{\delta }\sum _{i=0}^{\infty }{\displaystyle \frac{{\left(c\xi \right)}^i}{\Gamma (i+\delta +1)}},(\delta ,c,\xi \in \mathbb{C}).$$

(12)

In addition, suppose that $L_{\sigma ,\eta }\left(\xi \right)$ is the Mittag–Leffler mapping of two parameters [12], defined by

$$L_{\sigma ,\eta }\left(\xi \right)=\sum _{i=0}^{\infty }{\displaystyle \frac{{\xi }^i}{\Gamma (\sigma i+\eta )}},(\eta ,\sigma ,\xi \in \mathbb{C},Re\left(\sigma \right)>0,Re\left(\eta \right)>0).$$

(13)

If put $\eta =1$, from Equation (13), we derive Mittag–Leffler mapping of single parameter [13]:

$$L_{\sigma }\left(\xi \right)=\sum _{i=0}^{\infty }{\displaystyle \frac{{\xi }^i}{\Gamma (\sigma i+1)}}.$$

(14)

Various characteristics of Mittag–Leffler mapping and its generalized versionare reported in [14,15,16,17].

From Equations (12) and (13), the mapping of Miller–Ross can be expressed as

$${\Psi }_{\delta ,c}\left(\xi \right)={\xi }^{\delta }{L}_{1,1+\delta }\left(c\xi \right).$$

Ref. [18] recently presented a power series with coefficients derived from a Miller–Ross-type Poisson distribution as follows:

$$J_{\delta ,c}^v\left(\xi \right)=\xi +\sum _{i=2}^{\infty }{\displaystyle \frac{v^{\delta }{\left(cv\right)}^{i-1}}{\Gamma (i+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{\xi }^i,$$

where $\delta >-1,c>0$.

At this point, we examine the linear operator ${\ell }_{\delta ,c}^v:\Lambda ⟶\Lambda$, defined via the Hadamard multiplication:

$${\ell }_{\delta ,c}^v\hslash \left(\xi \right)=J_{\delta ,c}^v\left(\xi \right)\ast \hslash \left(\xi \right)=\xi +\sum _{i=2}^{\infty }{\displaystyle \frac{v^{\delta }{\left(cv\right)}^{i-1}}{\Gamma (i+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_i{\xi }^i,(\xi \in \mathbb{U}).$$

(15)

We now present a new category of bi-Bazilevič mappings that incorporates the Poisson distribution linked to Chebyshev polynomials and derive bounds for the Taylor–Maclaurin coefficients $\mid q_2 \mid$ and $\mid q_3 \mid$, as well as for Fekete–Szegö functional issues related to mappings within this category.

Definition 1.

A mapping $\hslash \left(\xi \right)\in \Sigma$ given by (2) is considered part of the class of bi-Bazilevič mappings ${\aleph }_{\Sigma }(s,\gamma ,\beta )$ if the subsequent subordinations hold true:

$${\left({\displaystyle \frac{\xi {({\ell }_{\delta ,c}^v\hslash \left(\xi \right))}^{\prime }}{{\ell }_{\delta ,c}^v\hslash \left(\xi \right)}}\right)}^{\beta }{\left({\displaystyle \frac{{\ell }_{\delta ,c}^v\hslash \left(\xi \right)}{\xi }}\right)}^{\gamma }\prec Q(\xi ,s)$$

(16)

and

$${\left({\displaystyle \frac{d{({\ell }_{\delta ,c}^v\hslash \left(d\right))}^{\prime }}{{\ell }_{\delta ,c}^v\hslash \left(d\right)}}\right)}^{\beta }{\left({\displaystyle \frac{{\ell }_{\delta ,c}^v\hslash \left(d\right)}{d}}\right)}^{\gamma }\prec Q(d,s),$$

(17)

where $\beta ,\gamma ⩾0$ and $ģ={\hslash }^{-1}$ are defined in (3), and Q represents the Chebyshev polynomials that generate the mapping likewise expressed in(4).

