Coefficient Estimates for the Functions with Respect to Symmetric Conjugate Points Connected with the Combination Binomial Series and Babalola Operator and Lucas Polynomials

Abstract

By using Lucas polynomials, we define a new subclass of analytic bi-univalent functions, class $\mathsf{\Sigma }$, in the open unit disc with respect to symmetric conjugate points connected with the combination Binomial series and Babalola operator. The bounds on the initial coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for the functions in this new subclass of $\mathsf{\Sigma }$ are investigated. Moreover, we obtain an estimation for the Fekete–Szego problem for the function subclass defined in this paper. Relevant connections of these results are presented here as corollaries.

1. Introduction

Let $\mathcal{A}$ represent the family of analytic functions whose members are

$$\mathcal{T}\left(\zeta \right)=\zeta +\sum _{j=2}^{\infty }{a}_j{\zeta }^j,(\Omega =\{\zeta :\left|\zeta \right|<1,\zeta \in \mathbb{C}\},$$

(1)

with the normalization condition $\mathcal{T}\left(0\right)=0,{\mathcal{T}}^{\prime }\left(0\right)=1,$ and $\mathcal{S}$ be a class of all functions belonging to $\mathcal{A},$ which are univalent functions. Furthermore, let $\mathcal{P}$ be the family of functions $p\left(\zeta \right)\in \mathcal{A}.$

If $\mathcal{T}$ and $\mathsf{{\rm Y}}$ are analytic functions in $\Omega$, then $\mathcal{T}$ is subordinate to $\mathsf{{\rm Y}}$, written $\mathcal{T}\prec \mathsf{{\rm Y}}$ if there exists an analytic Schwarz function w, in $\Omega$ with $w\left(0\right)=0$ and $\left|w\left(z\right)\right|<1$for all $\zeta \in \Omega ,$ such that $\mathcal{T}\left(\zeta \right)=\mathsf{{\rm Y}}\left(w\right(\zeta \left)\right).$ Moreover, if the function $\mathsf{{\rm Y}}$ is univalent in $\Omega ,$ then we have (see [1,2]):

$$\mathcal{T}\left(\zeta \right)\prec \mathsf{{\rm Y}}\left(\zeta \right)\iff \mathcal{T}\left(0\right)=\mathsf{{\rm Y}}\left(0\right)\mathrm{and}\mathcal{T}(\Omega )\subset \mathsf{{\rm Y}}(\Omega ).$$

Babalola defined the operator ${\mathcal{L}}_{\upsilon }^{\rho }:\Omega \to \Omega$as

$${\mathcal{L}}_{\upsilon }^{\sigma }\mathcal{T}\left(\zeta \right)=\left({\rho }_{\sigma }\ast {\rho }_{\sigma ,\upsilon }^{-1}\ast \mathcal{T}\right)\left(\zeta \right),$$

(2)

where

$${\rho }_{\sigma ,\upsilon }\left(\zeta \right)=\frac{\zeta }{{\left(1-\zeta \right)}^{\sigma -\upsilon +1}},\sigma -\upsilon +1>0,{\rho }_{\sigma }={\rho }_{\sigma ,0},$$

and ${\rho }_{\sigma ,\upsilon }^{-1}$ is

$$\left({\rho }_{\sigma ,\upsilon }\ast {\rho }_{\sigma ,\upsilon }^{-1}\right)\left(\zeta \right)=\frac{\zeta }{1-\zeta }\left(\sigma ,\upsilon \in \mathbb{N}=\{1,2,3,\dots \}\right).$$

For $\mathcal{T}\in \Omega ,$ then (2) is equivalent to

$${\mathcal{L}}_{\upsilon }^{\sigma }\mathcal{T}\left(\zeta \right)=\zeta +\sum _{j=2}^{\infty }\left[\frac{\mathsf{\Gamma }(\sigma +j)}{\mathsf{\Gamma }(\sigma +1)}·\frac{(\sigma -\upsilon )!}{(\sigma +j-\upsilon -1)!}\right]a_j{\zeta }^j.$$

Making use of the binomial series

$${\left(1-\delta \right)}^n=\sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right){\left(-1\right)}^i{\delta }^i\left(n\in \mathbb{N}\right),$$

for $\mathcal{T}\in \Omega ,$we introduce the linear differential operator as follows:

$${\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,0}\mathcal{T}\left(\zeta \right)=\mathcal{T}\left(\zeta \right),$$

