Bell Distribution Series Defined on Subclasses of Bi-Univalent Functions That Are Subordinate to Horadam Polynomials

Abstract

Several different subclasses of the bi-univalent function class $\mathsf{\Sigma }$ were introduced and studied by many authors using distribution series like Pascal distribution, Poisson distribution, Borel distribution, the Mittag-Leffler-type Borel distribution, Miller–Ross-Type Poisson Distribution. In the present paper, by making use of the Bell distribution, we introduce and investigate a new family ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\beta ,\gamma )$ of normalized bi-univalent functions in the open unit disk $\mathfrak{U}$, which are associated with the Horadam polynomials and estimate the second and the third coefficients in the Taylor-Maclaurin expansions of functions belonging to this class. Furthermore, we establish the Fekete–Szegö inequality for functions in the family ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\beta ,\gamma ).$ After specializing the parameters used in our main results, a number of new results are demonstrated to follow.

1. Introduction and Preliminaries

Orthogonal polynomials [1] are commonly employed in mathematical model solving to find solutions to ordinary differential equations that satisfy model requirements. Orthogonal polynomials are important for contemporary mathematics and have a wide range of uses in physics and engineering. It is common knowledge that these polynomials play a key role in approximation theory-related concerns. They can be found in differential equation theory, mathematical statistics, interpolation, approximation theory, probability theory, and quantum mechanics. They are also used in signal processing, image processing, and data analysis, where they are used to model and analyze complex systems and data sets. Their applications to automated control, quantum physics, signal analysis, scattering theory, and axially symmetric potential theory are also widely known [2,3].

Two polynomials ${\mathbb{S}}_{\rho }$ and ${\mathbb{S}}_{\sigma }$, of order $\rho$ and $\sigma ,$ respectively, are orthogonal if

$$\langle {\mathbb{S}}_{\rho },{\mathbb{S}}_{\sigma }\rangle ={\int }_c^d{\mathbb{S}}_{\rho }\left(x\right){\mathbb{S}}_{\sigma }\left(x\right)r\left(x\right)dx=0,\mathrm{for}\rho \ne \sigma ,$$

(1)

where $r\left(x\right)$ is a non-negative function in the interval $(c,d)$; therefore, all finite order polynomials ${\mathbb{S}}_{\rho }\left(x\right)$ have a well-defined integral.

Examples of well-known families of orthogonal polynomials include the Legendre polynomials, Hermite polynomials, Chebyshev polynomials, Jacobi polynomials, and Laguerre polynomials. Each family of orthogonal polynomials has its own weight function and interval, and they have many useful properties and applications.

Horadam polynomials are a family of polynomials defined by recurrence relations that generalize the Fibonacci and Lucas polynomials. They are named after Australian mathematician Murray S. Klamkin Horadam who introduced them in 1978.

Like the Fibonacci and Lucas polynomials, the Horadam polynomials have many interesting properties and connections to other areas of mathematics, including number theory, combinatorics, and algebraic geometry. They also satisfy various recurrence relations and identities, which can be used to derive closed-form expressions and study their properties.

Horadam polynomials have applications in various fields of science, including physics, engineering, and computer science. They have been used, for example, in modeling the behavior of certain physical systems, analyzing algorithms, and designing error-correcting codes.

The Horadam polynomials $h_n\left(x\right)$, which are provided by the recurrence relation as follows, were studied by Horzum and Kocer in 2009 [4].

$$h_n\left(x\right)=px{h}_{n-1}\left(x\right)+q{h}_{n-2}\left(x\right),(n\in \mathbb{N}\backslash \left\{1,2\right\}),$$

(2)

with

$$h_1\left(x\right)=a,h_2\left(x\right)=tx\mathrm{and}{h}_3\left(x\right)=pt{x}^2+aq,$$

(3)

for some real constant $a,$ t, p and $q.$

Remark 1.

For particular values of a, t, p and q, the Horadam polynomials $h_n\left(x\right)$ lead to various polynomials (see [4,5]), for example:

• If $a=t=p=q=1$, then we get the Fibonacci polynomials $F_n\left(x\right)$;
• If $a=2$ and $t=p=q=1$, then we get the Lucas polynomials $L_n\left(x\right)$;
• If $a=t=1$, $p=2$ and $q=-1$, then we get the Chebyshev polynomials $T_n\left(x\right)$ of the first kind;
• If $a=1$, $t=p=2$ and $q=-1$, then we get the Chebyshev polynomials $U_n\left(x\right)$ of the second kind;
• If $a=q=1$ and $t=p=2$, then we get the Pell polynomials $P_n\left(x\right)$;
• If $a=$$t=p=2$ and $q=1,$ then we get the Pell-Lucas polynomials $Q_n\left(x\right)$ of the first kind.

Numerous fields in the mathematical, physical, statistical, and engineering sciences depend heavily on the Fibonacci, Lucas, Chebyshev, and families of orthogonal polynomials and other special polynomials as well as their generalizations. Numerous articles have examined these kinds of polynomials from a theoretical standpoint.

