Certain Subclasses of Analytic Functions Associated with Generalized Telephone Numbers

Abstract

The goal of this article is to contemplate coefficient estimates for a new class of analytic functions f associated with generalized telephone numbers to originate certain initial Taylor coefficient estimates and Fekete–Szegö inequality for f in the new function class. Comparable results have been attained for the function $f^{-1}.$ Further application of our outcomes to certain functions demarcated by convolution products with certain normalized analytic functions in the open unit disk are specified, and we obtain Fekete–Szegö variations for this new function class defined over Poisson and Borel distribution series.

1. Introduction and Preliminaries

1.1. Telephonic Numbers:

The conventional telephone numbers (or involution numbers) are quantified by the recurrence relation

$$\mathfrak{T}\left(n\right)=\mathfrak{T}(n-1)+(n-1)\mathfrak{T}(n-2)forn\ge 2$$

with initial conditions

$$\mathfrak{T}\left(0\right)=\mathfrak{T}\left(1\right)=1.$$

In the year 1800, Heinrich August Rothe associated these numbers over symmetric groups and figured out that $\mathfrak{T}\left(n\right)$ is the number of involutions (self-inverse permutations) in the symmetric group (see, for example, [1,2]). Meanwhile, involutions resemble a standard Young tableaux, so it is correct that the nth involution number is also the number of the Young tableaux on the set $1,2,\dots ,n$ (for details, see [3]). According to John Riordan, the above recurrence relation yields the number of construction patterns in a telephone system with n subscribers (see [4]). Wlochand Wolowiec-Musial [5] familiarized generalized telephone numbers $\mathfrak{T}(\ell ,n)$ defined for integers $n\ge 0$ and $\ell \ge 1$ by the ensuing recursion:

$$\mathfrak{T}(\ell ,n)=\ell T(\ell ,n-1)+(n-1)\mathfrak{T}(\ell ,n-2)$$

with initial conditions

$$\mathfrak{T}(\ell ,0)=1,\mathfrak{T}(\ell ,1)=\ell ,$$

and deliberate few properties. Lately, Bednarz and Wolowiec-Musial [6] gave accessible generalization of telephone numbers by

$${\mathfrak{T}}_{\ell }\left(n\right)={\mathfrak{T}}_{\ell }(n-1)+\ell (n-1){\mathfrak{T}}_{\ell }(n-2)$$

for integers $n\ge 2$ and $\ell \ge 1$ with initial conditions

$${\mathfrak{T}}_{\ell }\left(0\right)={\mathfrak{T}}_{\ell }\left(1\right)=1$$

They presented the generating function, matrix generators and direct formula for these numbers. In addition, they demonstrated a few properties of these numbers related with congruence. Most recently, Deniz [7] derived the exponential generating function (or the summation formula) for ${\mathfrak{T}}_{\ell }\left(n\right)$ as follows:

$$e^{(x+\ell \frac{x}{2})}=\sum _{n=0}^{\infty }{\mathfrak{T}}_{\ell }\left(n\right)\frac{x^n}{n!}(\ell \ge 1)$$

As we can observe, if $\ell =1,$ then we obtain ${\mathfrak{T}}_{\ell }\left(n\right)\equiv \mathfrak{T}\left(n\right)$ classical telephone numbers. Clearly, ${\mathfrak{T}}_{\ell }\left(n\right)$ is for some values of n as

• ${\mathfrak{T}}_{\ell }\left(0\right)={\mathfrak{T}}_{\ell }\left(1\right)=1;$
• ${\mathfrak{T}}_{\ell }\left(2\right)=1+\ell ;$
• ${\mathfrak{T}}_{\ell }\left(3\right)=1+3\ell$;
• ${\mathfrak{T}}_{\ell }\left(4\right)=1+6\ell +3{\ell }^2$;
• ${\mathfrak{T}}_{\ell }\left(5\right)=1+10\ell +15{\ell }^2$;
• ${\mathfrak{T}}_{\ell }\left(6\right)=1+15\ell +45{\ell }^2+15{\ell }^3.$

We now consider the function

$$\begin{array}{ccc}\hfill \mathsf{\Psi }\left(\varsigma \right):& =& e^{(\varsigma +\ell \frac{{\varsigma }^2}{2})}\hfill \\ & =& 1+\varsigma +\frac{{\varsigma }^2}{2}+\frac{1+\ell }{6}{\varsigma }^3+\frac{1+3\ell }{24}{\varsigma }^4+\frac{3{\ell }^2+6\ell +1}{120}{\varsigma }^5+\cdots \hfill \end{array}$$

with its domain of definition as the open unit disk

$$\mathsf{\Delta }:=\left\{\varsigma :\varsigma \in \mathbb{C}\mathrm{and}\left|\varsigma \right|<1\right\}$$

(for details, see Deniz [7]). Congruently, $\mathsf{\Psi }\left(\varsigma \right)$ is an analytic in $\mathsf{\Delta },$ such that

• $\Re \left(\mathsf{\Psi }\right)>0\mathrm{in}\mathsf{\Delta }$;
• $\mathsf{\Psi }\left(0\right)=1,{\mathsf{\Psi }}^{\prime }\left(0\right)>0$;
• $\mathsf{\Psi }$ maps $\mathsf{\Delta }$ onto a region starlike with respect to 1 and symmetric with respect to the real axis.

1.2. The Analytic Functions

Let $\mathfrak{A}$ signify the collection of functions f that are holomorphic in $\mathsf{\Delta }$ and have the following normalized form:

$$f\left(\varsigma \right)=\varsigma +\sum _{n=2}^{\infty }{a}_n{\varsigma }^n,\varsigma \in \mathsf{\Delta },$$

(1)

with $f\left(0\right)=0=f^{\prime }\left(0\right)-1.$ Further, let $\mathfrak{S}$ be the sub-collection of the set $\mathfrak{A}$ comprising univalent functions. A function $f\in \mathfrak{S}$ is supposed to be starlike in $\mathsf{\Delta }$ if

$$\Re \left(\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}\right)>0,(\varsigma \in \mathsf{\Delta })$$

(2)

and a function $f\in \mathfrak{S}$ is supposed to be convex in $\mathsf{\Delta }$ if

$$\Re \left(\frac{{\left(\varsigma f^{\prime }\left(\varsigma \right)\right)}^{\prime }}{f^{\prime }\left(\varsigma \right)}\right)>0,(\varsigma \in \mathsf{\Delta })$$

(3)

holds. We correspondingly symbolize these function classes by ${\mathfrak{S}}^{*}$ and $\mathfrak{C}$ sustaining Principles (2) and (3).

For ${\mathsf{f}}_1,{\mathsf{f}}_2\in \mathfrak{A}$ and is analytic in $\mathsf{\Delta }$, so we say that ${\mathsf{f}}_1$ is subordinate to ${\mathsf{f}}_2$ if there exists a Schwarz function $w\left(\varsigma \right)$ that is holomorphic in $\mathsf{\Delta }$ with

$$w\left(0\right)=0\mathrm{and}\left|w\left(\varsigma \right)\right|<1(\varsigma \in \mathsf{\Delta }),$$

such that

$${\mathsf{f}}_1\left(\varsigma \right)={\mathsf{f}}_2\left(\omega \left(\varsigma \right)\right)(\varsigma \in \mathsf{\Delta }).$$

This subordination is symbolically written as follows:

$${\mathsf{f}}_1\prec {\mathsf{f}}_2\mathrm{or}{\mathsf{f}}_1\left(\varsigma \right)\prec {\mathsf{f}}_2\left(\varsigma \right)(z\in \mathsf{\Delta }).$$

In particular, if the function g is univalent in $\mathsf{\Delta },$ then the following equivalence holds true

$${\mathsf{f}}_1\left(0\right)={\mathsf{f}}_2\left(0\right)\mathrm{and}{\mathsf{f}}_1\left(\mathsf{\Delta }\right)\subset {\mathsf{f}}_2\left(\mathsf{\Delta }\right).$$