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,1}\mathcal{T}\left(\zeta \right)& =& {\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma }\mathcal{T}\left(\zeta \right)={\left(1-\delta \right)}^n{\mathcal{L}}_{\upsilon }^{\sigma }\mathcal{T}\left(\zeta \right)+\left[1-{\left(1-\delta \right)}^n\right]\zeta {\left({\mathcal{L}}_{\upsilon }^{\sigma }\mathcal{T}\right)}^{{}^{\prime }}\left(\zeta \right)\hfill \\ & =& z+\sum _{j=2}^{\infty }\left[1+\left(j-1\right)c^n\left(\delta \right)\right]\left[\frac{\mathsf{\Gamma }(\sigma +j)}{\mathsf{\Gamma }(\sigma +1)}·\frac{(\sigma -\upsilon )!}{(\sigma +j-\upsilon -1)!}\right]a_j{\zeta }^j\hfill \\ & & ⋮\hfill \\ \hfill {\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)& =& {\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma }\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m-1}\mathcal{T}\left(\zeta \right)\right)\hfill \\ & =& {\left(1-\delta \right)}^n{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m-1}\mathcal{T}\left(\zeta \right)+\left[1-{\left(1-\delta \right)}^n\right]\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m-1}\mathcal{T}\left(\zeta \right)\right)}^{{}^{\prime }}\hfill \\ & =& \zeta +\sum _{j=2}^{\infty }{\left[1+\left(j-1\right)c^n\left(\delta \right)\right]}^m\left[\frac{\mathsf{\Gamma }(\sigma +j)}{\mathsf{\Gamma }(\sigma +1)}·\frac{(\sigma -\upsilon )!}{(\sigma +j-\upsilon -1)!}\right]a_j{\zeta }^j\hfill \\ & =& \zeta +\sum _{j=2}^{\infty }{\psi }_j{a}_j{\zeta }^j,\hfill \\ & & \left(\delta >0;n,\sigma ,\upsilon \in \mathbb{N};m\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}\right),\hfill \end{array}$$

(3)

where

$${\psi }_j={\left[1+\left(j-1\right)c^n\left(\delta \right)\right]}^m\left[\frac{\mathsf{\Gamma }(\sigma +j)}{\mathsf{\Gamma }(\sigma +1)}·\frac{(\sigma -\upsilon )!}{(\sigma +j-\upsilon -1)!}\right],$$

(4)

and

$$c^n\left(\delta \right)=\sum _{i=1}^n\left(\begin{array}{c}n\\ i\end{array}\right){\left(-1\right)}^{i+1}{\delta }^i\left(n\in \mathbb{N}\right).$$

From (3), we obtain that

$$c^n\left(\delta \right)\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{{}^{\prime }}={\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m+1}\mathcal{T}\left(\zeta \right)-\left[1-c^n\left(\delta \right)\right]{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right).$$

(5)

Definition 1.

Let $p\left(\varkappa \right)$ and $r\left(\varkappa \right)$ be polynomials with real coefficients. The $(p,r)$-polynomials ${\mathcal{L}}_{p,r,n}\left(\varkappa \right)$ are given by the following recurrence relation (see [3,4]):

$${\mathcal{L}}_{p,r,n}\left(\varkappa \right)=p\left(\varkappa \right){\mathcal{L}}_{p,r,n-1}\left(\varkappa \right)+r\left(\varkappa \right){\mathcal{L}}_{p,r,n-2}\left(\varkappa \right)(n\in \mathbb{N}\setminus \left\{1\right\}),$$

with

$$\begin{array}{c}{\mathcal{L}}_{p,r,0}\left(\varkappa \right)=2,\hfill \\ {\mathcal{L}}_{p,r,1}\left(\varkappa \right)=p\left(\varkappa \right),\hfill \\ {\mathcal{L}}_{p,r,2}\left(\varkappa \right)=p^2\left(\varkappa \right)+2r\left(\varkappa \right),\hfill \\ {\mathcal{L}}_{p,r,3}\left(\varkappa \right)=p^3\left(\varkappa \right)+3p\left(x\right)r\left(\varkappa \right),\hfill \\ ⋮\hfill \end{array}$$

The generating function of the Lucas polynomials ${\mathcal{L}}_{p,r,n}\left(\varkappa \right)$ (see [5]) is given by:

$${\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}\left(\zeta \right):=\sum _{n=0}^{\infty }{\mathcal{L}}_{p,r,n}\left(\varkappa \right){\zeta }^n={\displaystyle \frac{2-p\left(\varkappa \right)\zeta }{1-p\left(\varkappa \right)\zeta -r\left(\varkappa \right){\zeta }^2}}.$$

(6)