The generator of the Horadam polynomials $h_n\left(x\right)$ is as follows:

$$\mathsf{\Omega }(x,\xi )={\displaystyle \sum }_{n=1}^{\infty }{h}_n\left(x\right){\xi }^{n-1}=\frac{a+(t-ap)x\xi }{1-px\xi -q{\xi }^2}.$$

(4)

Let $\mathcal{A}$ be the class of functions f of the form

$$f\left(\xi \right)=\xi +a_2{\xi }^2+a_3{\xi }^3+\cdots ,$$

(5)

that are analytic in the disk $\mathfrak{U}=\{\xi :\left|\xi \right|<1\}$. Also, we represent by $\mathcal{S}$ the subclass of $\mathcal{A}$ comprising functions of the Equation (5) which are also univalent in $\mathfrak{U}$.

The subordination of analytic functions f and g is denoted by $f\prec g$ if, for all $\xi \in \mathfrak{U}$, there exists a Schwarz function $\varpi$ with $\varpi \left(0\right)=0$ and $\left|\varpi \right(\xi \left)\right|<1$, such that

$$f\left(\xi \right)=g\left(\varpi \right(\xi \left)\right).$$

Moreover, if g is univalent in $\mathfrak{U}$, then

$$f\left(\xi \right)\prec g\left(\xi \right),\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if},f\left(0\right)=g\left(0\right)$$

and

$$f\left(\mathfrak{U}\right)\subset g\left(\mathfrak{U}\right).$$

According to the Koebe one-quarter theorem [6,7], every function $f\in \mathcal{S}$ has an inverse $f^{-1}$ defined by

$$f^{-1}\left(f\left(\xi \right)\right)=\xi (\xi \in \mathfrak{U})$$

and

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

where

$$g\left(w\right)=f^{-1}\left(w\right)=w-a_2{w}^2+(-a_3+2a_2^2)w^3-(a_4+5a_2^3-5a_3{a}_2)w^4+\cdots .$$

(6)

A function $f\in \mathcal{S}$ is said to be bi-univalent in $\mathfrak{U}$ if both $f\left(\xi \right)$ and $f^{-1}\left(\xi \right)$ are univalent in $\mathfrak{U}$.

Let $\mathsf{\Sigma }$ denote the class of bi-univalent functions in $\mathfrak{U}$ given by (5). Examples in the class $\mathsf{\Sigma }$ are

$$f_1\left(\xi \right)=\frac{\xi }{1-\xi },f_2\left(\xi \right)=log\frac{1}{1-\xi },$$

and their inverses,

$$f_1^{-1}\left(w\right)=\frac{w}{1+w},f_2^{-1}\left(w\right)=\frac{e^w-1}{e^w}$$

are in the class $\mathsf{\Sigma }$.

However, $\mathsf{\Sigma }$ does not include the well-known Koebe function. Additional typical instances of functions in $\mathfrak{U}$ include

$$\frac{2\xi -{\xi }^2}{2}\mathrm{and}\frac{\xi }{1-{\xi }^2}$$

are also not members of $\mathsf{\Sigma }$. For interesting subclasses of functions in the class $\mathsf{\Sigma }$, see ([8,9,10]).

Brannan and Taha [11] (see also [12]) introduced certain subclasses of the bi-univalent function class $\mathsf{\Sigma }$ similar to the familiar subclasses ${\mathcal{S}}^{*}\left(\alpha \right)$ and $\mathcal{K}\left(\alpha \right)$ of starlike and convex functions of order $\alpha (0\le \alpha <1),$ respectively (see [13]). Thus, following Brannan and Taha [11] (see also [12]), a function $f\in \mathcal{A}$ is in the class ${\mathcal{S}}_{\mathsf{\Sigma }}^{*}\left[\alpha \right]$ of strongly bi-starlike functions of order $\alpha (0<\alpha \le 1)$ if each of the following conditions is satisfied:

$$f\in \mathsf{\Sigma }\mathrm{and}\left|arg\left(\frac{\xi f^{\prime }\left(\xi \right)}{f\left(\xi \right)}\right)\right|<\frac{\alpha \pi }{2}(0<\alpha \le 1,\xi \in \mathfrak{U})$$

and

$$\left|arg\left(\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}\right)\right|<\frac{\alpha \pi }{2}(0<\alpha \le 1,w\in \mathfrak{U}),$$

where g is the extension of $f^{-1}$ to $\mathfrak{U}$. The classes ${\mathcal{S}}_{\mathsf{\Sigma }}^{*}\left(\alpha \right)$ and ${\mathcal{K}}_{\mathsf{\Sigma }}\left(\alpha \right)$ of bi-starlike functions of order $\alpha$ and bi-convex functions of order $\alpha$, corresponding (respectively) to the function classes ${\mathcal{S}}^{*}\left(\alpha \right)$ and $\mathcal{K}\left(\alpha \right)$, were also introduced analogously. For each of the function classes ${\mathcal{S}}_{\mathsf{\Sigma }}^{*}\left(\alpha \right)$ and ${\mathcal{K}}_{\mathsf{\Sigma }}\left(\alpha \right)$, they found non-sharp estimates on the first two Taylor–Maclaurin coefficients $|a_2|$ and $|a_3|$ (for details, see [11,12]). However, the coefficient problem for each of the succeeding Taylor–Maclaurin coefficients,

$$|a_n|\left(n\in \mathbb{N}\backslash \{1,2\}\right)$$

is still an open problem (see [11,12,13,14]).