Raina and Sokół [8] have defined a function class

$${\mathfrak{S}}^{\star }\left(\psi \right)=\{f\in \mathfrak{A}:\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}\prec \psi \left(\varsigma \right)=:\varsigma +\sqrt{1+{\varsigma }^2},\varsigma \in \mathsf{\Delta },\}$$

and discussed some coefficient inequalities for $f\in {\mathfrak{S}}^{\star }\left(\psi \right)$. Further in view of the Alexander result between the class $f\in \mathfrak{C}\left(\psi \right)\iff \varsigma f^{\prime }\left(\varsigma \right)\in {\mathfrak{S}}^{*}\left(\psi \right)$, Sokół and Thomas [9] improved the results for $f\in \mathfrak{C}\left(\psi \right).$ In recent years, several authors studied many families of holomorphic and univalent functions by varying the function $\psi$, as illustrated below:

• For $\psi =\frac{1+A\varsigma }{1+B\varsigma }(-1\le B<A\le 1)$, we obtain the class ${\mathcal{S}}^{*}\left(A,B\right)$, see [10];
• For different values of A and B, the class ${\mathcal{S}}^{*}\left(\alpha \right)={\mathcal{S}}^{*}\left(1-2\alpha ,-1\right)$ is shown in [11];
• For $\psi =1+\frac{2}{{\pi }^2}{\left(log\frac{1+\sqrt{\varsigma }}{1-\sqrt{\varsigma }}\right)}^2$, the class was defined and studied in [12];
• For $\psi =\sqrt{1+\varsigma }$, the class is denoted by ${\mathcal{S}}_L^{*},$ details can be seen in [13] and further considered in [14];
• If $\psi =1+\frac{4}{3}\varsigma +\frac{2}{3}{\varsigma }^2$, then such class denoted by ${\mathcal{S}}_C^{*}$ was introduced in [15] and further considered by [16];
• For $\psi =e^{\varsigma }$, the class ${\mathcal{S}}_e^{*}$ was defined and dicussed in [17,18];
• For $\psi =cosh\left(\varsigma \right)$, the class is represented by ${\mathcal{S}}_{cosh}^{*},$ see [19];
• For $\psi =1+sin\left(\varsigma \right)$, the class is signified by ${\mathcal{S}}_{sin}^{*}$, see [20].

We recall a generalized function class due to Guo and Liu [21], as given below. For $\vartheta \geqq 0,\kappa \geqq 0$ and $0\leqq \wp <1$ and $f\in \mathfrak{A}$, we let $f\in Q(\vartheta ,\kappa ,\wp )$ if

$$\Re \left\{\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}{\left(\frac{f\left(\varsigma \right)}{z}\right)}^{\vartheta }+\kappa \left[1+\frac{\varsigma f^{″}\left(\varsigma \right)}{f^{\prime }\left(\varsigma \right)}-\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}+\vartheta \left(\frac{z{f}^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}-1\right)\right]\right\}>\wp$$

holds. Inspired fundamentally by the aforementioned works (see [8,22,23,24]) on coefficient estimate problems, the Fekete–Szegö problem on certain subclasses of analytic functions (refers [25,26,27,28,29,30,31]), and Murugusundaramoorthy [32,33], in this paper, we describe the new class ${\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$ given in Definition 1, which amalgamates the various subclasses of ${\mathfrak{S}}^{*}$ and $\mathfrak{C}$ in connotation with generalized telephonic numbers. We shall find estimations of the first few coefficients of f given by Equation (1) belonging to ${\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$, and we show the Fekete–Szegö inequality and similarly for $f^{-1}\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell ).$ Furthermore, we confer a certain application of our outcomes to functions defined over Borel and Poisson distribution series.

Now, we define the new family ${\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$:

Definition 1.

For $\vartheta \geqq 0,\kappa \geqq 0;$ $f\in \mathfrak{A}$ is in the class ${\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$ if

$$\begin{array}{ccc}& & \left\{\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}{\left(\frac{f\left(\varsigma \right)}{\varsigma }\right)}^{\vartheta }+\kappa \left[\frac{{\left(\varsigma f^{\prime }\left(\varsigma \right)\right)}^{\prime }}{f^{\prime }\left(\varsigma \right)}-\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}+\vartheta \left(\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}-1\right)\right]\right\}\hfill \\ \hfill {\displaystyle }& & \prec e^{(\varsigma +\ell \frac{{\varsigma }^2}{2})}=:\mathsf{\Psi }\left(\varsigma \right);\varsigma =r{e}^{i\theta }\in \mathsf{\Delta }.\hfill \end{array}$$

(4)

By specializing the parameters $\vartheta$ and $\kappa$, we state the following new subclasses of $\mathfrak{S}$, as illustrated below, which are not discussed sofar for functions associated with telephonic numbers:

Remark 1.

1. 

${\displaystyle {\mathfrak{W}}_0^0(\mathsf{\Psi },\ell )\equiv {\mathfrak{S}}^{*}(\mathsf{\Psi },\ell )=\left\{f\in \mathfrak{A}:\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}\prec e^{(\varsigma +\ell \frac{{\varsigma }^2}{2})};\varsigma =r{e}^{i\theta }\in \mathsf{\Delta }.\right\}}$

2. 

${\displaystyle {\mathfrak{W}}_0^{\kappa }(\mathsf{\Psi },\ell )\equiv {\mathfrak{W}}_{\kappa }(\mathsf{\Psi },\ell )=\left\{f\in \mathfrak{A}:\left[\kappa \frac{{\left(\varsigma f^{\prime }\left(\varsigma \right)\right)}^{\prime }}{f^{\prime }\left(\varsigma \right)}+(1-\kappa )\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}\right]\right.}$

$\left.\prec e^{(\varsigma +\ell \frac{{\varsigma }^2}{2})};\varsigma =r{e}^{i\theta }\in \mathsf{\Delta }\right\}$

3. 

${\mathfrak{W}}_0^1(\mathsf{\Psi },\ell )\equiv \mathfrak{C}(\mathsf{\Psi },\ell )=\left\{f\in \mathfrak{A}:\frac{{\left(\varsigma f^{\prime }\left(\varsigma \right)\right)}^{\prime }}{f^{\prime }\left(\varsigma \right)}\prec e^{(\varsigma +\ell \frac{{\varsigma }^2}{2})};\varsigma =r{e}^{i\theta }\in \mathsf{\Delta }\right\}$

4. 

${\mathfrak{W}}_{\vartheta }^0(\mathsf{\Psi },\ell )\equiv {\mathfrak{B}}_{\vartheta }(\mathsf{\Psi },\ell )=\left\{f\in \mathfrak{A}:\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}{\left(\frac{f\left(\varsigma \right)}{z}\right)}^{\vartheta }\prec e^{(\varsigma +\ell \frac{{\varsigma }^2}{2})};\varsigma =r{e}^{i\theta }\in \mathsf{\Delta }\right\}$

5. 

${\mathfrak{W}}_1^0(\mathsf{\Psi },\ell )\equiv \mathfrak{R}(\mathsf{\Psi },\ell )=\left\{f\in \mathfrak{A}:f^{\prime }\left(\varsigma \right)\prec e^{(\varsigma +\ell \frac{{\varsigma }^2}{2})};\varsigma =r{e}^{i\theta }\in \mathsf{\Delta }\right\}$

2. Coefficient Estimates and Fekete–Szegö Inequality

To prove our main result, we need the following:

Let $\mathfrak{P}$ be the family of functions holomorphic in $\mathsf{\Delta }$ with $\mathsf{p}\left(0\right)=1$ and

$$\Re \left(\mathsf{p}\right(\varsigma \left)\right)>0(\varsigma \in \mathsf{\Delta }),$$

is given by

$$\mathsf{p}\left(\varsigma \right)=1+c_{1}z+c_2{\varsigma }^2+\cdots .$$

(5)

Lemma 1

([34]). If $p\left(\varsigma \right)\in \mathfrak{P}$ and assumed as in Equation (5), then

$$|c_2-v{c}_1^2|\leqq \left\{\begin{array}{ccc}-4v+2,\hfill & if\hfill & v\leqq 0,\hfill \\ 2,\hfill & if\hfill & 0\leqq v\leqq 1,\hfill \\ 4v-2,\hfill & if\hfill & v\geqq 1.\hfill \end{array}\right.$$

when $v<0$ or $v>1$, the equality holds true for ${\displaystyle p_1\left(\varsigma \right)=\frac{1+\varsigma }{1-\varsigma }}$ or one of its rotations. In addition, the equality holds if ${\displaystyle p_2\left(\varsigma \right)=\frac{1+{\varsigma }^2}{1-{\varsigma }^2},}$ when $0<v<1$ or one of its rotations. If $v=0$, the equality holds if and only if

$$p_3\left(\varsigma \right)=\left(\frac{1}{2}+\frac{1}{2}\eta \right)\frac{1+\varsigma }{1-\varsigma }+\left(\frac{1}{2}-\frac{1}{2}\eta \right)\frac{1-\varsigma }{1+\varsigma }(0\leqq \eta \leqq 1)$$

or one of its rotations. If $v=1$, the equality holds if and only if $p_1$ is the reciprocal of one of the functions such that the equality holds in the case of $v=0$.