By tailoring the parameters in class ${\aleph }_{\Sigma }(s,\gamma ,\beta )$, it serves as a broader version of several classes of Σ, as demonstrated in the examples below:

Example 1.

If $\gamma =0$ and $\beta =1$ in the class ${\aleph }_{\Sigma }(s,\gamma ,\beta )$, then we have $\aleph S_{\Sigma }\left(s\right)$, which refer to the class of mappings defined by (2) with the following conditions:

$${\displaystyle \frac{\xi {({\ell }_{\delta ,c}^v\hslash \left(\xi \right))}^{\prime }}{{\ell }_{\delta ,c}^v\hslash \left(\xi \right)}}\prec Q(\xi ,s)$$

and

$${\displaystyle \frac{d{({\ell }_{\delta ,c}^v\hslash \left(d\right))}^{\prime }}{{\ell }_{\delta ,c}^v\hslash \left(d\right)}}\prec Q(d,s),$$

where $ģ={\hslash }^{-1}$ is defined by (3), and Q represents the Chebyshev polynomials that generate the mapping likewise expressed in (4).

Example 2.

If $\gamma =1,$ and $\beta =0$ in the class ${\aleph }_{\Sigma }(s,\gamma ,\beta )$, then we have ${\aleph }_{\Sigma }\left(s\right)$, which refers to the class of mappings defined by (2), with the following conditions:

$${\displaystyle \frac{{\ell }_{\delta ,c}^v\hslash \left(\xi \right)}{\xi }}\prec Q(\xi ,s)$$

and

$${\displaystyle \frac{{\ell }_{\delta ,c}^v\hslash \left(d\right)}{d}}\prec Q(d,s),$$

where $ģ={\hslash }^{-1}$ is given by (3), and Q represents the Chebyshev polynomials that generate the mapping likewise expressed in (4).

Example 3.

If $\beta =1$ is in the class ${\aleph }_{\Sigma }(s,\gamma ,\beta )$, then ${\aleph }_{\Sigma }(s,\gamma )$ refers to the class of mappings defined by (2) with the following conditions:

$$\left({\displaystyle \frac{\xi {({\ell }_{\delta ,c}^v\hslash \left(\xi \right))}^{\prime }}{{\ell }_{\delta ,c}^v\hslash \left(\xi \right)}}\right){\left({\displaystyle \frac{{\ell }_{\delta ,c}^v\hslash \left(\xi \right)}{\xi }}\right)}^{\gamma }\prec Q(\xi ,s)$$

and

$$\left({\displaystyle \frac{d{({\ell }_{\delta ,c}^v\hslash \left(d\right))}^{\prime }}{{\ell }_{\delta ,c}^v\hslash \left(d\right)}}\right){\left({\displaystyle \frac{{\ell }_{\delta ,c}^v\hslash \left(d\right)}{d}}\right)}^{\gamma }\prec Q(d,s),$$

where $\gamma ⩾0$, the mapping $ģ={\hslash }^{-1}$ satisfies the representation (3), and Q represents the Chebyshev polynomials that generate the mapping likewise expressed in (4).

2. Main Results

The following theorem provides sharp bounds for the second and third coefficients of functions in the class ${\aleph }_{\Sigma }(s,\gamma ,\beta )$; these bounds are expressed explicitly in terms of parameters $s,\gamma$ and $\beta$, quantifying how the subordination condition restricts the initial coefficient.

Theorem 1.

If $\hslash \left(\xi \right)\in \Sigma$ satisfies the representation (2) and as an element of the class ${\aleph }_{\Sigma }(s,\gamma ,\beta )$, then

$$|q_2|\le {\displaystyle \frac{2s\sqrt{2s}{\Psi }_{\delta ,c}\left(v\right)\Gamma (2+\delta )}{\sqrt{|4s^2{I}_{\delta ,c}(v,\gamma ,\beta )+s{K}_{\delta ,c}\left(v\right)+{(\gamma +\beta )}^2{v}^{\delta }|c^2{v}^{\delta +2}}}}$$