Several subclasses of the bi-univalent function class $\mathsf{\Sigma }$ were introduced, inspired by the ground-breaking work of Srivastava et al. [15], and non-sharp estimates on the first two coefficients $\left|a_2\right|$ and $\left|a_3\right|$ in the Taylor-Maclaurin series expansion (5) were obtained in ([16,17,18,19,20,21,22,23,24,25,26,27,28]).

Fekete and Szegö [29] proved that the estimate

$$\left|a_3-\eta a_2^2\right|\le 1+2e^{\left(\frac{-2\eta }{1-\eta }\right)}$$

holds for any normalized univalent function f and $\eta \in \left[0,1\right].$ This inequality is sharp for each $\eta$ (see, [29]). Recently, many authors have obtained Fekete–Szegö inequalities for different classes of functions (see [30,31,32,33,34]).

In recent years, several studies have looked at crucial aspects of the geometric function theory including coefficient estimates, inclusion relations, and requirements for belonging to certain classes, using a variety of probability distributions, including the Poisson, Pascal, Borel, Mittag-Leffler-type Poisson distribution, etc. (see, [35,36,37,38,39,40]).

The Bell distribution, also known as the normal mixture distribution, is a probability distribution that arises in the context of statistical inference, signal processing, and other fields of science. The Bell distribution is a continuous probability distribution that is a mixture of normal distributions. In a Bell distribution, approximately $0.68$ of the data falls within one standard deviation of the mean, $0.95$ falls within two standard deviations, and $0.997$ falls within three standard deviations.

The Bell distribution has a symmetric bell-shaped probability density function that resembles a normal distribution but with heavier tails. The mixing parameter p controls the degree of asymmetry of the distribution, with p = 0.5 corresponding to a perfectly symmetric distribution. The Bell distribution has applications in a wide range of fields, including finance, physics, engineering, and biology. It has been used, for example, to model the distribution of stock returns, the properties of noisy signals, and the behavior of biological systems. The Bell curve has many important applications in statistics, such as hypothesis testing, confidence intervals, and regression analysis. It is also used in fields such as finance, economics, and psychology, where it is used to model the behavior of complex systems and to make predictions based on empirical data.

In 2018, Castellares et al. [41] introduced the Bell distribution, it is improved from the Bell numbers [42].

When a discrete random variable X follows the Bell distribution, its probability density function can be expressed as

$$P\left(X=m\right)=\frac{{\lambda }^m{e}^{e^{\left(-{\lambda }^2\right)+1}}{B}_m}{m!};m=1,2,3,\cdots ,$$

(7)

where

$$B_m=\frac{1}{e}{\displaystyle \sum }_{k=0}^{\infty }\frac{k^m}{m!}$$

(8)

is the Bell numbers, $m\ge 2$, and $\lambda >0$. The Bell number $B_m$ given in (8) is the mth moment of the Poisson distribution with parameter equal to 1. The first few Bell numbers are $B_2=2$, $B_3=5$, $B_4=15$ and $B_5=52$.

Now, we present the power series below, whose coefficients are from the Bell distribution.

$$\mathbb{B}(\lambda ,\xi )=\xi +{\displaystyle \sum }_{n=2}^{\infty }\frac{{\lambda }^{n-1}{e}^{e^{\left(-{\lambda }^2\right)+1}}{B}_n}{(n-1)!}{\xi }^n,\xi \in \mathfrak{U},\lambda >0.$$

(9)

Consider the linear operator ${\mathbb{P}}_{\lambda }:\mathcal{A}\to \mathcal{A}$ defined by the convolution

$${\mathbb{P}}_{\lambda }f\left(\xi \right)=\mathbb{B}(\lambda ,\xi )\ast f\left(\xi \right)=\xi +{\displaystyle \sum }_{n=2}^{\infty }\frac{{\lambda }^{n-1}{e}^{e^{\left(-{\lambda }^2\right)+1}}{B}_n}{(n-1)!}{a}_n{\xi }^n,\xi \in \mathfrak{U}.$$

(10)

Recently, a large number of researchers have investigated bi-univalent functions connected to orthogonal polynomials, few to mention ([43,44,45,46,47]). As far as we are aware, there hasn’t been any research on bi-univalent functions for Bell distribution subordinate to Horadam polynomials in the literature.

