Inclusion properties for bi-univalent functions of complex order defined by combining of Faber polynomial expansions and Fibonacci numbers

TL;DR

This paper introduces a new class of bi-univalent functions using fractional derivatives, Faber polynomial expansions, and Fibonacci numbers to establish bounds on their coefficients.

Contribution

It combines fractional derivatives, Faber polynomials, and Fibonacci numbers to define and analyze a novel class of bi-univalent functions with coefficient bounds.

Findings

Derived bounds for coefficients of the new bi-univalent function class

Established properties of functions using Faber polynomial expansions and Fibonacci numbers

Introduced a new function class using Tremblay fractional derivative operator

Abstract

In this present investigation, we introduce the new class R of bi-univalent functions defined by using the Tremblay fractional derivative operator. Additionally, we use the Faber polynomial expansions and Fibonacci numbers to derive bounds for the general coefficient an of the bi-univalent function class.

Equations

$$f(0)=f^{\prime }(0)-1=0.$$

$$f(z)=z+a_2{z}^2+a_3{z}^3+\cdots =z+\stackrel{\infty }{\sum _{n=2}}{a}_n{z}^n.$$

$$\varpi (z)=\stackrel{\infty }{\sum _{n=1}}{\mathfrak{c}}_n{z}^n\left(\varpi \left(0\right)=0,\left|\varpi \left(z\right)\right|<1\right),$$

$$f\left(z\right)=g\left(\varpi \left(z\right)\right)\left(z\in \mathbb{U}\right).$$

$$\begin{array}[]{l}g(w)=f^{-1}\left(w\right)=w~{}-a_{2}w^{2}+\left(2a_{2}^{2}-a_{3}\right)w^{3}\\ \\ \ \ \ \ \ \ \ \ \ \ \ -\left(5a_{2}^{3}-5a_{2}a_{3}+a_{4}\right)w^{4}+\cdots.\end{array}$$

$$D_z^{\rho }f(z)=\frac{1}{\Gamma (1-\rho )}\frac{d}{dz}\underset{0}{\overset{z}{\int }}\frac{f(\xi )}{(z-\xi {)}^{\rho }}d\xi (0\le \rho <1)$$

$$I_z^{\rho }f(z)=\frac{1}{\Gamma (\rho )}\underset{0}{\overset{z}{\int }}f(\xi )(z-\xi {)}^{\rho -1}d\xi (\rho >0).$$

$$I_z^{\mu ,\rho }f(z)=\frac{\Gamma (\rho )}{\Gamma (\mu )}{z}^{1-\rho }{D}_z^{\mu -\rho }{z}^{\mu -1}f(z)(0<\mu \le 1,0<\rho \le 1,\mu \ge \rho ,0<\mu -\rho <1).$$

$$I_z^{\mu ,\rho }f(z)=\frac{\mu }{\rho }z+\stackrel{\infty }{\sum _{n=2}}\frac{\Gamma (\rho )\Gamma (n+\mu )}{\Gamma (\mu )\Gamma (n+\rho )}{a}_n{z}^n.$$

$$g\left(w\right)=f^{-1}\left(w\right)=w+\stackrel{\infty }{\sum _{n=2}}\frac{1}{n}{K}_{n-1}^{-n}\left(a_2,a_3,...\right)w^n,$$

$$K_{n-1}^{-n}$$

$$\frac{1}{2}{K}_1^{-2}$$

$$\frac{1}{3}{K}_2^{-3}$$

$$\frac{1}{4}{K}_3^{-4}$$

$$K_n^p=p{a}_n+\frac{p\left(p-1\right)}{2}{\mathcal{G}}_n^2+\frac{p!}{\left(p-3\right)!3!}{\mathcal{G}}_n^3+...+\frac{p!}{\left(p-n\right)!n!}{\mathcal{G}}_n^n,$$

$${\mathcal{G}}_n^m\left(a_1,a_2,...,a_n\right)=\stackrel{\infty }{\sum _{n=1}}\frac{m!{\left(a_1\right)}^{{\delta }_1}...{\left(a_n\right)}^{{\delta }_n}}{{\delta }_1!...{\delta }_n!},$$