and

$$|q_3|\le {\displaystyle \frac{4s^2{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}{c^2{(\gamma +\beta )}^2{v}^{2\delta +2}}}+{\displaystyle \frac{2s\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}{c^2(\gamma +2\beta )v^{\delta +2}}},$$

where

$$I_{\delta ,c}(v,\gamma ,\beta )={\displaystyle \frac{(\gamma +2\beta )}{\Gamma (3+\delta )}}{\Psi }_{\delta ,c}\left(v\right){(\Gamma (2+\delta ))}^2-{(\gamma +\beta )}^2{v}^{\delta }$$

and

$$K_{\delta ,c}\left(v\right)=(\beta (\beta -3)+\gamma (\gamma -1)+2\beta \gamma )v^{\delta }.$$

Proof. 

Let $\hslash \left(\xi \right)\in {\aleph }_{\Sigma }(s,\gamma ,\beta )$. Then, in accordance with Definition 1, for specific analytic mappings d and k, where $d\left(0\right)=k\left(0\right)=0$ and $\mid d\left(\xi \right)\mid <1,\mid k\left(d\right)\mid <1$ for every $\xi ,d\in \mathbb{U}$, we can express the following:

$${({\displaystyle \frac{\xi {({\ell }_{\delta ,c}^v\hslash (\xi ))}^{\prime }}{{\ell }_{\delta ,c}^v\hslash (\xi )}})}^{\beta }{({\displaystyle \frac{{\ell }_{\delta ,c}^v\hslash (\xi )}{\xi }})}^{\gamma }=Q(d(\xi ),s)$$

(18)

and

$${\left({\displaystyle \frac{d{({\ell }_{\delta ,c}^v\hslash (d))}^{\prime }}{{\ell }_{\delta ,c}^v\hslash (d)}}\right)}^{\beta }{\left({\displaystyle \frac{{\ell }_{\delta ,c}^v\hslash (d)}{d}}\right)}^{\gamma }=Q(k(d),s).$$

(19)

It is quite recognized that if

$$\mid d\left(\xi \right)\mid =\mid m_1\xi +m_2{\xi }^2+m_3{\xi }^3+\dots \mid <1,(\xi \in \mathbb{U})$$

and

$$\mid k\left(d\right)\mid =\mid t_{1}d+t_2{d}^2+t_3{d}^3+\dots \mid <1,(d\in \mathbb{U}),$$

then

$$\mid m_n\mid \le 1,\mid t_n\mid \le 1,(\forall n\in \mathbb{N}).$$

(20)

From Equations (18) and (19), after expanding the left- hand side using the binomial expansion and simplifying both sides of the two equations, we obtain:

$$\left[1+{\displaystyle \frac{\beta c{v}^{\delta +1}}{\Gamma (2+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_2\xi +\left({\displaystyle \frac{2\beta c^2{v}^{\delta +2}}{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_3+{\displaystyle \frac{\beta (\beta -3)c^2{v}^{2\delta +2}}{2{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}}{q}_2^2\right){\xi }_2^2+\dots \right]\ast$$

$$\left[1+{\displaystyle \frac{\gamma c{v}^{\delta +1}}{\Gamma (2+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_2\xi +\left({\displaystyle \frac{\gamma c^2{v}^{\delta +2}}{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_3+{\displaystyle \frac{\gamma (\gamma -1)c^2{v}^{2\delta +2}}{2{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}}{q}_2^2\right){\xi }_2^2+\dots \right]$$

$$=1+H_1\left(s\right)m_1\xi +[H_1\left(s\right)m_2+H_2\left(s\right)m_1^2]{\xi }^2+\dots$$

(21)

and

$$\left[1+{\displaystyle \frac{\beta c{v}^{\delta +1}}{\Gamma (2+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_{2}d+\left({\displaystyle \frac{2\beta c^2{v}^{\delta +2}}{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_3+{\displaystyle \frac{\beta (\beta -3)c^2{v}^{2\delta +2}}{2{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}}{q}_2^2\right)d_2^2+\dots \right]\ast$$

$$\left[1+{\displaystyle \frac{\gamma c{v}^{\delta +1}}{\Gamma (2+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_{2}d+\left({\displaystyle \frac{\gamma c^2{v}^{\delta +2}}{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_3+{\displaystyle \frac{\gamma (\gamma -1)c^2{v}^{2\delta +2}}{2{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}}{q}_2^2\right)d_2^2+\dots \right]$$

$$=1+H_1\left(s\right)t_{1}d+[H_1\left(s\right)t_2+H_2\left(s\right)t_1^2]d^2+\dots$$