The rest of this article is organized as follows. In Section 2 we introduce a new subclass ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\beta ,\gamma )$ of $\mathsf{\Sigma }$ involving the Bell distribution linked to Horadam polynomials, and deriving bounds for the second and the third coefficients in the Taylor-Maclaurin expansions. Section 3 deals with the estimation of Fekete–Szegö inequality for functions in the family ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\beta ,\gamma )$. Relevant connections of some of the special cases of the main results are pointed out in Section 4. Section 5 closes up the paper with some conclusions.

2. Bounds of the Class ${\mathfrak{G}}_{\mathsf{\Sigma }}^{\mathit{t}}(\mathit{x},\mathit{p},\mathit{q},\mathit{\lambda },\mathit{\beta },\mathit{\gamma })$

This section begins with a definition of a new subclass associated with the Bell distribution series.

Definition 1.

If the following subordinations are met, a function $f\in \mathsf{\Sigma }$ given by (5) is said to belong to the class ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\beta ,\gamma )$:

$$(1-\gamma )\frac{{\mathbb{P}}_{\lambda }f\left(\xi \right)}{\xi }+\gamma {\left({\mathbb{P}}_{\lambda }f\left(\xi \right)\right)}^{\prime }+\beta \xi {\left({\mathbb{P}}_{\lambda }f\left(\xi \right)\right)}^{″}\prec \mathsf{\Omega }(x,\xi )+1-a$$

(11)

and

$$(1-\gamma )\frac{{\mathbb{P}}_{\lambda }f\left(w\right)}{w}+\gamma {\left({\mathbb{P}}_{\lambda }f\left(w\right)\right)}^{\prime }+\beta w{\left({\mathbb{P}}_{\lambda }f\left(w\right)\right)}^{″}\prec \mathsf{\Omega }(x,w)+1-a,$$

(12)

where $\xi ,w\in \mathfrak{U}$, $\gamma ,\beta \ge 0$, $x\in \mathbb{R}$, and the function $g=f^{-1}$ is given by (6).

Example 1.

For $\beta =0,$ we have, ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,0,\gamma )={\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\gamma )$, in which ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\gamma )$ indicates the group of functions $f\in \mathsf{\Sigma }$ given by (5) and satisfying the criterion below.

$$(1-\gamma )\frac{{\mathbb{P}}_{\lambda }f\left(\xi \right)}{\xi }+\gamma {\left({\mathbb{P}}_{\lambda }f\left(\xi \right)\right)}^{\prime }\prec \mathsf{\Omega }(x,\xi )+1-a$$

(13)

and

$$(1-\gamma )\frac{{\mathbb{P}}_{\lambda }f\left(w\right)}{w}+\gamma {\left({\mathbb{P}}_{\lambda }f\left(w\right)\right)}^{\prime }\prec \mathsf{\Omega }(x,w)+1-a,$$

(14)

where $\xi ,w\in \mathfrak{U}$, $\gamma \ge 0$, $x\in \mathbb{R}$, and the function $g=f^{-1}$ is given by (6).

Example 2.

For $\beta =0$ and $\gamma =1,$ we have, ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,1)={\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda )$, in which ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda )$ denotes the class of functions $f\in \mathsf{\Sigma }$ given by (5) and satisfying the following condition

$${\left({\mathbb{P}}_{\lambda }f\left(\xi \right)\right)}^{\prime }\prec \mathsf{\Omega }(x,\xi )+1-a$$

(15)

and

$${\left({\mathbb{P}}_{\lambda }f\left(w\right)\right)}^{\prime }\prec \mathsf{\Omega }(x,w)+1-a,$$

(16)

where $\xi ,w\in \mathfrak{U}$, $x\in \mathbb{R}$, and the function $g=f^{-1}$ is given by (6).

Example 3.

For $\beta =0$ and $\gamma =0,$ we have, ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,0,0)={\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,0)$, in which ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,0)$ indicates the group of functions $f\in \mathsf{\Sigma }$ given by (5) and satisfying the criterion below.

$$\frac{{\mathbb{P}}_{\lambda }f\left(\xi \right)}{\xi }\prec \mathsf{\Omega }(x,\xi )+1-a$$

(17)

and

$$\frac{{\mathbb{P}}_{\lambda }f\left(w\right)}{w}\prec \mathsf{\Omega }(x,w)+1-a,$$

(18)

where $\xi ,w\in \mathfrak{U}$, $x\in \mathbb{R}$, and the function $g=f^{-1}$ is given by (6).

Example 4.

For $\lambda =1,$ we have, ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,1,\beta ,\gamma )={\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\beta ,\gamma )$, in which ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\beta ,\gamma )$ indicates the group of functions $f\in \mathsf{\Sigma }$ given by (5) and satisfying the criterion below.

$$(1-\gamma )\frac{{\mathbb{P}}_{1}f\left(\xi \right)}{\xi }+\gamma {\left({\mathbb{P}}_{1}f\left(\xi \right)\right)}^{\prime }+\beta \xi {\left({\mathbb{P}}_{1}f\left(\xi \right)\right)}^{″}\prec \mathsf{\Omega }(x,\xi )+1-a$$

(19)

and

$$(1-\gamma )\frac{{\mathbb{P}}_{1}f\left(w\right)}{w}+\gamma {\left({\mathbb{P}}_{1}f\left(w\right)\right)}^{\prime }+\beta w{\left({\mathbb{P}}_{1}f\left(w\right)\right)}^{″}\prec \mathsf{\Omega }(x,w)+1-a,$$

(20)

where $\xi ,w\in \mathfrak{U}$, $\gamma \ge 0$, $x\in \mathbb{R}$, and the function $g=f^{-1}$ is given by (6).