Note that for particular values of p and r, the $(p,r)$-polynomial ${\mathcal{L}}_{p,r,n}\left(\varkappa \right)$ includes various polynomials; among these, we have the following cases (see [5] for more details; also [6,7]):

• Putting $p\left(\varkappa \right)=\varkappa$ and $r\left(\varkappa \right)=1,$ we obtain the Lucas polynomials ${\mathcal{L}}_n\left(\varkappa \right)$;
• Putting $p\left(\varkappa \right)=2\varkappa$ and $r\left(\varkappa \right)=1,$ we attain the Pell–Lucas polynomials $Q_n\left(\varkappa \right)$;
• Putting $p\left(\varkappa \right)=1$ and $r\left(\varkappa \right)=2\varkappa ,$ we attain the Jacobsthal–Lucas polynomials $j_n\left(\varkappa \right)$;
• Putting $p\left(\varkappa \right)=3\varkappa$ and $r\left(\varkappa \right)=-2,$ we attain the Fermat–Lucas polynomials $f_n\left(\varkappa \right)$;
• Putting $p\left(\varkappa \right)=2\varkappa$ and $r\left(\varkappa \right)=-1,$ we have the Chebyshev polynomials $T_n\left(\varkappa \right)$ of the first kind.

By the Koebe one quarter theorem (see [8]), we have that the image of $\Omega$ under every univalent function $\mathcal{T}\in \mathcal{S}$ contains a disk of radius $\frac{1}{4}.$Therefore, every function $\mathcal{T}\in \mathcal{S}$ has an inverse ${\mathcal{T}}^{-1}$ that satisfies

$${\mathcal{T}}^{-1}\left(\mathcal{T}\left(\zeta \right)\right)=\zeta \left(\zeta \in \Omega \right)$$

and

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

where

$${\mathcal{T}}^{-1}\left(w\right)=w-a_2{w}^2+\left(2a_2^2-a_3\right)w^3-\left(5a_2^3-5a_2{a}_3+a_4\right)w^4+\dots$$

(7)

A function $\mathcal{T}\in \mathcal{A}$ is said to be bi-univalent in $\Omega$ if both $\mathcal{T}\left(\zeta \right)$ and ${\mathcal{T}}^{-1}\left(\zeta \right)$ are univalent in $\Omega$. Let $\mathsf{\Sigma }$ denote the class of bi-univalent functions in $\Omega$ given by (1). For a brief history and interesting examples in the class $\mathsf{\Sigma }$, see [9]. Brannan and Taha [10] (see also [11,12,13]) introduced certain subclasses of the bi-univalent functions class $\mathsf{\Sigma }$ similar to the familiar subclasses ${\mathcal{S}}^{\ast }\left(\delta \right)$ and $\mathcal{K}\left(\delta \right)$ of starlike and convex functions of order $\delta \left(0\le \delta <1\right)$, respectively (see [9]). The classes ${\mathcal{S}}_{\mathsf{\Sigma }}^{\ast }\left(\delta \right)$ and ${\mathcal{K}}_{\mathsf{\Sigma }}\left(\delta \right)$ of bi-starlike functions of order $\delta$ and bi-convex functions of order $\delta$$\left(0<\delta \le 1\right),$ corresponding to the function classes ${\mathcal{S}}^{\ast }\left(\delta \right)$ and $\mathcal{K}\left(\delta \right),$were also introduced analogously. For each of the function classes ${\mathcal{S}}_{\mathsf{\Sigma }}^{\ast }\left(\delta \right)$ and ${\mathcal{K}}_{\mathsf{\Sigma }}\left(\delta \right),$ they found non-sharp estimates on the first two Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ (for details, see [10,13]).

Note that the functions

$$f_1\left(\zeta \right)={\displaystyle \frac{\zeta }{1-\zeta }},f_2\left(\zeta \right)={\displaystyle \frac{1}{2}}log{\displaystyle \frac{1+\zeta }{1-\zeta }},f_3\left(\zeta \right)=-log(1-\zeta )$$

(8)

with their corresponding inverses

$$f_1^{-1}\left(w\right)={\displaystyle \frac{w}{1+w}},f_2^{-1}\left(w\right)={\displaystyle \frac{e^{2w}-1}{e^{2w}+1}},f_3^{-1}\left(w\right)={\displaystyle \frac{e^w-1}{e^w}}$$