(22)

Consequently, when we examine the related coefficients in (21) and (22), we find

$${\displaystyle \frac{(\gamma +\beta )c{v}^{\delta +1}}{\Gamma (2+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_2=H_1\left(s\right)m_1,$$

(23)

$${\displaystyle \frac{(\gamma +2\beta )c^2{v}^{\delta +2}}{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_3+{\displaystyle \frac{[\beta (\beta -3)+\gamma (\gamma -1)+2\beta \gamma ]c^2{v}^{2\delta +2}}{2{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}}{q}_2^2=H_1\left(s\right)m_2+H_2\left(s\right)m_1^2.$$

(24)

$$-{\displaystyle \frac{(\gamma +\beta )c{v}^{\delta +1}}{\Gamma (2+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_2=H_1\left(s\right)t_1,$$

(25)

$${\displaystyle \frac{(\gamma +2\beta )c^2{v}^{\delta +2}}{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}}[2q_2^2-q_3]+{\displaystyle \frac{[\beta (\beta -3)+\gamma (\gamma -1)+2\beta \gamma ]c^2{v}^{2\delta +2}}{2{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}}{q}_2^2=H_1\left(s\right)t_2+H_2\left(s\right)t_1^2.$$

(26)

From adding the two Equations (23) and (25), it can be concluded that we have

$$m_1=-t_1$$

(27)

and

$${\displaystyle \frac{2{(\gamma +\beta )}^2{c}^2{v}^{2\delta +2}}{{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}}{q}_2^2={\left[H_1\left(s\right)\right]}^2(m_1^2+t_1^2).$$

(28)

By adding Equation (24) to Equation (26), we obtain

$${\displaystyle \frac{2(\gamma +2\beta )c^2{v}^{\delta +2}}{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}}{q}_2^2+{\displaystyle \frac{[\beta (\beta -3)+\gamma (\gamma -1)+2\beta \gamma ]c^2{v}^{2\delta +2}}{{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}}{q}_2^2=H_1\left(s\right)(m_2+t_2)+H_2\left(s\right)(m_1^2+t_1^2).$$

(29)

By replacing the value of $(m_1^2+t_1^2)$ from (28) into the right side of (29), we conclude that

$$\left[{\displaystyle \frac{2(\gamma +2\beta )}{\Gamma (3+\delta )}}+{\displaystyle \frac{\left([\beta (\beta -3)+\gamma (\gamma -1)+2\beta \gamma ]{\left(H_1\left(s\right)\right)}^2-2{(\gamma +\beta )}^2{H}_2\left(s\right)\right)v^{\delta }}{{(\Gamma (2+\delta ))}^2{\Psi }_{\delta ,c}\left(v\right){\left(H_1\left(s\right)\right)}^2}}\right]{\displaystyle \frac{c^2{v}^{\delta +2}}{{\Psi }_{\delta ,c}\left(v\right)}}{q}_2^2$$

$$=H_1\left(s\right)(m_2+t_2).$$

(30)

$$q_2^2={\displaystyle \frac{H_1\left(s\right)(m_2+t_2){\Psi }_{\delta ,c}\left(v\right)}{\left[{\displaystyle \frac{2(\gamma +2\beta )}{\Gamma (3+\delta )}}+{\displaystyle \frac{\left([\beta (\beta -3)+\gamma (\gamma -1)+2\beta \gamma ]{\left(H_1\left(s\right)\right)}^2-2{(\gamma +\beta )}^2{H}_2\left(s\right)\right)v^{\delta }}{{(\Gamma (2+\delta ))}^2{\Psi }_{\delta ,c}\left(v\right){\left(H_1\left(s\right)\right)}^2}}\right]c^2{v}^{\delta +2}}}$$