First, we give the coefficient estimates for the class ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\beta ,\gamma )$ given in Definition 1.

Theorem 1.

Let $f\in \mathsf{\Sigma }$ given by (5) belongs to the class ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\beta ,\gamma ).$ Then

$$\begin{array}{cc}\hfill \left|a_2\right|& \le \hfill \\ \hfill & \frac{tx\sqrt{2tx}}{\lambda e^{e^{\frac{1}{2}\left(1-{\lambda }^2\right)}}\sqrt{\left|\left[5\left(1+2\gamma +6\beta \right){\left(tx\right)}^2-8e^{e^{\left(1-{\lambda }^2\right)}}{\left(1+\gamma +2\beta \right)}^2\left(pt{x}^2+aq\right)\right]\right|}},\hfill \end{array}$$

and

$$\left|a_3\right|\le \frac{t^2{x}^2}{4{\lambda }^2{\left(1+\gamma +2\beta \right)}^2{e}^{2e^{\left(1-{\lambda }^2\right)}}}+\frac{2tx}{5{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}}.$$

Proof. 

Let $f\in {\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\beta ,\gamma ).$ From Definition 1, we can write

$$(1-\gamma )\frac{{\mathbb{P}}_{\lambda }f\left(\xi \right)}{\xi }+\gamma {\left({\mathbb{P}}_{\lambda }f\left(\xi \right)\right)}^{\prime }+\beta \xi {\left({\mathbb{P}}_{\lambda }f\left(\xi \right)\right)}^{″}=\mathsf{\Omega }(x,\varkappa \left(\xi \right))+1-a$$

(21)

and

$$(1-\gamma )\frac{{\mathbb{P}}_{\lambda }f\left(w\right)}{w}+\gamma {\left({\mathbb{P}}_{\lambda }f\left(w\right)\right)}^{\prime }+\beta w{\left({\mathbb{P}}_{\lambda }f\left(w\right)\right)}^{″}=\mathsf{\Omega }(x,\tau \left(w\right))+1-a,$$

(22)

where the analytical functions $\varkappa$ and $\tau$ have the form

$$\varkappa \left(\xi \right)=c_1\xi +c_2{\xi }^2+c_3{\xi }^3+\cdots ,(\xi \in \mathfrak{U})$$

and

$$\tau \left(w\right)=d_{1}w+d_2{w}^2+d_3{w}^3+\cdots ,(w\in \mathfrak{U}),$$

such that $\varkappa \left(0\right)=\tau \left(0\right)=0$ and $\left|\varkappa \right(\xi \left)\right|<1,$ $\left|\tau \right(w\left)\right|<1$ for all $\xi ,w\in \mathfrak{U}.$

From the equalities (21) and (22), we get

$$(1-\gamma )\frac{{\mathbb{P}}_{\lambda }f\left(\xi \right)}{\xi }+\gamma {\left({\mathbb{P}}_{\lambda }f\left(\xi \right)\right)}^{\prime }+\beta \xi {\left({\mathbb{P}}_{\lambda }f\left(\xi \right)\right)}^{″}=1+h_2\left(x\right)c_1\xi +\left[h_2\left(x\right)c_2+h_3\left(x\right)c_1^2\right]{\xi }^2+\cdots$$

(23)

and

$$(1-\gamma )\frac{{\mathbb{P}}_{\lambda }f\left(w\right)}{w}+\gamma {\left({\mathbb{P}}_{\lambda }f\left(w\right)\right)}^{\prime }+\beta w{\left({\mathbb{P}}_{\lambda }f\left(w\right)\right)}^{″}=1+h_2\left(x\right)d_{1}w+\left[h_2\left(x\right)d_2+h_3\left(x\right)d_1^2\right]w^2+\cdots .$$

(24)

It is common knowledge that if

$$\left|\varkappa \left(\xi \right)\right|=\left|c_1\xi +c_2{\xi }^2+c_3{\xi }^3+\cdots \right|<1,(\xi \in \mathfrak{U})$$

and

$$\left|\tau \left(w\right)\right|=\left|d_{1}w+d_2{w}^2+d_3{w}^3+\cdots \right|<1,(w\in \mathfrak{U}),$$

then

$$|c_j|\le 1\mathrm{and}|d_j|\le 1\mathrm{for}\mathrm{all}j\in \mathbb{N}.$$

(25)

Equating the coefficients of both sides in (23) and (24), we get

$$2\lambda \left(1+\gamma +2\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}{a}_2=h_2\left(x\right)c_1,$$