Lemma 2

([35]). If $p\left(\varsigma \right)\in \mathfrak{P}$ and assumed as in Equation (5), then

$$\begin{array}{cc}\hfill \mid c_n\mid & \le 2,(n\ge 1)\mathrm{and}|c_2-\frac{c_1^2}{2}|\le 2-\frac{|c_1{|}^2}{2}.\hfill \end{array}$$

Lemma 3

([36]). If $p\left(\varsigma \right)\in \mathfrak{P}$ and assumed as in Equation (5), and $vs.\in \mathbb{C}$, then

$$|c_2-vs.c_1^2|\leqq 2max\{1,|2v-1|\}$$

(6)

and is sharp for the functions

$$p_1\left(\varsigma \right)=\frac{1+\varsigma }{1-\varsigma },p_2\left(\varsigma \right)=\frac{1+{\varsigma }^2}{1-{\varsigma }^2}.$$

Lemma 4

([37]). If $p\left(\varsigma \right)\in \mathfrak{P}$ and is assumed as in Equation (5), then for any $\hslash \in \mathbb{C}$,

$$\begin{array}{cc}\hfill |c_2-\hslash \frac{c_1^2}{2}|& \le max(2,2|\hslash -1\left|\right)=\left\{\begin{array}{cc}2,\hfill & 0\le \hslash \le 2;\hfill \\ 2|\hslash -1|,\hfill & elsewhere.\hfill \end{array}\right.\hfill \end{array}$$

the result is sharp for $p_1\left(\varsigma \right)=\frac{1+\varsigma }{1-\varsigma }$ or $p_2\left(\varsigma \right)=\frac{1+{\varsigma }^2}{1-{\varsigma }^2}$.

Theorem 1.

Let $\vartheta \geqq 0\mathrm{and}\kappa \geqq 0$ and f be assumed by Equation (1). If $f\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$, then

$$\begin{array}{ccc}\hfill |a_2|& \leqq & \frac{1}{(1+\vartheta )(1+\kappa )},\hfill \\ \hfill |a_3|& \leqq & \frac{1}{(\vartheta +2)(1+2\kappa )}max\{1,\frac{1}{2}|\left(\frac{{\displaystyle {\vartheta }^2+\vartheta -2(\vartheta +3)\kappa -2}}{{\left((1+\vartheta )(1+\kappa )\right)}^2}\right)-1-\ell |\}.\hfill \end{array}$$

These results are sharp.

Proof. 

Express the function $\mathsf{p}\left(\varsigma \right)$ by

$$\begin{array}{ccc}\hfill \mathsf{p}\left(\varsigma \right):& =& \frac{1+w\left(\varsigma \right)}{1-w\left(\varsigma \right)}=1+c_1\varsigma +c_2{\varsigma }^2+\cdots .\hfill \end{array}$$

It is informed to see that

$$\begin{array}{ccc}\hfill w\left(\varsigma \right)& =& \frac{\mathsf{p}\left(\varsigma \right)-1}{\mathsf{p}\left(\varsigma \right)+1}\hfill \\ & =& \frac{1}{2}\left[c_1\varsigma +\left(c_2-\frac{c_1^2}{2}\right){\varsigma }^2+\left(c_3-c_1{c}_2+\frac{c_1^3}{4}\right){\varsigma }^3+\cdots \right],\hfill \end{array}$$

(7)

where $w\left(\varsigma \right)$, $w\left(0\right)=0;\left|w\right(\varsigma \left)\right|<1$ in $\mathsf{\Delta }$ is a Schwarz function and regular in $\mathsf{\Delta }$. We perceive that $\Re \left(\mathsf{p}\right(\varsigma \left)\right)>0$ and $\mathsf{p}\left(0\right)=1$; thus

$$\begin{array}{ccc}\hfill \mathsf{\Psi }\left(w\right(\varsigma \left)\right)& =& e^{(\frac{\mathsf{p}\left(\varsigma \right)-1}{\mathsf{p}\left(\varsigma \right)+1}+\ell \frac{{[\frac{\mathsf{p}\left(\varsigma \right)-1}{\mathsf{p}\left(\varsigma \right)+1}]}^2}{2})}\hfill \\ & =& 1+\frac{c_1}{2}\varsigma +\left(\frac{c_2}{2}+\frac{(\ell -1)c_1^2}{8}\right){\varsigma }^2+\left(\frac{c_3}{2}+(\ell -1)\frac{c_1{c}_2}{4}+\frac{(1-3\ell )}{48}{c}_1^3\right){\varsigma }^3+.....\hfill \end{array}$$

(8)

If $f\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$, then there is a Schwarz function $w\left(\varsigma \right)$ such that

$$\begin{array}{ccc}\hfill & & \frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}{\left(\frac{f\left(\varsigma \right)}{\varsigma }\right)}^{\vartheta }+\kappa \left[\frac{{\left(\varsigma f^{\prime }\left(\varsigma \right)\right)}^{\prime }}{f^{\prime }\left(\varsigma \right)}-\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}+\vartheta \left(\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}-1\right)\right]\hfill \\ & & =\mathsf{\Psi }\left(w\right(\varsigma \left)\right)\hfill \\ & & =e^{(w\left(\varsigma \right)+\kappa \frac{{\left[w\left(\varsigma \right)\right]}^2}{2})}.\hfill \end{array}$$

(9)

For assumed $f\left(\varsigma \right)$ as in Equation (1), a computation confirms that

$$\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}=1+a_2\varsigma +(2a_3-a_2^2){\varsigma }^2+(3a_4+a_2^3-3a_3{a}_2){\varsigma }^3+\cdots .$$

Similarly, we have

$$1+\frac{\varsigma f^{″}\left(\varsigma \right)}{f^{\prime }\left(\varsigma \right)}=1+2a_2\varsigma +(6a_3-4a_2^2){\varsigma }^2+\cdots .$$

Let us describe the function $\mathfrak{W}\left(\varsigma \right)$ by

$$\begin{array}{ccc}\hfill \mathfrak{W}\left(\varsigma \right):& =& \frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}{\left(\frac{f\left(\varsigma \right)}{\varsigma }\right)}^{\vartheta }+\kappa \left[\frac{{\left(\varsigma f^{\prime }\left(\varsigma \right)\right)}^{\prime }}{f^{\prime }\left(\varsigma \right)}-\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}+\vartheta \left(\frac{\varsigma f^{\prime }\left(\varsigma \right)}{f\left(\varsigma \right)}-1\right)\right]\hfill \\ \hfill \end{array}$$

(10)

An easy computation shows that

$$\begin{array}{ccc}\mathfrak{W}\left(\varsigma \right)\hfill & =\hfill & {\displaystyle 1+(1+\vartheta )(1+\kappa )a_2\varsigma +(\vartheta +2)(1+2\kappa )a_3{\varsigma }^2}\hfill \\ \hfill & +\hfill & \left({\displaystyle \frac{{\vartheta }^2+\vartheta }{2}-(\vartheta +3)\kappa -1}\right)a_2^2{\varsigma }^2+\cdots \hfill \\ \hfill & =\hfill & 1+b_1\varsigma +b_2{\varsigma }^2+\cdots .\hfill \end{array}$$

(11)

Now by Equations (8) and (11),

$$b_1=\frac{c_1}{2}\mathrm{and}{b}_2=\frac{c_2}{2}+\frac{(\ell -1)c_1^2}{8}.$$

(12)

In view of Equations (11) and (12), we see that

$$\begin{array}{ccc}\hfill b_1& =& (1+\vartheta )(1+\kappa )a_2,\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill b_2& =& (\vartheta +2)(1+2\kappa )a_3+\left({\displaystyle \frac{{\vartheta }^2+\vartheta }{2}-(\vartheta +3)\kappa -1}\right)a_2^2\hfill \end{array}$$