(9)

are elements of $\mathsf{\Sigma }.$

El-Ashwah and Thomas [14] introduced the class ${\mathcal{S}}_{sc}^{\ast }$ of starlike with respect to symmetric conjugate points as follows:

$$\Re \left\{\frac{\zeta {\mathcal{T}}^{{}^{\prime }}\left(\zeta \right)}{\mathcal{T}\left(\zeta \right)-\overline{\mathcal{T}(-\overline{\zeta })}}\right\}>0,\zeta \in \Omega .$$

Ping and Janteng [15] introduced the class ${\mathcal{S}}_{sc}^{\ast }$$\left(A,B\right)$of starlike with respect to symmetric conjugate points as follows:

$$\frac{2\zeta {\mathcal{T}}^{{}^{\prime }}\left(\zeta \right)}{\mathcal{T}\left(\zeta \right)-\overline{\mathcal{T}(-\overline{\zeta })}}\prec \frac{1+A\zeta }{1+B\zeta },\zeta \in \Omega .$$

It is perceived that this formalism facilitates further mathematical exploration and also helps us to understand the symmetric and geometric properties of such operators better. Inspired by aforementioned works on bi-univalent functions and by using the combination of binomial series and the Babalola operator in the present paper, we define a new subclass as in Definition 2 of the function class $\mathsf{\Sigma }$ and determine the estimates of the coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for the function in this new subclass of the function class $\mathsf{\Sigma }$. We also discuss the Fekete–Szego inequalities’ results [16] for f$\in {\mathcal{S}}_{sc}^{\gamma }\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)$.

Definition 2.

A function $\mathcal{T}\left(\zeta \right)\in \mathcal{A}$ is said to be in the class ${\mathcal{S}}_{sc}^{\gamma }\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)(0\le \gamma \le 1;\delta >0;n,\sigma ,\upsilon \in \mathbb{N};m\in {\mathbb{N}}_0)$ if

$$\begin{array}{c}{\displaystyle \frac{2\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{\prime }}{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta })}}}+{\displaystyle \frac{2{\left(\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{\prime }\right)}^{\prime }}{{({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta }))}^{\prime }}}\\ -{\displaystyle \frac{2\gamma {\zeta }^2{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{″}+2\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{\prime }}{\gamma \zeta ({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta }){)}^{{}^{\prime }}}+(1-\gamma )({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta }))}}}\\ \prec {\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}\left(\zeta \right)-1,\end{array}$$

(10)

and

$$\begin{array}{c}{\displaystyle \frac{2w{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{\prime }}{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w})}}}+{\displaystyle \frac{2{\left(w{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{\prime }\right)}^{\prime }}{{({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w})})}^{\prime }}}\\ -{\displaystyle \frac{2\gamma w^2{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{″}+2w{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{\prime }}{\gamma w{({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w})})}^{\prime }+(1-\gamma )({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w}))}}}\\ \prec {\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}\left(w\right)-1,\end{array}$$

(11)

with$0\le \gamma \le 1;\delta >0;n,\sigma ,\upsilon \in \mathbb{N};m\in {\mathbb{N}}_0,$where the function $\mathcal{G}={\mathcal{T}}^{-1}$ is given by (7).

Example 1.

A function $\mathcal{T}\left(\zeta \right)\in \mathcal{A}$ is said to be in the class ${\mathcal{S}}_{sc}^0\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)\equiv {\mathcal{M}}_{sc}\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)$ if

$$\frac{2{\left(\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{\prime }\right)}^{\prime }}{{({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(-\overline{\zeta }\right))}^{\prime }}\prec {\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}\left(\zeta \right)-1,$$

and

$$\frac{2{\left(w{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{\prime }\right)}^{\prime }}{{({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w}))})}^{\prime }}\prec {\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}\left(w\right)-1,$$

where $\delta >0;n,\sigma ,\upsilon \in \mathbb{N};m\in {\mathbb{N}}_0,$ where the function $\mathcal{G}={\mathcal{T}}^{-1}$ is given by (7).

Example 2.

A function $\mathcal{T}\left(\zeta \right)\in \mathcal{A}$ is said to be in the class ${\mathcal{S}}_{sc}^1\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)\equiv {\mathcal{N}}_{sc}\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)$ if

$${\displaystyle \frac{2\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{\prime }}{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta })}}}\prec {\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}\left(\zeta \right)-1,$$

(12)

and

$${\displaystyle \frac{2w{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{\prime }}{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w})}}}\prec {\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}\left(w\right)-1,$$

(13)

where$\delta >0;n,\sigma ,\upsilon \in \mathbb{N};m\in {\mathbb{N}}_0,$where the function $\mathcal{G}={\mathcal{T}}^{-1}$ is given by (7).

The following lemma will be needed later.