$${\delta }_1+{\delta }_2+...+{\delta }_n$$

$${\delta }_1+2{\delta }_2+...+n{\delta }_n$$

$${\mathcal{G}}_n^1=a_n{\mathcal{G}}_n^n=a_1^n.$$

$$\begin{array}[]{l}\mathfrak{u}\left(z\right)=\sum_{n=1}^{\infty}t_{n}z^{n}\ \ \left(\left|z\right|<1,\ z\in\mathbb{U}\right)\ \ \ ,\\ \\ \mathfrak{v}\left(w\right)=\sum_{n=1}^{\infty}s_{n}w^{n}\ \ \left(\left|w\right|<1,\ w\in\mathbb{U}\right).\end{array}$$

$$\left|t_1\right|\le 1,\left|t_2\right|\le 1-{\left|t_1\right|}^2,\left|s_1\right|\le 1,\left|s_2\right|\le 1-{\left|s_1\right|}^2.$$

$${\mathfrak{R}}_{\Sigma ,\gamma }^{\mu ,\rho }\left(\tilde{\mathfrak{p}}\right)(\gamma \in \mathbb{C}\backslash \{0\},0<\mu \le 1,0<\rho \le 1,z,w\in \mathbb{U})$$

$$\left[1+\frac{1}{\gamma }\left(\frac{\rho {\left(I_z^{\mu ,\rho }f(z)\right)}^{\prime }}{\mu }-1\right)\right]\prec \tilde{\mathfrak{p}}\left(z\right)=\frac{1+{\tau }^2{z}^2}{1-\tau z-{\tau }^2{z}^2}$$

$$\left[1+\frac{1}{\gamma }\left(\frac{\rho {\left(I_z^{\mu ,\rho }g(w)\right)}^{\prime }}{\mu }-1\right)\right]\prec \tilde{\mathfrak{p}}\left(w\right)=\frac{1+{\tau }^2{w}^2}{1-\tau w-{\tau }^2{w}^2},$$

$$\frac{1}{\left|\tau \right|}=\frac{\left|\tau \right|}{1-\left|\tau \right|}$$

$${\Lambda }_0=0,{\Lambda }_1=1,{\Lambda }_{n+2}={\Lambda }_n+{\Lambda }_{n+1}(n\in {\mathbb{N}}_0=\left\{0,1,2,\dots \right\}),$$

$${\Lambda }_n=\frac{(1-\tau {)}^n-{\tau }^n}{\sqrt{5}},\tau =\frac{1-\sqrt{5}}{2}.$$

$$\tilde{\mathfrak{p}}\left(z\right)$$

$$\widetilde{\mathfrak{p}}_{n}=\left\{\begin{array}[]{ll}\tau&\left(n=1\right)\\ &\\ 3\tau^{2}&\left(n=2\right)\\ &\\ \tau\widetilde{\mathfrak{p}}_{n-1}+\tau^{2}\widetilde{\mathfrak{p}}_{n-2}&\left(n=3,4,\ldots\right)\end{array}\right..$$

$$\left[1+\frac{1}{\gamma }\left(f^{\prime }(z)-1\right)\right]\prec \tilde{\mathfrak{p}}\left(z\right)$$

$$\left[1+\frac{1}{\gamma }\left(g^{\prime }(w)-1\right)\right]\prec \tilde{\mathfrak{p}}\left(w\right),$$

$$f^{\prime }(z)\prec \tilde{\mathfrak{p}}\left(z\right)$$

Full text

Keywords: Bi-univalent functions, subordination, Faber polynomials, Fibonacci numbers, Tremblay fractional derivative operator.

2010, Mathematics Subject Classification: 30C45, 33D15.

1 Introduction, Definitions and Notations

Let $\mathbb{C}$ be the complex plane and $\mathbb{U}=\left\{z:z\in\mathbb{C}\text{ and }\left|z\right|<1\right\}$ be open unit disc in $\mathbb{C}$. Further, let $\mathcal{A}$ represent the class of functions analytic in $\mathbb{U}$, satisfying the condition

[TABLE]

Then each function $f$ in $\mathcal{A}$ has the following Taylor series expansion

[TABLE]

The class of this kind of functions is represented by $\mathcal{S}$.

With a view to reminding the rule of subordination for analytic functions, let the functions $f,g$ be analytic in $\mathbb{U}$. A function $f$ is subordinate to $g,$ indited as $f\prec g,$ if there exists a Schwarz function

[TABLE]

analytic in $\mathbb{U}$ such that

[TABLE]

For the Schwarz function $\mathbf{\varpi}\left(z\right)$ we know that $\left|\mathfrak{c}_{n}\right|<1$ (see [9]).