(14)

or equivalently, we have

$$\begin{array}{ccc}\hfill a_2& =& \frac{c_1}{2(1+\vartheta )(1+\kappa )},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill a_3& =& \frac{1}{(\vartheta +2)(1+2\kappa )}\left(\frac{c_2}{2}-\frac{c_1^2}{8}\left[{\displaystyle 1-\ell +\frac{{\displaystyle {\vartheta }^2+\vartheta -2(\vartheta +3)\kappa -2}}{{\left((1+\vartheta )(1+\kappa )\right)}^2}}\right]\right)\hfill \\ & =& \frac{1}{2(\vartheta +2)(1+2\kappa )}\left(c_2-\frac{c_1^2}{4}\left[{\displaystyle 1-\ell +\frac{{\displaystyle {\vartheta }^2+\vartheta -2(\vartheta +3)\kappa -2}}{{\left((1+\vartheta )(1+\kappa )\right)}^2}}\right]\right).\hfill \end{array}$$

(16)

Therefore, by Lemma 2, we have

$$a_2\le \frac{1}{(1+\vartheta )(1+\kappa )}$$

and by means of Approximation (6) specified in Lemma 3, we have

$$\begin{array}{ccc}\hfill |a_3|& \leqq & \frac{1}{(2+\vartheta )(1+2\kappa )}max\{1,|2\times \frac{1}{4}\left({\displaystyle 1-\ell +\frac{{\displaystyle {\vartheta }^2+\vartheta -2(\vartheta +3)\kappa -2}}{{\left((1+\vartheta )(1+\kappa )\right)}^2}}\right)-1|\}\hfill \\ & =& \frac{1}{(2+\vartheta )(1+2\kappa )}max\{1,\frac{1}{2}|\left(\frac{{\displaystyle {\vartheta }^2+\vartheta -2(\vartheta +3)\kappa -2}}{{\left((1+\vartheta )(1+\kappa )\right)}^2}\right)-1-\ell |\}.\hfill \end{array}$$

To show that the bounds are sharp, we express $K_{{\mathsf{\Psi }}_n}\left(\varsigma \right)$, ${\mathsf{\Psi }}_n=\mathsf{\Psi }\left({\varsigma }^{n-1}\right)(n\geqq 2)$ with $K_{{\mathsf{\Psi }}_n}\left(0\right)=0={\left[K_{{\mathsf{\Psi }}_n}\right]}^{\prime }\left(0\right)-1$, by

$$\begin{array}{cc}& \frac{\varsigma K_{{\mathsf{\Psi }}_n}^{\prime }\left(\varsigma \right)}{K_{{\mathsf{\Psi }}_n}\left(\varsigma \right)}{\left(\frac{K_{{\mathsf{\Psi }}_n}\left(\varsigma \right)}{\varsigma }\right)}^{\vartheta }+\kappa \left[\frac{{\left(\varsigma K_{{\mathsf{\Psi }}_n}^{\prime }\left(\varsigma \right)\right)}^{\prime }}{K_{{\mathsf{\Psi }}_n}^{\prime }\left(\varsigma \right)}-\frac{\varsigma K_{{\mathsf{\Psi }}_n}^{\prime }\left(\varsigma \right)}{K_{{\mathsf{\Psi }}_n}\left(\varsigma \right)}+\vartheta \left(\frac{\varsigma K_{{\mathsf{\Psi }}_n}^{\prime }\left(\varsigma \right)}{K_{{\mathsf{\Psi }}_n}\left(\varsigma \right)}-1\right)\right]\\ & =\mathsf{\Psi }\left({\varsigma }^{n-1}\right).\end{array}$$

Evidently, $K_{{\mathsf{\Psi }}_n}\in {\mathfrak{W}}_{\vartheta ,\kappa }(\mathsf{\Psi },\ell )$. We inscribe $K_{\mathsf{\Psi }}:=K_{{\mathsf{\Psi }}_2}$. That is, when $n=2,\mathrm{we}\mathrm{obtain}\mathsf{\Psi }\left(\varsigma \right)=e^{\varsigma +\frac{\ell {\varsigma }^2}{2}}.$ Thus,

$$\begin{array}{cc}\hfill f\left(\varsigma \right)& ={\int }_0^{\varsigma }\mathsf{\Psi }\left(t\right)dt\hfill \\ & ={\int }_0^z{e}^{t+\frac{\ell t^2}{2}}dt\hfill \\ & =\varsigma +\frac{{\varsigma }^2}{2}+\frac{1+\ell }{6}{\varsigma }^3+\frac{1+3\ell }{24}{\varsigma }^4+\frac{3{\ell }^2+6\ell +1}{120}{\varsigma }^5+\cdots .\hfill \end{array}$$

Here, we have $b_1=1$ and $b_2=1/2.$ By means of Equations (13) and (12), we obtain

$$|a_2|=\frac{1}{(1+\vartheta )(1+\kappa )}$$

and by using Equations (12) and (14), we have

$$c_2+\frac{(\ell -1)c_1^2}{4}=(\vartheta +2)(1+2\kappa )a_3+\left({\displaystyle \frac{{\vartheta }^2+\vartheta }{2}-(\vartheta +3)\kappa -1}\right)a_2^2.$$

Substituting for $a_2=\frac{1}{(1+\vartheta )(1+\kappa )}$ modest computation and taking the modulus yields

$$|a_3|=\frac{1}{2(2+\vartheta )(1+2\kappa )}\left|\frac{{\displaystyle {\vartheta }^2+\vartheta -2(\vartheta +3)\kappa -2}}{({\left(1+\vartheta \left)\right(1+\kappa )\right)}^2}-\ell -1\right|.$$

□

Remark 2.

Let $\vartheta =0and\kappa \geqq 0.$ If $f\in {\mathfrak{W}}_{\kappa }(\mathsf{\Psi },\ell )$ is given by Equation (1), then

$$\begin{array}{ccc}\hfill |a_2|& \leqq & \frac{1}{1+\kappa },\hfill \\ \hfill |a_3|& \leqq & \frac{1}{2(1+2\kappa )}max\{1,|\frac{{\kappa }^2+8\kappa +3}{2{(1+\kappa )}^2}+\ell |\}=\frac{1}{2(1+2\kappa )}\left(\frac{{\kappa }^2+8\kappa +3}{2(1+2\kappa ){(1+\kappa )}^2}+\ell \right).\hfill \end{array}$$

Remark 3.

Fixing $\vartheta =0and\kappa =0.$ If $f\in {\mathfrak{W}}_0^0(\mathsf{\Psi },\ell )\equiv {\mathfrak{S}}^{*}(\mathsf{\Psi },\ell )$ given by Equation (1), then

$$\begin{array}{c}\hfill |a_2|\leqq 1,and|a_3|\leqq \frac{1}{2}max\{1,|\frac{3}{2}+\ell |\}=\frac{1}{2}(\frac{3}{2}+\ell ).\end{array}$$

Remark 4.

Assuming $\vartheta =0and\kappa =1$, and if $f\in {\mathfrak{W}}_0^1(\mathsf{\Psi },\ell )\equiv \mathfrak{C}(\mathsf{\Psi },\ell )$ is given by Equation (1), then

$$\begin{array}{c}\hfill |a_2|\leqq \frac{1}{2},and\left|a_3\right|\leqq \frac{1}{6}max\{1,|\frac{1}{2}+\ell |\}=\frac{1}{6}(\frac{1}{2}+\ell ).\end{array}$$

Remark 5.

Taking $\kappa =0,$ and if $f\in {\mathfrak{W}}_{\vartheta }^0(\mathsf{\Psi },\ell )\equiv {\mathfrak{B}}_{\vartheta }(\mathsf{\Psi },\ell )$ given by (1), then

$$\begin{array}{ccc}\hfill |a_2|& \leqq & \frac{1}{1+\vartheta },\hfill \\ \hfill |a_3|& \leqq & \frac{1}{\vartheta +2}max\{1,\frac{1}{2}|\frac{{\vartheta }^2+\vartheta -2}{{\left(1+\vartheta \right)}^2}-1-\ell |\}=\frac{1}{2(1+\vartheta )}\left(\frac{\vartheta +3}{{(1+\vartheta )}^2}+\ell \right).\hfill \end{array}$$

Remark 6.