Lemma 1

([17]). If $\omega \left(\zeta \right)=\sum _{j=1}^{\infty }{p}_j{\zeta }^j$ is a Schwarz function for $\zeta \in \Omega$, then

$$|p_1|\le 1,|p_j|\le 1-|p_1{|}^2,j\ge 1.$$

2. Coefficient Bounds for the Function Class ${\mathcal{S}}_{\mathit{sc}}^{\mathbf{\gamma }}\left(\mathit{n},\mathbf{\delta },\mathbf{\upsilon },\mathbf{\sigma },\mathit{m},\mathit{p}\left(\mathbf{\varkappa }\right),\mathit{r}\left(\mathbf{\varkappa }\right)\right)$

Throughout this paper, we shall assume that $0\le \gamma \le 1;\delta >0;n,\sigma ,\upsilon \in \mathbb{N};m\in {\mathbb{N}}_0.$

Theorem 1.

If the function $\mathcal{T}$ given by (1) is in the class ${\mathcal{S}}_{sc}^{\gamma }\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)$, then

$$\left|a_2\right|\le \frac{\left|p\left(\varkappa \right)\right|\sqrt{\left|p\left(\varkappa \right)\right|}}{\sqrt{2\left|\left(3-2\gamma \right)p^2\left(\varkappa \right){\psi }_3-2(p^2\left(\varkappa \right)+2r\left(\varkappa \right)){\left(2-\gamma \right)}^2{\psi }_2^2\right|}},$$

and

$$\left|a_3\right|\le \frac{\left|p\right(\varkappa \left)\right|}{2\left(3-2\gamma \right){\psi }_3}+\frac{p^2\left(\varkappa \right)}{4{\left(2-\gamma \right)}^2{\psi }_2^2},$$

where ${\psi }_j$, $j\in \{2,3\}$, are given by (4).

Proof. 

Let $\mathcal{T}\in {\mathcal{S}}_{sc}^{\gamma }\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)$. Then, there exist two analytic functions $\mathcal{R}$ and $\mathcal{S}$ in $\Omega$ with $\mathcal{R}\left(0\right)=\mathcal{S}\left(0\right)=0$, and $\left|\mathcal{R}\right(\zeta \left)\right|<1$, $\left|\mathcal{S}\right(w\left)\right|<1$ for all $\zeta ,w\in \Omega$ given by

$$\mathcal{R}\left(\zeta \right)=\sum _{j=1}^{\infty }{v}_j{\zeta }^j\mathrm{and}\mathcal{S}\left(w\right)=\sum _{j=1}^{\infty }{s}_j{w}^j,\zeta ,w\in \Omega ,$$

from Lemma 1, we have

$$\left|v_j\right|\le 1\mathrm{and}\left|s_j\right|\le 1,j\in \mathbb{N}.$$

(14)

In view of (10) and (11), we obtain

$${\displaystyle \frac{2\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{\prime }}{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta })}}}+{\displaystyle \frac{2{\left(\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{\prime }\right)}^{\prime }}{{({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta })})}^{\prime }}}$$

$$\begin{array}{ccc}& & -{\displaystyle \frac{2\gamma {\zeta }^2{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{″}+2\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{\prime }}{\gamma \zeta {({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta })})}^{\prime }+(1-\gamma )({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta }))}}}\hfill \\ & =& {\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}\left(\mathcal{R}\left(\zeta \right)\right)-1,\hfill \end{array}$$

(15)

and

$${\displaystyle \frac{2w{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{\prime }}{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w})}}}+{\displaystyle \frac{2{\left(w{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{\prime }\right)}^{\prime }}{{({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w})})}^{\prime }}}$$

$$\begin{array}{ccc}& & -{\displaystyle \frac{2\gamma w^2{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{″}+2w{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{\prime }}{\gamma w{({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w})})}^{\prime }+(1-\gamma )({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w}))}}}\hfill \\ & =& {\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}\left(\mathcal{S}\left(w\right)\right)-1.\hfill \end{array}$$

(16)

After some basic calculations, we obtain

$${\displaystyle \frac{2\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{\prime }}{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta })}}}+{\displaystyle \frac{2{\left(\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{\prime }\right)}^{\prime }}{{({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta })})}^{\prime }}}$$

$$\begin{array}{ccc}& & -{\displaystyle \frac{2\gamma {\zeta }^2{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{″}+2\zeta {\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)\right)}^{\prime }}{\gamma \zeta {({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta })})}^{\prime }+(1-\gamma )({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}\left(\zeta \right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{T}(-\overline{\zeta }))}}}\hfill \\ & =& 1+2(2-\gamma ){\psi }_2{a}_2\zeta +2\left(3-2\gamma \right){\psi }_3{a}_3{\zeta }^2+\dots ,\hfill \end{array}$$