(31)

Additionally, by applying computations (10), (11), (20), and (31), we discover that

$$q_2^2={\displaystyle \frac{{\left(H_1\left(s\right)\right)}^3(m_2+t_2){\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2{(\Gamma (2+\delta ))}^2}{{\displaystyle \frac{{\left(H_1\left(s\right)\right)}^2{\Psi }_{\delta ,c}\left(v\right){(\Gamma (2+\delta ))}^2}{{\displaystyle \frac{\Gamma (3+\delta )}{2(\gamma +2\beta )c^2{v}^{\delta +2}}}}}+([\beta (\beta -3)+\gamma (\gamma -1)+2\beta \gamma ]H_1\left(s\right)-2{(\gamma +\beta )}^2{H}_2\left(s\right))c^2{v}^{2\delta +2}}}.$$

(32)

$$|q_2|\le {\displaystyle \frac{2s\sqrt{2s}{\Psi }_{\delta ,c}\left(v\right)\Gamma (2+\delta )}{\sqrt{|4s^2{I}_{\delta ,c}(v,\gamma ,\beta )+s{K}_{\delta ,c}\left(v\right)+{(\gamma +\beta )}^2{v}^{\delta }|c^2{v}^{\delta +2}}}}.$$

By subtracting Equation (26) from Equation (24)

$$2\left[{\displaystyle \frac{(\gamma +2\beta )c^2{v}^{\delta +2}}{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}}\right](q_3-q_2^2)=H_1\left(s\right)(m_2-t_2)+H_2\left(s\right)(m_1^2-t_1^2).$$

(33)

Considering Equations (27) and (28), Equation (33) becomes

$$q_3={\displaystyle \frac{{\left(H_1\left(s\right)\right)}^2(m_1^2+t_1^2){(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}{2c^2{(\gamma +\beta )}^2{v}^{2\delta +2}}}+{\displaystyle \frac{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)H_1\left(s\right)(m_2-t_2)}{2c^2(\gamma +2\beta )v^{\delta +2}}}.$$

Thus, applying (10), we obtain

$$|q_3|\le {\displaystyle \frac{4s^2{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}{c^2{(\gamma +\beta )}^2{v}^{2\delta +2}}}+{\displaystyle \frac{2s\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}{c^2(\gamma +2\beta )v^{\delta +2}}}.$$

This completes the proof. □

Theorem 2.

If $\hslash \left(\xi \right)\in \Sigma$ satisfies the representation (2) and as an element of the class ${\aleph }_{\Sigma }(s,\gamma ,\beta )$, then

$$|q_3-\vartheta q_2^2|\le \left[\begin{array}{cc}{\displaystyle \frac{2s{\Psi }_{\delta ,c}\left(v\right)\Gamma (3+\delta )}{(\gamma +2\beta )c^2{v}^{\delta +2}}}& \mid \vartheta -1\mid \le \rho \\ & \\ {\displaystyle \frac{8s^3{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2{(\Gamma (3+\delta ))}^2(1-\vartheta )}{[4s^2{I}_{\delta ,c}(v,\gamma ,\beta )+s{K}_{\delta +c}\left(v\right)+{(\gamma +\beta )}^2{v}^{\delta }]c^2{v}^{2\delta +2}}}& \mid \vartheta -1\mid \ge \rho ,\end{array}\right]$$

where

$$\rho =|1-{\displaystyle \frac{\Gamma (3+\delta ){(\gamma +\beta )}^2(4s^2-1)v^{\delta }-s[\beta (\beta -3)+\gamma (\gamma -1)+2\beta \gamma ]v^{\delta }\Gamma (3+\delta )}{4s^2(\gamma +2\beta ){\Psi }_{\delta ,c}\left(v\right){(\Gamma (2+\delta ))}^2}}|.$$

Proof. 