Let $\vartheta =1and\kappa =0.$ If $f\left(\varsigma \right)$ given by (1) belongs to ${\mathfrak{W}}_0^0(\mathsf{\Psi },\ell )=\mathfrak{R}(\mathsf{\Psi },\ell )$, then

$$\begin{array}{ccc}\hfill |a_2|& \leqq & \frac{1}{1+\kappa },\hfill \\ \hfill |a_3|& \leqq & \frac{1}{3}max\{1,\frac{1}{2}|1+\ell |\}=\frac{1}{6}\left(1+\ell \right).\hfill \end{array}$$

Theorem 2.

Let $0\leqq \mu \leqq 1,\vartheta \geqq 0,\kappa \geqq 0$ and if $f\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$ be given in (1), then

$$\begin{array}{ccc}\hfill |a_3-\mu a_2^2|& \leqq & \left\{\begin{array}{ccc}{\displaystyle \frac{1}{2\mathbf{q}}\left(1+\ell -\frac{\aleph }{{\mathbf{w}}^2}\right),}\hfill & if\hfill & \mu \leqq {\sigma }_1,\hfill \\ {\displaystyle \frac{1}{\mathbf{q}},}\hfill & if\hfill & {\sigma }_1\leqq \mu \leqq {\sigma }_2,\hfill \\ {\displaystyle \frac{1}{2\mathbf{q}}\left(-1-\ell +\frac{\aleph }{{\mathbf{w}}^2}\right),}\hfill & if\hfill & \mu \geqq {\sigma }_2,\hfill \end{array}\right.\hfill \end{array}$$

where, for convenience,

$$\begin{array}{c}\hfill {\sigma }_1=\frac{(\ell -1){\mathbf{w}}^2+2(\vartheta +3)\kappa -\mathbf{m}}{2\mathbf{q}};{\sigma }_2=\frac{(3+\ell ){\mathbf{w}}^2+2(\vartheta +3)\kappa -\mathbf{m}}{2\mathbf{q}};\end{array}$$

$$\aleph :=\mathbf{m}-2(\vartheta +3)\kappa +2\mu \mathbf{q},$$

(17)

$$\mathbf{m}:={\vartheta }^2+\vartheta -2,$$

(18)

$$\mathbf{q}:=(\vartheta +2)(1+2\kappa ),$$

(19)

and

$$\mathbf{w}:=(1+\vartheta )(1+\kappa ).$$

(20)

Proof. 

Using Equations (15) and (16), we obtain

$$\begin{array}{ccc}\hfill a_3-\mu a_2^2& =& \frac{1}{(\vartheta +2)(1+2\kappa )}\left(\frac{c_2}{2}-\frac{c_1^2}{8}\times \right.\hfill \\ & & \left.\left[1-\ell +\frac{{\vartheta }^2+\vartheta -2-2(\vartheta +3)\kappa +2\mu (\vartheta +2)(1+2\kappa )}{{\left((1+\vartheta )(1+\kappa )\right)}^2}\right]\right)\hfill \\ & =& \frac{1}{2(\vartheta +2)(1+2\kappa )}\left(c_2-\frac{c_1^2}{4}\right.\hfill \\ & & \left.\left[1-\ell +\frac{{\vartheta }^2+\vartheta -2-2(\vartheta +3)\kappa +2\mu (\vartheta +2)(1+2\kappa )}{{\left((1+\vartheta )(1+\kappa )\right)}^2}\right]\right)\hfill \\ & =& \frac{1}{2(\vartheta +2)(1+2\kappa )}\left(c_2-v{c}_1^2\right)\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill v:& =& \frac{1}{4}\left[1-\ell +\frac{{\vartheta }^2+\vartheta -2-2(\vartheta +3)\kappa +2\mu (\vartheta +2)(1+2\kappa )}{{\left((1+\vartheta )(1+\kappa )\right)}^2}\right]\hfill \\ & =& \frac{1}{4}\left[1-\ell +\frac{\mathbf{m}-2(\vartheta +3)\kappa +2\mu \mathbf{q}}{{\mathbf{w}}^2}\right].\hfill \end{array}$$

To illustrate that the bounds are sharp, we express $F_{\eta }$ and $G_{\eta }$$(0\leqq \eta \leqq 1)$, correspondingly, with $F_{\eta }\left(0\right)=0=F_{\eta }^{\prime }\left(0\right)-1$ and $G_{\eta }\left(0\right)=0=G_{\eta }^{\prime }\left(0\right)-1$ by

$$\begin{array}{ccc}\hfill \frac{\varsigma F_{\eta }^{\prime }\left(\varsigma \right)}{F_{\eta }\left(\varsigma \right)}{\left(\frac{F_{\eta }\left(\varsigma \right)}{\varsigma }\right)}^{\vartheta }& +& \kappa \left[\frac{{\left(\varsigma F_{\eta }^{\prime }\left(\varsigma \right)\right)}^{\prime }}{F_{\eta }^{\prime }\left(\varsigma \right)}-\frac{\varsigma F_{\eta }^{\prime }\left(\varsigma \right)}{F_{\eta }\left(\varsigma \right)}+\vartheta \left(\frac{\varsigma F_{\eta }^{\prime }\left(\varsigma \right)}{F_{\eta }\left(\varsigma \right)}-1\right)\right]\hfill \\ & =& {\rm Y}\left(\frac{\varsigma (\eta +\varsigma )}{1+\eta \varsigma }\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \frac{\varsigma G_{\eta }^{\prime }\left(\varsigma \right)}{G_{\eta }\left(\varsigma \right)}{\left(\frac{G_{\eta }\left(\varsigma \right)}{\varsigma }\right)}^{\vartheta }& +& \kappa \left[\frac{{\left(\varsigma G_{\eta }^{\prime }\left(\varsigma \right)\right)}^{\prime }}{G_{\eta }^{\prime }\left(\varsigma \right)}-\frac{\varsigma G_{\eta }^{\prime }\left(\varsigma \right)}{G_{\eta }\left(\varsigma \right)}+\vartheta \left(\frac{\varsigma G_{\eta }^{\prime }\left(\varsigma \right)}{G_{\eta }\left(\varsigma \right)}-1\right)\right]\hfill \\ & =& {\rm Y}\left(-\frac{\varsigma (\eta +\varsigma )}{1+\eta \varsigma }\right),\hfill \end{array}$$

respectively.

Clearly, the functions $K_{{{\rm Y}}_n}=\mathsf{\Psi }\left({\varsigma }^{n-1}\right),F_{\eta },G_{\eta }\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$. In addition, we write $K_{{{\rm Y}}_2}:=\mathsf{\Psi }\left(\varsigma \right)=e^{(\varsigma +\ell \frac{{\varsigma }^2}{2})}$.

If $\mu <{\sigma }_1$ or $\mu >{\sigma }_2$, then the equality holds if and only if f is $K_{{{\rm Y}}_2}$ or one of its rotations. The equality holds if and only if f is $K_{{{\rm Y}}_3}=\mathsf{\Psi }\left({\varsigma }^2\right)=e^{({\varsigma }^2+\ell \frac{{\varsigma }^4}{2})},$ i.e., $\mathsf{\Psi }\left({\varsigma }^2\right)=1+{\varsigma }^2+\frac{(1+\ell ){\varsigma }^4}{2}+\cdots ,$ when ${\sigma }_1<\mu <{\sigma }_2$ or one of its rotations. Here, we have $b_1=0;b_2=1$; thus, by substituting these values into Equations (13) and (14), we obtain sharp results.

The equivalence is true if and only if f is $F_{\eta }$ when $\mu ={\sigma }_1$ or one of its turnings. If $\mu ={\sigma }_2$, the equivalence holds if and only if f is $G_{\eta }$ or one of its rotation. □

By Lemma 3, we state the following:

Theorem 3.