(26)

$$\frac{5}{2}{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}{a}_3=h_2\left(x\right)c_2+h_3\left(x\right)c_1^2,$$

(27)

$$-2\lambda \left(1+\gamma +2\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}{a}_2=h_2\left(x\right)d_1,$$

(28)

and

$$\frac{5}{2}{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}\left[2a_2^2-a_3\right]=h_2\left(x\right)d_2+h_3\left(x\right)d_1^2.$$

(29)

It follows from (26) and (28) that

$$c_1=-d_1$$

(30)

and

$$8{\lambda }^2{\left(1+\gamma +2\beta \right)}^2{e}^{2e^{\left(1-{\lambda }^2\right)}}{a}_2^2={\left[h_2\left(x\right)\right]}^2(c_1^2+d_1^2).$$

(31)

If we add (27) and (29), we get

$$5{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}{a}_2^2=h_2\left(x\right)\left(c_2+d_2\right)+h_3\left(x\right)(c_1^2+d_1^2).$$

(32)

Replacing the value of $\left(c_1^2+d_1^2\right)$ from (31) in the right hand side of (32), we have

$$\begin{array}{cc}\hfill & \left[5\left(1+2\gamma +6\beta \right)-8{\left(1+\gamma +2\beta \right)}^2{e}^{e^{\left(1-{\lambda }^2\right)}}\frac{h_3\left(x\right)}{{\left[h_2\left(x\right)\right]}^2}\right]{\lambda }^2{e}^{e^{\left(1-{\lambda }^2\right)}}{a}_2^2\hfill \\ \hfill & =h_2\left(x\right)\left(c_2+d_2\right).\hfill \end{array}$$

(33)

Using (3), (25) and (33), we find that

$$\begin{array}{cc}\hfill \left|a_2\right|& \le \hfill \\ \hfill & \frac{tx\sqrt{2tx}}{\lambda e^{e^{\frac{1}{2}\left(1-{\lambda }^2\right)}}\sqrt{\left|\left[5\left(1+2\gamma +6\beta \right){\left[tx\right]}^2-8e^{e^{\left(1-{\lambda }^2\right)}}{\left(1+\gamma +2\beta \right)}^2\left(pt{x}^2+aq\right)\right]\right|}}.\hfill \end{array}$$

Moreover, if we subtract (29) from (27), we obtain

$$5{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}(a_3-a_2^2)=h_2\left(x\right)\left(c_2-d_2\right)+h_3\left(x\right)(c_1^2-d_1^2).$$

(34)

Then, in view of (30) and (31), Equation (34) becomes

$$\begin{array}{cc}\hfill a_3& =\frac{{\left[h_2\left(x\right)\right]}^2}{8{\lambda }^2{\left(1+\gamma +2\beta \right)}^2{e}^{2e^{\left(1-{\lambda }^2\right)}}}(c_1^2+d_1^2)\hfill \\ \hfill & +\frac{h_2\left(x\right)}{5{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}}\left(c_2-d_2\right).\hfill \end{array}$$

By applying (3), we conclude that

$$\left|a_3\right|\le \frac{t^2{x}^2}{4{\lambda }^2{\left(1+\gamma +2\beta \right)}^2{e}^{2e^{\left(1-{\lambda }^2\right)}}}+\frac{2tx}{5{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}}.$$

□

3. Fekete-Szegö Inequalities

Using the values of $a_2^2$ and $a_3$, we prove the functional $\left|a_3-\eta a_2^2\right|$ for class functions ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\beta ,\gamma )$.

Theorem 2.

Let $f\in \mathsf{\Sigma }$ given by (5) belongs to the class ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\beta ,\gamma )$. Then

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{c}\frac{2\left|tx\right|}{5{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}},\\ \\ \frac{2{\left(tx\right)}^3\left|1-\eta \right|}{{\lambda }^2{e}^{e^{\left(1-{\lambda }^2\right)}}\left|\left[5\left(1+2\gamma +6\beta \right)t^2{x}^2-8{\left(1+\gamma +2\beta \right)}^2{e}^{e^{\left(1-{\lambda }^2\right)}}\left(pt{x}^2+aq\right)\right]\right|},\end{array}\right.\begin{array}{c}\left|\eta -1\right|\le \delta \\ \\ \left|\eta -1\right|\ge \delta ,\end{array}$$

where

$$\delta =\left|1-\frac{8{\left(1+\gamma +2\beta \right)}^2{e}^{e^{\left(1-{\lambda }^2\right)}}\left(pt{x}^2+aq\right)}{5\left(1+2\gamma +6\beta \right)t^2{x}^2}\right|.$$

Proof. 