By utilizing Equations (27) and (33) we obtain

$$q_3=q_2^2+{\displaystyle \frac{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)H_1\left(s\right)}{2(\gamma +2\beta )c^2{v}^{\delta +2}}}(m_2-t_2).$$

(34)

Now,

$$q_3-\vartheta q_2^2=(1-\vartheta )q_2^2+{\displaystyle \frac{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)H_1\left(s\right)}{2(\gamma +2\beta )c^2{v}^{\delta +2}}}(m_2-t_2).$$

By using the substitution in (32), we obtain

$$={\displaystyle \frac{(1-\vartheta ){(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2{\left(H_1\left(s\right)\right)}^3(m_2+t_2)}{\left[{\displaystyle \frac{2(\gamma +2\beta )}{\Gamma (3+\delta )}}{\Psi }_{\delta ,c}\left(v\right){\left(H_1\left(s\right)\right)}^2{(\Gamma (2+\delta ))}^2+k_{\delta ,c}\left(v\right)H_1\left(s\right)v^{\delta }-2{(\gamma +\beta )}^2{H}_2\left(s\right)v^{\delta }\right]c^2{v}^{\delta +2}}}$$

$$+{\displaystyle \frac{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)H_1\left(s\right)}{2(\gamma +2\beta )c^2{v}^{\delta +2}}}(m_2-t_2)$$

$$=H_1\left(s\right)\left[R\left(\vartheta \right)+{\displaystyle \frac{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}{2(\gamma +2\beta )c^2{v}^{\delta +2}}}\right]m_2+H_1\left(s\right)\left[R\left(\vartheta \right)-{\displaystyle \frac{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}{2(\gamma +2\beta )c^2{v}^{\delta +2}}}\right]t_2,$$

where $R\left(\vartheta \right)=$

$$={\displaystyle \frac{(1-\vartheta ){(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2{\left(H_1\left(s\right)\right)}^2(m_2+t_2)}{\left[{\displaystyle \frac{2(\gamma +2\beta )}{\Gamma (3+\delta )}}{\Psi }_{\delta ,c}\left(v\right){\left(H_1\left(s\right)\right)}^2{(\Gamma (2+\delta ))}^2+k_{\delta ,c}\left(v\right)H_1\left(s\right)v^{\delta }-2{(\gamma +\beta )}^2{H}_2\left(s\right)v^{\delta }\right]c^2{v}^{\delta +2}}}.$$

From (10) and (11), considering that $H_1\left(s\right)=2s$ and $H_2\left(s\right)=4s^2-1$, we can deduce that

$|q_3-\vartheta q_2^2|\le \left[\begin{array}{cc}{\displaystyle \frac{{\Psi }_{\delta ,c}\left(v\right)\Gamma (3+\delta )|H_1\left(s\right)|}{(\gamma +2\beta )c^2{v}^{\delta +2}}}& 0⩽\mid R\left(\vartheta \right)\mid \le {\displaystyle \frac{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}{2(\gamma +2\beta )c^2{v}^{\delta +2}}},\\ & \\ 2\mid H_1\left(s\right)\mid \mid R\left(\vartheta \right)\mid & \mid R\left(\vartheta \right)\mid ⩾{\displaystyle \frac{\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}{2(\gamma +2\beta )c^2{v}^{\delta +2}}}.\end{array}\right]$

This completes the proof. □

Corresponding primarily to Examples 1 and 2, Theorems 1 and 2 provide the ensuing corollaries.

Corollary 1.

If $\hslash \left(\xi \right)\in \Sigma$ satisfies the representation (2) and as an element of the class $\aleph S_{\Sigma }\left(s\right)$, then

$$|q_2|\le {\displaystyle \frac{2s\sqrt{2s}{\Psi }_{\delta ,c}\left(v\right)\Gamma (2+\delta )}{\sqrt{|2s^2{I}_{\delta ,c}(v,0,1)-2s{v}^{\delta }+v^{\delta }|c^2{v}^{\delta +2}}}},$$

$$|q_3|\le {\displaystyle \frac{4s^2{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}{c^2{v}^{2\delta +2}}}+{\displaystyle \frac{2s\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}{2c^2{v}^{\delta +2}}},$$