Let $0\leqq \vartheta \leqq 1,and0\leqq \kappa \leqq 1.$ For any $\mu \in \mathbb{C},$ and if $f\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell ),$ then

$$\begin{array}{ccc}\hfill |a_3-\mu a_2^2|& \le & \frac{1}{(\vartheta +2)(1+2\kappa )}\hfill \\ & \times & max\left\{1,\frac{1}{2}\left|-1-\ell +\frac{{\vartheta }^2+\vartheta -2-2(\vartheta +3)\kappa +2\mu (\vartheta +2)(1+2\kappa )}{{\left((1+\vartheta )(1+\kappa )\right)}^2}\right|\right\}\hfill \\ & \le & \frac{1}{\mathbf{q}}max\left\{1,\frac{1}{2}\left|-1-\ell +\frac{\mathbf{m}-2(\vartheta +3)\kappa +2\mu \mathbf{q}}{{\mathbf{w}}^2}\right|\right\}\hfill \end{array}$$

holds.

3. Coefficient Inequalities for the Function $f^{-1}$

Theorem 4.

If $f\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$ and $f^{-1}\left(\varpi \right)=\varpi +{\displaystyle \sum _{n=2}^{\infty }}{d}_n{\varpi }^n$ is the inverse function of f with $\left|\varpi \right|<r_0;r_0\ge \frac{1}{4}$ the radius of the K$\stackrel{″}{o}$ebe domain of the class $f\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$, then

$$\begin{array}{cc}\hfill |d_2|& \le \frac{1}{2(1+\vartheta )(1+\kappa )}\hfill \\ \hfill |d_2|& \le \frac{1}{2\mathbf{q}}max\left\{1,\mid \frac{-(1+\ell ){\mathbf{w}}^2+4\mathbf{q}+\mathbf{m}-2(\vartheta +3)\kappa }{2{\mathbf{w}}^2}\mid \right\}.\hfill \end{array}$$

For any complex number μ, we have

$$\begin{array}{c}\hfill \mid d_3-\mu d_2^2\mid \le \frac{1}{\mathbf{q}}max\left\{1,\mid \frac{-(1+\ell ){\mathbf{w}}^2+\mathbf{m}-2(\vartheta +3)\kappa +(4-2\mu )\mathbf{q}}{2{\mathbf{w}}^2}\mid \right\}.\end{array}$$

(21)

Proof. 

As

$$f^{-1}\left(\varpi \right)=\varpi +\sum _{n=2}^{\infty }{d}_n{\varpi }^n,$$

(22)

it can be understood that

$$f^{-1}\left(f\left(\varsigma \right)\right)=f\left\{f^{-1}\left(\varsigma \right)\right\}=\varsigma .$$

(23)

From Equations (1) and (23), we note that

$$f^{-1}(\varsigma +\sum _{n=2}^{\infty }{a}_n{\varsigma }^n)=\varsigma .$$

(24)

From Equations (23) and (24), one can obtain

$$\varsigma +(a_2+d_2){\varsigma }^2+(a_3+2a_2{d}_2+d_3){\varsigma }^3+.........=\varsigma .$$

(25)

By associating the coefficients of $\varsigma$ and ${\varsigma }^2$ from Relation (25), we understand that

$$\begin{array}{cc}\hfill d_2& =-a_2\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill d_3& =2a_2^2-a_3.\hfill \end{array}$$

(27)

From Relations (15), (16), (26), and (27)

$$\begin{array}{cc}\hfill d_2& =-\frac{c_1}{2(1+\vartheta )(1+\kappa )};\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill d_3& =\frac{c_1^2}{2{\left[(1+\vartheta )(1+\kappa )\right]}^2}-\frac{1}{2(\vartheta +2)(1+2\kappa )}\hfill \\ & \times \left(c_2-\frac{c_1^2}{4}\left[{\displaystyle 1-\ell +\frac{{\displaystyle {\vartheta }^2+\vartheta -2(\vartheta +3)\kappa -2}}{{\left((1+\vartheta )(1+\kappa )\right)}^2}}\right]\right);\hfill \\ & =-\frac{1}{2\mathbf{q}}\left(c_2-\frac{(1-\ell ){\mathbf{w}}^2+4\mathbf{q}+\mathbf{m}-2(\vartheta +3)\kappa }{4{\mathbf{w}}^2}{c}_1^2\right).\hfill \end{array}$$

(29)

For any complex number $\mu$, consider

$$\begin{array}{c}\hfill d_3-\mu d_2^2=-\frac{1}{2\mathbf{q}}\left(c_2-\frac{(1-\ell ){\mathbf{w}}^2+\mathbf{m}-2(\vartheta +3)\kappa +(4-2\mu )\mathbf{q}}{4{\mathbf{w}}^2}{c}_1^2\right)\end{array}$$

(30)

Taking the modulus and by using Lemma 3 on the right-hand side of Equation (30), one can acquire the result as in Equation (21). Hence, this concludes the proof. □

Remark 7.

By appropriately specifying $\vartheta ,\kappa$ in Theorem 4, one can easily deduce the results for the function classes listed in Remark 1, which are new and have not been discussed so far in association with generalized telephone numbers.

4. Application to Functions Defined by Certain Distributions Based on Convolution

The distributions, such as Binomial, Poisson, Pascal, logarithm, and hypergeometric, and their applications to the class of univalent functions have been intensively studied by various researchers in different prospectives.

Let ${\displaystyle \phi \left(\varsigma \right)=\varsigma +\sum _{n=2}^{\infty }{\phi }_n{\varsigma }^n,({\phi }_n>0)}$ and $f\in \mathfrak{A}$, then

$${\displaystyle \mathcal{F}\left(\varsigma \right)=(f\ast \phi )\left(\varsigma \right)=\varsigma +\sum _{n=2}^{\infty }{\phi }_n{a}_n{\varsigma }^n.}$$

We define the class ${\mathfrak{W}}_{\vartheta ,\kappa }^{\phi }(\mathsf{\Psi },\ell )$ in the following way:

$${\mathfrak{W}}_{\vartheta ,\kappa }^{\phi }:=\{f\in \mathfrak{A}\mathrm{and}\mathcal{F}\left(\varsigma \right)\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )\}$$

(31)

where ${\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$ is as given in Definition 1.

In this section, for $f\in {\mathfrak{W}}_{\vartheta ,\kappa }^{\phi }(\mathsf{\Psi },\ell )$, we obtain $a_2$ and $a_3$ corresponding to $f\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$. Applying Theorem 2 to the function

$$\mathcal{F}\left(\varsigma \right)=(f\ast \phi )\left(\varsigma \right)=\varsigma +{\phi }_2{a}_2{\varsigma }^2+{\phi }_3{a}_3{\varsigma }^3+\cdots$$

(32)

we obtain the following Theorems 5 and 6:

Theorem 5.

Let $0\leqq \vartheta \leqq 1,and0\leqq \kappa \leqq 1.$ If $f\in {\mathfrak{W}}_{\vartheta ,\kappa }^{\phi }(\mathsf{\Psi },\ell ),$ then for $\mu \in \mathbb{C}$, we have

$$\begin{array}{ccc}\hfill |a_3-\mu a_2^2|& & =\frac{2}{(\vartheta +2)(1+2\kappa ){\phi }_3}\hfill \\ & & max\left\{1,\frac{1}{2}\left|-1-\ell +\frac{{\vartheta }^2+\vartheta -2-2(\vartheta +3)\kappa }{{\left((1+\vartheta )(1+\kappa )\right)}^2}+\frac{2\mu (\vartheta +2)(1+2\kappa ){\phi }_3}{{\left((1+\vartheta )(1+\kappa ){\phi }_2\right)}^2}\right|\right\}.\hfill \end{array}$$

Proof. 