From (33) and (34)

$$\begin{array}{cc}\hfill & a_3-\eta a_2^2\hfill \\ \hfill & =\left(1-\eta \right)\frac{{\left[h_2\left(x\right)\right]}^3\left(c_2+d_2\right)}{{\lambda }^2{e}^{e^{\left(1-{\lambda }^2\right)}}\left[5\left(1+2\gamma +6\beta \right){\left[h_2\left(x\right)\right]}^2-8{\left(1+\gamma +2\beta \right)}^2{e}^{e^{\left(1-{\lambda }^2\right)}}{h}_3\left(x\right)\right]}\hfill \\ \hfill & +\frac{h_2\left(x\right)}{5{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}}\left(c_2-d_2\right)\hfill \\ \hfill & =h_2\left(x\right)\left[h\left(\eta \right)+\frac{1}{5{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}}\right]c_2\hfill \\ \hfill & +h_2\left(x\right)\left[h\left(\eta \right)-\frac{1}{5{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}}\right]d_2,\hfill \end{array}$$

where

$$\mathsf{{\rm Y}}\left(\eta \right)=\frac{{\left[h_2\left(x\right)\right]}^2\left(1-\eta \right)}{{\lambda }^2{e}^{e^{\left(1-{\lambda }^2\right)}}\left[5\left(1+2\gamma +6\beta \right){\left[h_2\left(x\right)\right]}^2-8{\left(1+\gamma +2\beta \right)}^2{e}^{e^{\left(1-{\lambda }^2\right)}}{h}_3\left(x\right)\right]},$$

Then, in view of (3), we conclude that

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{c}\frac{2\left|h_2\left(x\right)\right|}{5{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}}\\ \\ 2\left|h_2\left(x\right)\right|\left|\mathsf{{\rm Y}}\left(\eta \right)\right|\end{array}\right.\begin{array}{c}0\le \left|\mathsf{{\rm Y}}\left(\eta \right)\right|\le \frac{1}{5{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}},\\ \\ \left|\mathsf{{\rm Y}}\left(\eta \right)\right|\ge \frac{1}{5{\lambda }^2\left(1+2\gamma +6\beta \right)e^{e^{\left(1-{\lambda }^2\right)}}}.\end{array}$$

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4. Special Cases and Consequences

By specializing the parameters $\beta$, $\lambda$ and $\gamma$ in the above theorems, we obtain the following corollaries.

Corollary 1.

Let $f\in \mathsf{\Sigma }$ given by (5) belongs to the class ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\gamma ).$ Then

$$\begin{array}{cc}\hfill \left|a_2\right|& \le \hfill \\ \hfill & \frac{tx\sqrt{2tx}}{\lambda e^{e^{\frac{1}{2}\left(1-{\lambda }^2\right)}}\sqrt{\left|\left[5\left(1+2\gamma \right){\left(tx\right)}^2-8e^{e^{\left(1-{\lambda }^2\right)}}{\left(1+\gamma \right)}^2\left(pt{x}^2+aq\right)\right]\right|}},\hfill \end{array}$$

$$\left|a_3\right|\le \frac{t^2{x}^2}{4{\lambda }^2{\left(1+\gamma \right)}^2{e}^{2e^{\left(1-{\lambda }^2\right)}}}+\frac{2tx}{5{\lambda }^2\left(1+2\gamma \right)e^{e^{\left(1-{\lambda }^2\right)}}},$$

and

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{c}\frac{2\left|tx\right|}{5{\lambda }^2\left(1+2\gamma \right)e^{e^{\left(1-{\lambda }^2\right)}}},\\ \\ \frac{2{\left(tx\right)}^3\left|1-\eta \right|}{{\lambda }^2{e}^{e^{\left(1-{\lambda }^2\right)}}\left|\left[5\left(1+2\gamma \right)t^2{x}^2-8{\left(1+\gamma \right)}^2{e}^{e^{\left(1-{\lambda }^2\right)}}\left(pt{x}^2+aq\right)\right]\right|},\end{array}\right.\begin{array}{c}\left|\eta -1\right|\le \mathsf{\Phi }\\ \\ \left|\eta -1\right|\ge \mathsf{\Phi },\end{array}$$

where

$$\mathsf{\Phi }=\left|1-\frac{8{\left(1+\gamma \right)}^2{e}^{e^{\left(1-{\lambda }^2\right)}}\left(pt{x}^2+aq\right)}{5\left(1+2\gamma \right)t^2{x}^2}\right|.$$

Corollary 2.

Let $f\in \mathsf{\Sigma }$ given by (5) belongs to the class ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ).$ Then

$$\begin{array}{cc}\hfill \left|a_2\right|& \le \hfill \\ \hfill & \frac{tx\sqrt{2tx}}{\lambda e^{e^{\frac{1}{2}\left(1-{\lambda }^2\right)}}\sqrt{\left|\left[15{\left(tx\right)}^2-32e^{e^{\left(1-{\lambda }^2\right)}}\left(pt{x}^2+aq\right)\right]\right|}},\hfill \end{array}$$

$$\left|a_3\right|\le \frac{t^2{x}^2}{16{\lambda }^2{e}^{2e^{\left(1-{\lambda }^2\right)}}}+\frac{2tx}{15{\lambda }^2{e}^{e^{\left(1-{\lambda }^2\right)}}},$$