where

$$I_{\delta ,c}(v,0,1)={\displaystyle \frac{2}{\Gamma (3+\delta )}}{\Psi }_{\delta ,c}\left(v\right){(\Gamma (2+\delta ))}^2-v^{\delta },$$

and

$$|q_3-\vartheta q_2^2|\le \left[\begin{array}{cc}{\displaystyle \frac{2s{\Psi }_{\delta ,c}\left(v\right)\Gamma (3+\delta )}{2c^2{v}^{\delta +2}}}& \mid \vartheta -1\mid \le \rho \\ & \\ {\displaystyle \frac{8s^3{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2{(\Gamma (3+\delta ))}^2(1-\vartheta )}{[2s^2{I}_{\delta ,c}(v,0,1)-2s{v}^{\delta }+v^{\delta }]c^2{v}^{2\delta +2}}}& \mid \vartheta -1\mid \ge \rho ,\end{array}\right]$$

where

$$\rho =|1-{\displaystyle \frac{\Gamma (3+\delta )(4s^2-1)v^{\delta }+2s{v}^{\delta }\Gamma (3+\delta )}{8s^2{\Psi }_{\delta ,c}\left(v\right){(\Gamma (2+\delta ))}^2)}}|.$$

Corollary 2.

If $\hslash \left(\xi \right)\in \Sigma$ satisfies the representation (2) and as an element of the class ${\aleph }_{\Sigma }\left(s\right)$, then

$$|q_2|\le {\displaystyle \frac{2s\sqrt{2s}{\Psi }_{\delta ,c}\left(v\right)\Gamma (2+\delta )}{\sqrt{|4s^2{I}_{\delta ,c}(v,1,0)+v^{\delta }|c^2{v}^{\delta +2}}}}$$

$$|q_3|\le {\displaystyle \frac{4s^2{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}{c^2{v}^{2\delta +2}}}+{\displaystyle \frac{2s\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}{c^2{v}^{\delta +2}}},$$

where

$$I_{\delta ,c}(v,1,0)={\displaystyle \frac{1}{\Gamma (3+\delta )}}{\Psi }_{\delta ,c}\left(v\right){(\Gamma (2+\delta ))}^2-v^{\delta }$$

and

$$|q_3-\vartheta q_2^2|\le \left[\begin{array}{cc}{\displaystyle \frac{2s{\Psi }_{\delta ,c}\left(v\right)\Gamma (3+\delta )}{c^2{v}^{\delta +2}}}& \mid \vartheta -1\mid \le \rho \\ & \\ {\displaystyle \frac{8s^3{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2{(\Gamma (3+\delta ))}^2(1-\vartheta )}{[4s^2{I}_{\delta ,c}(v,1,0)+v^{\delta }]c^2{v}^{2\delta +2}}}& \mid \vartheta -1\mid \ge \rho ,\end{array}\right]$$

where

$$\rho =|1-{\displaystyle \frac{\Gamma (3+\delta )(4s^2-1)v^{\delta }}{4s^2{\Psi }_{\delta ,c}\left(v\right){(\Gamma (2+\delta ))}^2}}|.$$

Corollary 3.

If $\hslash \left(\xi \right)\in \Sigma$ satisfies the representation (2) and as an element of the class ${\aleph }_{\Sigma }(s,\gamma )$, then

$$|q_2|\le {\displaystyle \frac{2s\sqrt{2s}{\Psi }_{\delta ,c}\left(v\right)\Gamma (2+\delta )}{\sqrt{|4s^2{I}_{\delta ,c}(v,\gamma ,1)+s{K}_{\delta ,c}\left(v\right)+{(\gamma +1)}^2{v}^{\delta }|c^2{v}^{\delta +2}}}}$$

$$|q_3|\le {\displaystyle \frac{4s^2{(\Gamma (2+\delta ))}^2{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2}{c^2{(\gamma +1)}^2{v}^{2\delta +2}}}+{\displaystyle \frac{2s\Gamma (3+\delta ){\Psi }_{\delta ,c}\left(v\right)}{c^2(\gamma +2)v^{\delta +2}}},$$