According to the Koebe-One Quarter Theorem, every univalent function $f\in\mathcal{A}$ has an inverse $f^{-1}$ satisfying $f^{-1}\left(f\left(z\right)\right)=z~{}~{}\left(z\in\mathbb{U}\right)$ and $f\left(f^{-1}\left(w\right)\right)=w~{}$ $\left(\left|w\right|<r_{0}\left(f\right);~{}~{}r_{0}\left(f\right)\geq\frac{1}{4}\right),$ where

[TABLE]

A function $f\in\mathcal{A}$ is said to be bi-univalent in $\mathbb{U}$ if both $f$ and $f^{-1}$ are univalent in $\mathbb{U}.~{}$Let $\Sigma$ denote the class of bi-univalent functions in $\mathbb{U}$ given by (1). For a brief historical account and for several notable investigation of functions in the class $\Sigma,$ see the pioneering work on this subject by Srivastava et al. [20] (see also [6, 7, 14, 15]). The interest on estimates for the first two coefficients $\left|a_{2}\right|$, $\left|a_{3}\right|$ of the bi-univalent functions keep on by many researchers (see, for example, [4, 12, 13, 16, 21]). However, in the literature, there are only a few works (by making use of the Faber polynomial expansions) determining the general coefficient bounds $\left|a_{n}\right|$ for bi-univalent functions ([5, 10, 11, 17]). The coefficient estimate problem for each of $\left|a_{n}\right|$ $\left(\ n\in\mathbb{N}\backslash\left\{1,2\right\};\ \ \mathbb{N}=\left\{1,2,3,...\right\}\right)$ is still an open problem.

Now, we recall to a notion of $q$-operators that play a major role in Geometric Function Theory. The application of the $q$-calculus in the context of Geometric Function Theory was actually provided and the basic (or $q$-) hypergeometric functions were first used in Geometric Function Theory in a book chapter by Srivastava [18]. For the convenience, we provide some basic notation details of $q$-calculus which are used in this paper.

Definition 1

(See [19]) For a function $f$ (analytic in a simply-connected region of $\mathbb{C}$), the fractional derivative of order $\rho$ is stated by

[TABLE]

and the fractional integral of order $\rho$ is stated by

[TABLE]

Definition 2

(See [17]) The Tremblay fractional derivative operator of the function $f$ is defined as

[TABLE]

From (3), we deduce that

[TABLE]

In this paper, we study the new class $\mathfrak{R}_{\Sigma,\gamma}^{\mu,\rho}\left(\widetilde{\mathfrak{p}}\right)$ of bi-univalent functions established by using the Tremblay fractional derivative operator. Further, we use the Faber polynomial expansions and Fibonacci numbers to derive bounds for the general coefficient $\left|a_{n}\right|$ of the bi-univalent function class.

2 Preliminaries

By utilizing the Faber polynomial expansions for functions $f$ $\in\mathcal{A}$ of the form (1), the coefficients of its inverse map $g=f$ -1 may be stated by [2, 3]:

[TABLE]

where

[TABLE]

such that $V_{j}$ $\left(7\leq j\leq n\right)$ is a homogeneous polynomial in the variables $a_{2},a_{3},...,a_{n}$. In the following, the first three terms of $K_{n-1}^{-n}$ are stated by

[TABLE]

In general, the expansion of $K_{n}^{p}$ $(p\in\mathbb{Z}=\left\{0,\pm 1,\pm 2,\ldots\right\})$ is stated by

[TABLE]

where $\mathcal{G}_{n}^{p}=$ $\mathcal{G}_{n}^{p}\left(a_{1},a_{2},...\right)$ and by [1],

[TABLE]

while $a_{1}=1$, the sum is taken over all nonnegative integers $\delta_{1},...,\delta_{n}$ satisfying

[TABLE]

The first and the last polynomials are

[TABLE]

For two analytic functions $\mathfrak{u}\left(z\right)$, $\mathfrak{v}\left(w\right)$ \left(\mathfrak{u}\left(0\right)=\mathfrak{v}\left(0\right)=0,\ \left|\mathfrak{u}\left(z\right)\right|<1,\ \left|\mathfrak{v}\left(w\right)\right|<1\right),\suppose that