For $f\left(\varsigma \right)\in {\mathfrak{W}}_{\vartheta ,\kappa }^{\phi }(\mathsf{\Psi },\ell )$, then by Equations (31) and (32), we have

$$\begin{array}{ccc}p\left(\varsigma \right):\hfill & =\hfill & \frac{\varsigma {\mathcal{F}}^{\prime }\left(\varsigma \right)}{\mathcal{F}\left(\varsigma \right)}{\left(\frac{\mathcal{F}\left(\varsigma \right)}{\varsigma }\right)}^{\vartheta }+\kappa \left[\frac{\varsigma {\left({\mathcal{F}}^{\prime }\left(\varsigma \right)\right)}^{\prime }}{{\mathcal{F}}^{\prime }\left(\varsigma \right)}-\frac{\varsigma {\mathcal{F}}^{\prime }\left(\varsigma \right)}{\mathcal{F}\left(\varsigma \right)}+\vartheta \left(\frac{\varsigma {\mathcal{F}}^{\prime }\left(\varsigma \right)}{\mathcal{F}\left(\varsigma \right)}-1\right)\right]\hfill \\ \hfill & =\hfill & 1+b_1\varsigma +b_2{\varsigma }^2+\cdots .\hfill \end{array}$$

(33)

Proceeding online similar to Theorem 1, we obtain

$$\begin{array}{c}\hfill \begin{array}{c}{\displaystyle p\left(\varsigma \right)=1+(1+\vartheta )(1+\kappa ){\phi }_2{a}_2\varsigma +(\vartheta +2)(1+2\kappa ){\phi }_3{a}_3{\varsigma }^2}\hfill \\ +\left({\displaystyle \frac{{\vartheta }^2+\vartheta }{2}-(\vartheta +3)\kappa -1}\right){\phi }_2^2{a}_2^2{\varsigma }^2+\cdots .\hfill \end{array}\end{array}$$

(34)

From Equations (13)–(16) and from Equation (34), we obtain

$$\begin{array}{ccc}& & a_2=\frac{c_1}{2(1+\vartheta )(1+\kappa ){\phi }_2}\hfill \end{array}$$

(35)

$$\begin{array}{ccc}& & a_3=\frac{1}{(\vartheta +2)(1+2\kappa ){\phi }_3}\left(\frac{c_2}{2}-\frac{c_1^2}{8}\left({\displaystyle 1-\ell +\frac{{\displaystyle {\vartheta }^2+\vartheta -2(\vartheta +3)\kappa -2}}{{\left((1+\vartheta )(1+\kappa ){\phi }_2\right)}^2}}\right)\right)\hfill \\ & & a_3-\mu a_2^2\hfill \\ & & =\frac{1}{(\vartheta +2)(1+2\kappa ){\phi }_3}\left(\frac{c_2}{2}-\frac{c_1^2}{8}\left({\displaystyle 1-\ell +\frac{{\displaystyle {\vartheta }^2+\vartheta -2(\vartheta +3)\kappa -2}}{{\left((1+\vartheta )(1+\kappa ){\phi }_2\right)}^2}}\right)\right)\hfill \\ & & -\mu \frac{c_1^2}{4{(1+\vartheta )}^2{(1+\kappa )}^2{\phi }_2^2}\hfill \\ & & =\frac{1}{(\vartheta +2)(1+2\kappa ){\phi }_3}\left(\frac{c_2}{2}-\frac{c_1^2}{8}\left({\displaystyle 1-\ell +\frac{{\displaystyle {\vartheta }^2+\vartheta -2(\vartheta +3)\kappa -2}}{{\left((1+\vartheta )(1+\kappa ){\phi }_2\right)}^2}}\right.\right.\hfill \\ & & -\left.\left.\mu \frac{2(\vartheta +2)(1+2\kappa ){\phi }_3}{{(1+\vartheta )}^2{(1+\kappa )}^2{\phi }_2^2}\right)\right).\hfill \end{array}$$

(36)

Thus, our result follows by taking the absolute value and applying Lemma 3. The equality holds for the functions given as

$$\frac{\varsigma {\mathcal{F}}^{\prime }\left(\varsigma \right)}{\mathcal{F}\left(\varsigma \right)}{\left(\frac{\mathcal{F}\left(\varsigma \right)}{z}\right)}^{\vartheta }+\kappa \left[\frac{{\left(\varsigma {\mathcal{F}}^{\prime }\left(\varsigma \right)\right)}^{\prime }}{{\mathcal{F}}^{\prime }\left(\varsigma \right)}-\frac{\varsigma {\mathcal{F}}^{\prime }\left(\varsigma \right)}{\mathcal{F}\left(\varsigma \right)}+\vartheta \left(\frac{\varsigma {\mathcal{F}}^{\prime }\left(\varsigma \right)}{\mathcal{F}\left(\varsigma \right)}-1\right)\right]=\mathsf{\Psi }\left(\varsigma \right)$$

and

$$\frac{\varsigma {\mathcal{F}}^{\prime }\left(\varsigma \right)}{\mathcal{F}\left(\varsigma \right)}{\left(\frac{\mathcal{F}\left(\varsigma \right)}{\varsigma }\right)}^{\vartheta }+\kappa \left[\frac{{\left(\varsigma {\mathcal{F}}^{\prime }\left(\varsigma \right)\right)}^{\prime }}{{\mathcal{F}}^{\prime }\left(\varsigma \right)}-\frac{z{\mathcal{F}}^{\prime }\left(\varsigma \right)}{\mathcal{F}\left(\varsigma \right)}+\vartheta \left(\frac{\varsigma {\mathcal{F}}^{\prime }\left(\varsigma \right)}{\mathcal{F}\left(\varsigma \right)}-1\right)\right]=\mathsf{\Psi }\left({\varsigma }^2\right)$$

□

Theorem 6.

Let $\vartheta \geqq 0,\kappa \geqq 0,0\leqq \mu \leqq 1$ and ${\phi }_n>0$ and f be given by Equation (1). If $f\in {\mathfrak{W}}_{\vartheta ,\kappa }^{\phi }(\mathsf{\Psi },\ell )$, then

$$\begin{array}{ccc}\hfill |a_3-\mu a_2^2|& \leqq & \left\{\begin{array}{ccc}{\displaystyle \frac{1}{2\mathbf{q}{\phi }_3}\left(1+\ell -\frac{{\aleph }_2}{2{\mathbf{w}}^2}\right),}\hfill & if\hfill & \mu \leqq {\sigma }_1,\hfill \\ {\displaystyle \frac{1}{\mathbf{q}{\phi }_3},}\hfill & if\hfill & {\sigma }_1\leqq \mu \leqq {\sigma }_2,\hfill \\ {\displaystyle \frac{1}{2\mathbf{q}{\phi }_3}\left(-1-\ell +\frac{{\aleph }_2}{2{\mathbf{w}}^2}\right),}\hfill & if\hfill & \mu \geqq {\sigma }_2,\hfill \end{array}\right.\hfill \end{array}$$

where, for convenience,

$$\begin{array}{ccc}\hfill {\sigma }_1& :=& \frac{{\phi }_2^2}{{\phi }_3}\left[\frac{(\ell -1){\mathbf{w}}^2+2(\vartheta +3)\kappa -\mathbf{m}}{2\mathbf{q}}\right],{\sigma }_2=\frac{{\phi }_2^2}{{\phi }_3}\left[\frac{(3+\ell ){\mathbf{w}}^2+2(\vartheta +3)\kappa -\mathbf{m}}{2\mathbf{q}}\right],\hfill \\ \hfill {\aleph }_2& :=& \mathbf{m}-2(\vartheta +3)\kappa +2\mu \frac{{\phi }_3}{{\phi }_2^2}\mathbf{q},\hfill \end{array}$$

and $\mathbf{m},\mathbf{q},\mathbf{w}$ are as defined in Equations (18)–(20).

Proof. 

Using Equations (35) and (36) and following the steps in Theorems 2 and 5, we obtain the desired result. □

Suppose $\mathcal{X}$ is the Poisson distributed variable, and $\mathcal{X}$ takes the values $0,1,2,3,\cdots$ with probabilities $e^{-m}$, $m\frac{e^{-m}}{1!}$, $m^2\frac{e^{-m}}{2!}$, $m^3\frac{e^{-m}}{3!},...$ correspondingly, where $m>0.$ Thus,

$$P(\mathcal{X}=r)=\frac{e^{-m}{m}^r}{r!},r=0,1,2,3,\cdots .$$

In [38], Porwal presented a power series whose coefficients are probabilities of the Poisson distribution

$$\mathcal{K}(m,\varsigma )=\varsigma +\sum _{n=2}^{\infty }\frac{e^{-m}{m}^{n-1}}{(n-1)!}{\varsigma }^n,\varsigma \in \mathsf{\Delta },$$

where $m>0.$ Through a ratio test, the radius of convergence of the above series is infinity. By means of the Hadamard product, Porwal [38] (see also, [22,39,40] presented a new linear operator ${\mathcal{I}}^m\left(\varsigma \right):\mathcal{A}\to \mathcal{A}$ defined by

$$\begin{array}{ccc}\hfill {\mathcal{I}}^{m}f=\mathcal{K}(m,\varsigma )\ast f\left(\varsigma \right)& =& \varsigma +\sum _{n=2}^{\infty }\frac{m^{n-1}}{(n-1)!}{e}^{-m}{a}_n{\varsigma }^n,\hfill \\ & =& \varsigma +\sum _{n=2}^{\infty }{\psi }_m{a}_n{\varsigma }^n,\hfill \\ & =& \varsigma +{\psi }_2{a}_2{\varsigma }^2+{\psi }_3{a}_3{\varsigma }^3+\cdots \varsigma \in \mathsf{\Delta },\hfill \end{array}$$

where ${\psi }_n=\frac{m^{n-1}}{(n-1)!}{e}^{-m}.$ Precisely,

$${\psi }_2=m{e}^{-m}$$

(37)

and

$${\psi }_3=\frac{m^2}{2}{e}^{-m}.$$

(38)

From Equations (37) and (38), by taking ${\phi }_2=m{e}^{-m}$ and ${\phi }_3=\frac{m^2}{2}{e}^{-m}$ in Theorems 5 and 6, we state the results in Theorem 7 and 8.