where

$$I_{\delta ,c}(v,\gamma ,1)={\displaystyle \frac{(\gamma +2)}{\Gamma (3+\delta )}}{\Psi }_{\delta ,c}\left(v\right){(\Gamma (2+\delta ))}^2-{(\gamma +1)}^2{v}^{\delta },$$

$$K_{\delta ,c}\left(v\right)=({\gamma }^2+\gamma -2)v^{\delta },$$

and

$$|q_3-\vartheta q_2^2|\le \left[\begin{array}{cc}{\displaystyle \frac{2s{\Psi }_{\delta ,c}\left(v\right)\Gamma (3+\delta )}{(\gamma +2)c^2{v}^{\delta +2}}}& \mid \vartheta -1\mid \le \rho \\ & \\ {\displaystyle \frac{8s^3{\left({\Psi }_{\delta ,c}\left(v\right)\right)}^2{(\Gamma (3+\delta ))}^2(1-\vartheta )}{[4s^2{I}_{\delta ,c}(v,\gamma ,1)+s{K}_{\delta +c}\left(v\right)+{(\gamma +1)}^2{v}^{\delta }]c^2{v}^{2\delta +2}}}& \mid \vartheta -1\mid \ge \rho ,\end{array}\right]$$

where

$$\rho =|1-{\displaystyle \frac{\Gamma (3+\delta ){(\gamma +1)}^2(4s^2-1)v^{\delta }-s{K}_{\delta ,c}\left(v\right)\Gamma (3+\delta )}{4s^2(\gamma +2){\Psi }_{\delta ,c}\left(v\right){(\Gamma (2+\delta ))}^2}}|.$$

Remark 1.

The present study generalizes the class of functions introduced in [2] in several ways. First, our definition of the bi-Bazilevič subclass reduces exactly to that of [2] when the Miller–Ross-type Poisson distribution is omitted. Second, all coefficient bounds and inequality of Fekete–Szegő obtained in [2] are recovered as special instance of our more general results (Theorems 1 and 2) by fixing the additional parameters to suitable values. Moreover, the incorporation of the Poisson distribution series of the Miller–Ross-type allows us to extend the analysis further to wider families, including starlike, bi-starlike, and bi-Bazilevič functions, which were not covered in [2]. Thus, the work in [2] can be viewed as a particular instance within the broader framework established here.

3. Conclusions

This study successfully introduced and explored a new class of analytic mappings, namely the bi-Bazilevič mappings related to Chebyshev polynomials and the Miller–Ross-type Poisson distribution series. This research has made important advances in geometric mapping theory by broadening the traditional structure of Bazilevič mappings with the addition of probabilistic distribution series and orthogonal polynomials. These bounds generalize the corresponding estimates in [2], which are recovered as special cases when the Miller–Ross-type Poisson distribution is omitted. The key findings of this research can be outlined as follows. Initially, precise evaluations of the initial Taylor–Maclaurin coefficients $\mid q_2\mid$ and $\mid q_3\mid$ were obtained for functions belonging to the newly defined class ${\aleph }_{\Sigma }(s,\gamma ,\beta )$, together with the Fekete–Szegő inequality. Moreover, the classical Fekete–Szegő inequality was extended to this category, enhancing the comprehension of the distortion characteristics of these mappings. Third, the integration of Chebyshev polynomials with Miller–Ross-type Poisson series was shown to be a productive framework for creating new subclasses of analytic functions, highlighting a successful interaction between orthogonal polynomials and probabilistic techniques. Furthermore, the proposed framework extends naturally to starlike, bistarlike, and biBazilevič functions, which were not covered in previous studies.

An evident path for upcoming research is to substitute Chebyshev polynomials with Gegenbauer polynomials, which extend Chebyshev and Legendre polynomials, thus facilitating the exploration of wider subclasses of bi-Bazilevič functions and the formulation of related coefficient estimates and the Fekete–Szegő inequality.