and

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{c}\frac{2\left|tx\right|}{15{\lambda }^2{e}^{e^{\left(1-{\lambda }^2\right)}}},\\ \\ \frac{2{\left(tx\right)}^3\left|1-\eta \right|}{{\lambda }^2{e}^{e^{\left(1-{\lambda }^2\right)}}\left|\left[15t^2{x}^2-32e^{e^{\left(1-{\lambda }^2\right)}}\left(pt{x}^2+aq\right)\right]\right|},\end{array}\right.\begin{array}{c}\left|\eta -1\right|\le \left|1-\frac{32e^{e^{\left(1-{\lambda }^2\right)}}\left(pt{x}^2+aq\right)}{15t^2{x}^2}\right|\\ \\ \left|\eta -1\right|\ge \left|1-\frac{32e^{e^{\left(1-{\lambda }^2\right)}}\left(pt{x}^2+aq\right)}{15t^2{x}^2}\right|.\end{array}$$

Corollary 3.

Let $f\in \mathsf{\Sigma }$ given by (5) belongs to the class ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,0).$ Then

$$\begin{array}{cc}\hfill \left|a_2\right|& \le \hfill \\ \hfill & \frac{tx\sqrt{2tx}}{\lambda e^{e^{\frac{1}{2}\left(1-{\lambda }^2\right)}}\sqrt{\left|\left[5{\left(tx\right)}^2-8e^{e^{\left(1-{\lambda }^2\right)}}\left(pt{x}^2+aq\right)\right]\right|}},\hfill \end{array}$$

$$\left|a_3\right|\le \frac{t^2{x}^2}{4{\lambda }^2{e}^{2e^{\left(1-{\lambda }^2\right)}}}+\frac{2tx}{5{\lambda }^2{e}^{e^{\left(1-{\lambda }^2\right)}}},$$

and

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{c}\frac{2\left|tx\right|}{5{\lambda }^2{e}^{e^{\left(1-{\lambda }^2\right)}}},\\ \\ \frac{2{\left(tx\right)}^3\left|1-\eta \right|}{{\lambda }^2{e}^{e^{\left(1-{\lambda }^2\right)}}\left|\left[5t^2{x}^2-8e^{e^{\left(1-{\lambda }^2\right)}}\left(pt{x}^2+aq\right)\right]\right|},\end{array}\right.\begin{array}{c}\left|\eta -1\right|\le \mathsf{\Phi }\\ \\ \left|\eta -1\right|\ge \mathsf{\Phi },\end{array}$$

where

$$\mathsf{\Phi }=\left|1-\frac{8e^{e^{\left(1-{\lambda }^2\right)}}\left(pt{x}^2+aq\right)}{5t^2{x}^2}\right|.$$

Corollary 4.

Let $f\in \mathsf{\Sigma }$ given by (5) belongs to the class ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\beta ,\gamma )$. Then

$$\begin{array}{cc}\hfill \left|a_2\right|& \le \hfill \\ \hfill & \frac{tx\sqrt{2tx}}{e\sqrt{\left|\left[5\left(1+2\gamma +6\beta \right){\left(tx\right)}^2-8e{\left(1+\gamma +2\beta \right)}^2\left(pt{x}^2+aq\right)\right]\right|}},\hfill \end{array}$$

$$\left|a_3\right|\le \frac{t^2{x}^2}{4e^2{\left(1+\gamma +2\beta \right)}^2}+\frac{2tx}{5e\left(1+2\gamma +6\beta \right)}.$$

and

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{c}\frac{2\left|tx\right|}{5e\left(1+2\gamma +6\beta \right)},\\ \\ \frac{2{\left(tx\right)}^3\left|1-\eta \right|}{e\left|\left[5\left(1+2\gamma +6\beta \right)t^2{x}^2-8e{\left(1+\gamma +2\beta \right)}^2\left(pt{x}^2+aq\right)\right]\right|},\end{array}\right.\begin{array}{c}\left|\eta -1\right|\le \delta \\ \\ \left|\eta -1\right|\ge \delta ,\end{array}$$

where

$$\delta =\left|1-\frac{8e{\left(1+\gamma +2\beta \right)}^2\left(pt{x}^2+aq\right)}{5\left(1+2\gamma +6\beta \right)t^2{x}^2}\right|.$$

5. Conclusions

In this study, we introduced a new class of normalized analytics and bi-univalent functions connected to the Bell distribution series denoted by ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\beta ,\gamma )$. We have derived estimates for the Taylor-Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ and Fekete-Szegö functional problems. Additionally, by appropriately specializing the parameters $\beta$ and $\gamma$, one may determine the outcomes for the subclasses ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,\gamma )$, ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda )$, ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\lambda ,0)$ and ${\mathfrak{G}}_{\mathsf{\Sigma }}^t(x,p,q,\beta ,\gamma )$ specified in Examples 1, 2, 3 and 4, respectively, and linked to the Bell distribution series. Making use of Bell distribution series (10) could inspire researchers to derive the estimates of the Taylor-Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ and Fekete-Szegö functional problems for functions belonging to new subclasses of bi-univalent functions defined by means of Horadam polynomials associated with this distribution series.