Theorem 7.

Let $0\leqq \kappa \leqq 1,and0\leqq \vartheta \leqq 1.$ For $\mu \in \mathbb{C}$ and if $f\in {\mathfrak{W}}_{\vartheta ,\kappa }^m\left(q\right),$ then,

$$\begin{array}{ccc}& & |a_3-\mu a_2^2|\hfill \\ & & =\frac{4}{(\vartheta +2)(1+2\kappa )m^2{e}^{-m}}\times \hfill \\ & & max\left\{1,\frac{1}{2}\left|-1-\ell +\frac{{\vartheta }^2+\vartheta -2-2(\vartheta +3)\kappa }{{\left((1+\vartheta )(1+\kappa )\right)}^2}+\frac{\mu (\vartheta +2)(1+2\kappa )}{{\left((1+\vartheta )(1+\kappa )\right)}^2{e}^{-m}}\right|\right\}.\hfill \end{array}$$

Theorem 8.

Let $0\leqq \mu \leqq 1,\vartheta \geqq 0,\kappa \geqq 0$ and ${\psi }_n>0.$ If $f\in {\mathfrak{W}}_{\vartheta ,\kappa }^m\left(q\right)$ be as given in Equation (1), then

$$\begin{array}{ccc}\hfill |a_3-\mu a_2^2|& \leqq & \left\{\begin{array}{ccc}{\displaystyle \frac{1}{\mathbf{q}{m}^2{e}^{-m}}\left(1+\ell -\frac{{\aleph }_2}{2{\mathbf{w}}^2}\right),}\hfill & if\hfill & \mu \leqq {\sigma }_1,\hfill \\ {\displaystyle \frac{2}{\mathbf{q}{m}^2{e}^{-m}},}\hfill & if\hfill & {\sigma }_1\leqq \mu \leqq {\sigma }_2,\hfill \\ {\displaystyle \frac{1}{\mathbf{q}{m}^2{e}^{-m}}\left(-1-\ell +\frac{{\aleph }_2}{2{\mathbf{w}}^2}\right),}\hfill & if\hfill & \mu \geqq {\sigma }_2,\hfill \end{array}\right.\hfill \end{array}$$

where, for convenience,

$$\begin{array}{ccc}\hfill {\sigma }_1& :=& e^{-m}\left[\frac{(\ell -1){\mathbf{w}}^2+2(\vartheta +3)\kappa -\mathbf{m}}{2\mathbf{q}}\right],{\sigma }_2=e^{-m}\left[\frac{(3+\ell ){\mathbf{w}}^2+2(\vartheta +3)\kappa -\mathbf{m}}{2\mathbf{q}}\right],\hfill \\ \hfill {\aleph }_2& :=& \mathbf{m}-2(\vartheta +3)\kappa +\frac{\mu }{e^{-m}}\mathbf{q},\hfill \end{array}$$

and $\mathbf{m},\mathbf{q},$ and $\mathbf{w}$ are as defined in Equations (18)–(20).

Recently, there has been an interest in the application of a Borel distribution to the results for various subclasses of analytic and univalent functions (see [41,42,43]). Now, we state certain functional inequalities for $f\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$.

A discrete random variable X is said to follow a Borel distribution with parameter $\varrho$ if its probability mass function $p\left(x\right)$ is given by

$$p(x=r)=\frac{{\left(\varrho r\right)}^{r-1}{e}^{-\varrho r}}{r!}r=1,2,3,\cdots .$$

(39)

Recently, for $0\le \varrho \le 1,$ Wanas and Khuttar [44] presented a power series

$$\mathfrak{B}(\varrho ,\varsigma )=\varsigma +\sum _{n=2}^{\infty }\frac{{\left(\varrho (n-1)\right)}^{n-2}{e}^{-\varrho (n-1)}}{(n-1)!}{\varsigma }^n(\varsigma \in \mathsf{\Delta })$$

(40)

whose coefficients are the probabilities of the Borel distribution. It has be shown that the radius of convergence of the above series is infinity through a ratio test.

Let us introduce a linear operator $L_{\varrho }:\mathcal{A}⟶\mathcal{A}$ defined by

$$\begin{array}{ccc}\hfill L_{\varrho }f\left(\varsigma \right)=\mathfrak{B}(\varrho ,\varsigma )\ast f\left(\varsigma \right)& =& \varsigma +\sum _{n=2}^{\infty }\frac{{\left(\varrho (n-1)\right)}^{n-2}{e}^{-\varrho (n-1)}}{(n-1)!}{a}_n{\varsigma }^n\hfill \\ & =& \varsigma +\sum _{n=2}^{\infty }{\mathrm{\Lambda }}_n{a}_n{\varsigma }^n\hfill \\ & =& \varsigma +{\mathrm{\Lambda }}_2{a}_2{\varsigma }^2+{\mathrm{\Lambda }}_3{a}_3{\varsigma }^3+\cdots ,\hfill \end{array}$$

(41)

where ${\mathrm{\Lambda }}_n={\mathrm{\Lambda }}_n\left(\varrho \right)=\frac{{\left(\varrho (n-1)\right)}^{n-2}{e}^{-\varrho (n-1)}}{(n-1)!}.$ In particular, by fixing $n=2$ and $n=3$, we have

$${\mathrm{\Lambda }}_2=e^{-\varrho }{\mathrm{\Lambda }}_3=\varrho e^{-2\varrho }.$$

Now, by taking

$${\phi }_2={\mathrm{\Lambda }}_2=e^{-\varrho }{\phi }_3={\mathrm{\Lambda }}_3=\varrho e^{-2\varrho }.$$

one can easily state the results associated with the Borel distribution.

Remark 8.

By suitably fixing the values for $\vartheta ,\kappa$ in Theorems 7 and 8, we derive the results for the functionals listed in Remark 1 by subordinating with telephonic numbers that are related with Poisson (or Borel) distribution series by convolution, which yields new results not yet discussed.

5. Conclusions

In the current paper, we mainly obtain the upper bounds of the initial Taylors coefficients of the generalized function class ${\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell )$ defined $\mathsf{\Delta }$ and obtained initial coefficient estimates for $f\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell ).$ Furthermore, we find the Fekete–Szegö inequalities for function in $f\in {\mathfrak{W}}_{\vartheta }^{\kappa }(\mathsf{\Psi },\ell ).$ Several consequences of the results are also pointed out by suitably fixing the parameters in Theorem 1 as a remark. In Theorem 2, we deduced the Fekete–Szegö functional for these function classes. Comparable results have been attained for the function $f^{-1}.$ We acquire Fekete–Szegö variations for certain subclasses of functions defined over Poisson and Borel distribution series. Then, we applied our outcomes to certain functions demarcated by convolution products. Furthermore, motivating further research on this subject, we have chosen to draw the attention of the interested readers toward a considerably large number of related recent publications and developments in the area of mathematical analysis based on hypergeometric functions [45]. In conclusion, we choose to reiterate an important observation, which was presented in the recently published review-cum-expository review article by Srivastava ([31], p. 340), new $q-$ analogues can easily (and possibly trivially) be translated into the corresponding results for the so-called $\left(q\right)-$ analogues (with $0<\left|q\right|\le 1$) by applying some obvious parametric variations.