(17)

$${\displaystyle \frac{2w{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{\prime }}{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w})}}}+{\displaystyle \frac{2{\left(w{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{\prime }\right)}^{\prime }}{{({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w})})}^{\prime }}}$$

$$\begin{array}{ccc}& & -{\displaystyle \frac{2\gamma w^2{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{″}+2w{\left({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)\right)}^{\prime }}{\gamma w{({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w})})}^{\prime }+(1-\gamma )({\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}\left(w\right)-\overline{{\mathcal{D}}_{n,\delta ,\upsilon }^{\sigma ,m}\mathcal{G}(-\overline{w}))}}}\hfill \\ & =& 1-2(2-\gamma ){\psi }_2{a}_{2}w+2\left(3-2\gamma \right)\left(2a_2^2-a_3\right){\psi }_3{w}^2+\dots ,\hfill \end{array}$$

(18)

$${\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}\left(\mathcal{R}\left(\zeta \right)\right)-1=1+{\mathcal{L}}_{p,r,1}\left(\varkappa \right)v_1\zeta +\left({\mathcal{L}}_{p,r,1}\left(\varkappa \right)v_2+{\mathcal{L}}_{p,r,2}\left(\varkappa \right)v_1^2\right){\zeta }^2+\dots ,$$

(19)

and

$${\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}\left(\mathcal{S}\left(w\right)\right)-1=1+{\mathcal{L}}_{p,r,1}\left(\varkappa \right)s_{1}w+\left({\mathcal{L}}_{p,r,1}\left(\varkappa \right)s_2+{\mathcal{L}}_{p,r,2}\left(\varkappa \right)s_1^2\right)w^2+\dots$$

(20)

Next, using (17) and (19), equating the corresponding coefficients of $\zeta$ in (15), we obtain

$$2(2-\gamma ){\psi }_2{a}_2={\mathcal{L}}_{p,r,1}\left(\varkappa \right)v_1,$$

(21)

$$2\left(3-2\gamma \right){\psi }_3{a}_3={\mathcal{L}}_{p,r,1}\left(\varkappa \right)v_2+{\mathcal{L}}_{p,r,2}\left(\varkappa \right)v_1^2.$$

(22)

By using (18) and (20), equating the corresponding coefficients of w in (16), we obtain

$$-2(2-\gamma ){\psi }_2{a}_2={\mathcal{L}}_{p,r,1}\left(\varkappa \right)s_1,$$

(23)

$$2\left(3-2\gamma \right)\left(2a_2^2-a_3\right){\psi }_3={\mathcal{L}}_{p,r,1}\left(\varkappa \right)s_2+{\mathcal{L}}_{p,r,2}\left(\varkappa \right)s_1^2.$$

(24)

From (21) and (23), we have

$$v_1=-s_1.$$

(25)

Squaring (21) and (23), and then adding the new relations, we obtain

$$8{\left(2-\gamma \right)}^2{a}_2^2{\psi }_2^2={\mathcal{L}}_{p,r,1}^2\left(\varkappa \right)\left(v_1^2+s_1^2\right).$$

(26)

Furthermore, by adding (22) and (24), we have

$$4\left(3-2\gamma \right){\psi }_3{a}_2^2={\mathcal{L}}_{p,r,1}\left(\varkappa \right)\left(v_2+s_2\right)+{\mathcal{L}}_{p,r,2}\left(\varkappa \right)\left(v_1^2+s_1^2\right).$$

(27)

Now, (26) can be written as

$$v_1^2+s_1^2=\frac{8{\left(2-\gamma \right)}^2}{{\mathcal{L}}_{p,r,1}^2\left(\varkappa \right)}{a}_2^2{\psi }_2^2.$$

Using the above equation and (27), we have

$$a_2^2=\frac{{\mathcal{L}}_{p,r,1}^3\left(\varkappa \right)\left(v_2+s_2\right)}{4\left(3-2\gamma \right){\mathcal{L}}_{p,r,1}^2\left(\varkappa \right){\psi }_3-8{\mathcal{L}}_{p,r,2}\left(\varkappa \right){\left(2-\gamma \right)}^2{\psi }_2^2},$$

(28)

and from (2) and (14), we have

$$\left|a_2\right|\le \frac{\left|p\left(\varkappa \right)\right|\sqrt{\left|p\left(\varkappa \right)\right|}}{\sqrt{2\left|\left(3-2\gamma \right)p^2\left(\varkappa \right){\psi }_3-2(p^2\left(\varkappa \right)+2r\left(\varkappa \right)){\left(2-\gamma \right)}^2{\psi }_2^2\right|}},$$

which gives the bound for $\left|a_2\right|$.

To derive the coefficient bound of $\left|a_3\right|$, subtracting (24) from (22), we obtain

$$4\left(3-2\gamma \right){\psi }_3\left(a_3-a_2^2\right)={\mathcal{L}}_{p,r,1}\left(\varkappa \right)\left(v_2-s_2\right)+{\mathcal{L}}_{p,r,2}\left(\varkappa \right)\left(v_1^2-s_1^2\right).$$

(29)

Form (25), (26) and (29), we obtain

$$a_3=\frac{{\mathcal{L}}_{p,r,1}\left(\varkappa \right)\left(v_2-s_2\right)}{4\left(3-2\gamma \right){\psi }_3}+\frac{{\mathcal{L}}_{p,r,1}^2\left(\varkappa \right)\left(v_1^2+s_1^2\right)}{8{\left(2-\gamma \right)}^2{\psi }_2^2}.$$

(30)

Again, using (2) and (14), we obtain

$$\left|a_3\right|\le \frac{\left|p\right(\varkappa \left)\right|}{2\left(3-2\gamma \right){\psi }_3}+\frac{p^2\left(\varkappa \right)}{4{\left(2-\gamma \right)}^2{\psi }_2^2}.$$

□

Corollary 1.

Putting $\gamma =0$ in Theorem 1, we obtain the following corollary: ${\mathcal{M}}_{sc}(n,\delta ,\upsilon ,\sigma ,m,$$p\left(\varkappa \right),r\left(\varkappa \right))$

$$\left|a_2\right|\le \frac{\left|p\left(\varkappa \right)\right|\sqrt{\left|p\left(\varkappa \right)\right|}}{\sqrt{2\left|3p^2\left(\varkappa \right){\psi }_3-8(p^2\left(\varkappa \right)+2r\left(\varkappa \right)){\psi }_2^2\right|}},$$

and

$$\left|a_3\right|\le \frac{\left|p\right(\varkappa \left)\right|}{6{\psi }_3}+\frac{p^2\left(\varkappa \right)}{16{\psi }_2^2},$$

where ${\psi }_j$, $j\in \{2,3\}$, are given by (4).

Putting $\gamma =1$ in Theorem 1, we obtain the following corollary: ${\mathcal{N}}_{sc}(n,\delta ,\upsilon ,\sigma ,m,$$p\left(\varkappa \right),r\left(\varkappa \right))$.

Corollary 2.

If the function $\mathcal{T}$ given by (1) is in the class ${\mathcal{N}}_{sc}\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)$, then, where ${\psi }_j$, $j\in \{2,3\}$ are given by (4).

$$\left|a_2\right|\le \frac{\left|p\left(\varkappa \right)\right|\sqrt{\left|p\left(\varkappa \right)\right|}}{\sqrt{2\left|p^2\left(\varkappa \right){\psi }_3-2(p^2\left(\varkappa \right)+2r\left(\varkappa \right)){\psi }_2^2\right|}},$$

and

$$\left|a_3\right|\le \frac{\left|p\right(\varkappa \left)\right|}{2{\psi }_3}+\frac{p^2\left(\varkappa \right)}{4{\psi }_2^2},$$

where ${\psi }_j$, $j\in \{2,3\}$, are given by (4).

3. Fekete–Szego Problem for the Function Class ${\mathcal{S}}_{\mathit{sc}}^{\mathbf{\gamma }}\left(\mathit{n},\mathbf{\delta },\mathbf{\upsilon },\mathbf{\sigma },\mathit{m},\mathit{p}\left(\mathbf{\varkappa }\right),\mathit{r}\left(\mathbf{\varkappa }\right)\right)$

Theorem 2.

If the function $\mathcal{T}$ given by (1) is in the class ${\mathcal{S}}_{sc}^{\gamma }\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)$, then

$$\left|a_3-\mu a_2^2\right|\le \left|p\left(\varkappa \right)\right|\left(|\mathcal{K}+\mathcal{L}|+|\mathcal{K}-\mathcal{L}|\right),$$

(31)

where

$$\mathcal{K}=\frac{\left(1-\mu \right)p^2\left(\varkappa \right)}{2[\left(3-2\gamma \right)p^2\left(\varkappa \right){\psi }_3-(p^2\left(\varkappa \right)+2r\left(\varkappa \right)){\left(2-\gamma \right)}^2{\psi }_2^2]},$$

(32)

and

$$\mathcal{L}=\frac{1}{2\left(3-2\gamma \right){\psi }_3},$$

where $\mu \in \mathbb{C}$, and ${\psi }_j$, $j\in \{2,3\}$, are given by (4).

Proof. 

If $\mathcal{T}\in {\mathcal{S}}_{sc}^{\gamma }\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)$. From (25) and (29), we have

$$a_3-a_2^2=\frac{{\mathcal{L}}_{p,r,1}\left(\varkappa \right)\left(v_2-s_2\right)}{4\left(3-2\gamma \right){\psi }_3}.$$

(33)

Multiplying (28) by $(1-\mu )$, we obtain

$$\begin{array}{ccc}& & \left(1-\mu \right)a_2^2\hfill \\ & =& \frac{\left(1-\mu \right){\mathcal{L}}_{p,r,1}^3\left(\varkappa \right)\left(v_2+s_2\right)}{4\left(3-2\gamma \right){\mathcal{L}}_{p,r,1}^2\left(\varkappa \right){\psi }_3-8{\mathcal{L}}_{p,r,2}\left(\varkappa \right){\left(2-\gamma \right)}^2{\psi }_2^2}.\hfill \end{array}$$

(34)

Adding (33) and (34), we obtain

$$a_3-\mu a_2^2={\mathcal{L}}_{p,r,1}\left(\varkappa \right)\left[\left(\mathcal{K}+\mathcal{L}\right)v_2+\left(\mathcal{K}-\mathcal{L}\right)s_2\right],$$

(35)

where $\mathcal{K}$ and $\mathcal{L}$ are given by (32), and taking the absolute value of (35), from (14), we obtain the inequality (31). The proof is complete. □

Remark 1.

A simple computation shows that the inequality $\left|\mathcal{K}\right|\le \mathcal{L}$ is equivalent to

$$\begin{array}{ccc}& & \left|\mu -1\right|\hfill \\ & \le & \left|1-\frac{\left(p^2\left(\varkappa \right)+2r\left(\varkappa \right)\right){\left(2-\gamma \right)}^2{\psi }_2^2}{\left(3-2\gamma \right)p^2\left(\varkappa \right){\psi }_3}\right|,\hfill \end{array}$$

and then from Theorem 2, we obtain the next result:

(i) If the function $\mathcal{T}$ given by (1) belongs to the class ${\mathcal{S}}_{sc}^{\gamma }\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)$, then

$$\left|a_3-\mu a_2^2\right|\le \frac{\left|p\left(\varkappa \right)\right|}{\left(3-2\gamma \right){\psi }_3},$$

where $\mu \in \mathbb{C}$, with

$$\left|\mu -1\right|\le \left|1-\frac{\left(p^2\left(\varkappa \right)+2r\left(\varkappa \right)\right){\left(2-\gamma \right)}^2{\psi }_2^2}{\left(3-2\gamma \right)p^2\left(\varkappa \right){\psi }_3}\right|,$$

and ${\psi }_j$, $j\in \{2,3\}$, are given by (4);

(ii) If the function $\mathcal{T}$ given by (1) belongs to the class ${\mathcal{S}}_{sc}^{\gamma }\left(n,\delta ,\upsilon ,\sigma ,m,p\left(\varkappa \right),r\left(\varkappa \right)\right)$, then

$$\left|a_3-\mu a_2^2\right|\le \frac{\left|p\left(\varkappa \right)\right|}{3{\psi }_3},$$

where $\mu \in \mathbb{C}$, with

$$\left|\mu -1\right|\le \left|1-\frac{4\left(p^2\left(\varkappa \right)+2r\left(\varkappa \right)\right){\psi }_2^2}{3p^2\left(\varkappa \right){\psi }_3}\right|,$$

and ${\psi }_j$, $j\in \{2,3\}$, are given by (4).

4. Conclusions

We defined a subclass of bi-univalent functions connected with the combination binomial series and Babalola operator and Lucas polynomials. We obtained non-sharp bounds for the initial coefficients and the Fekete–Szegö inequalities for the functions in this new class. Some interesting corollaries and applications of the results were also discussed. One can take special cases of Lucas’s polynomial ${\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}$, to give special results for Definition 2. In conclusion, another opportunity to further this topic is provided by substituting Lucas’s polynomial ${\mathcal{G}}_{{\mathcal{L}}_{p,r,n}\left(\varkappa \right)}$ for the Horadam polynomials $h_n\left(x\right).$