[TABLE]

It is well known that

[TABLE]

Definition 3

A function $f\in\Sigma$ is said to be in the class

[TABLE]

if the following subordination relationships are satisfied:

[TABLE]

and

[TABLE]

where the function $g$ is given by (2) and $\tau=\frac{1-\sqrt{5}}{2}\approx-0.618.$

Remark 4

The function $\widetilde{\mathfrak{p}}\left(z\right)$ is not univalent in $\mathbb{U}$, but it is univalent in the disc $\left|z\right|<\frac{3-\sqrt{5}}{2}\approx-0.38$. For example, $\widetilde{\mathfrak{p}}\left(0\right)=\widetilde{\mathfrak{p}}\left(-\frac{1}{2\tau}\right)$ and $\widetilde{\mathfrak{p}}\left(e^{\pm i\arccos(1/4)}\right)=\frac{\sqrt{5}}{5}$. Also, it can be written as

[TABLE]

which indicates that the number $\left|\tau\right|$ divides $\left[0,1\right]$ such that it fulfills the golden section (see for details Dziok et al. [8]).

Additionally, Dziok et al. [8] indicate a useful connection between the function $\widetilde{\mathfrak{p}}\left(z\right)$ and the Fibonacci numbers. Let $\left\{\Lambda_{n}\right\}$ be the sequence of Fibonacci numbers

[TABLE]

then

[TABLE]

If we set

[TABLE]

then the coefficients $\widetilde{\mathfrak{p}}_{n}$ satisfy

[TABLE]

Specializing the parameters $\gamma,\mu$ and $\rho$, we state the following definitions.

Definition 5

For $\mu=\rho=1,$ a function $f\in\Sigma$ is said to be in the class $\mathfrak{R}_{\Sigma,\gamma}\left(\widetilde{\mathfrak{p}}\right)\left(\gamma\in\mathbb{C}\backslash\{0\}\right)$ if it satisfies the following conditions respectively:

[TABLE]

and

[TABLE]

where $g=f^{-1}.$

Definition 6

For $\gamma=\mu=\rho=1,$ a function $f\in\Sigma$ is said to be in the class $\mathfrak{R}_{\Sigma}\left(\widetilde{\mathfrak{p}}\right)$ if it satisfies the following conditions respectively:

[TABLE]

and

[TABLE]

where $g=f^{-1}.$

3 Main Result and its consequences

Theorem 7

For $\gamma\in\mathbb{C}\backslash\{0\}$, let $f\in\mathfrak{R}_{\Sigma,\gamma}^{\mu,\rho}\left(\widetilde{\mathfrak{p}}\right)$. If $a_{m}=0~{}\left(2\leq m\leq n-1\right)$, then

[TABLE]

Proof. Let $f$ be given by (1). By the definition of subordination yields

[TABLE]

and

[TABLE]

Now, an application of Faber polynomial expansion to the power series $\mathfrak{R}_{\Sigma,\gamma}^{\mu,\rho}\left(\widetilde{\mathfrak{p}}\right)$ (e.g. see [2] or [[3], equation (1.6)]) yields

[TABLE]

where

[TABLE]

[TABLE]

In particular, the first two terms are, $\mathcal{F}_{1}=\frac{2(\mu+1)}{\gamma(\rho+1)}a_{2},\mathcal{F}_{1}=\frac{3(\mu+1)(\mu+2)}{\gamma(\rho+1)(\rho+2)}a_{3}.$

By the same token, for its inverse map $g=f^{-1}$, it is seen that

[TABLE]

Next, the equations (6) and (7) lead to

[TABLE]

and

[TABLE]

Comparing the corresponding coefficients of (6) and (7) yields

[TABLE]

[TABLE]

For $a_{m}=0\ \left(2\leq m\leq n-1\right),$ we get $b_{n}=-a_{n}$ and so

[TABLE]

and

[TABLE]

Now taking the absolute values of either of the above two equations and from (4), we obtain

[TABLE]

Corollary 8

For $\gamma\in\mathbb{C}\backslash\{0\}$, suppose that $f\in\mathfrak{R}_{\Sigma,\gamma}\left(\widetilde{\mathfrak{p}}\right)$. If $a_{m}=0~{}\left(2\leq m\leq n-1\right)$, then

[TABLE]

Corollary 9

Suppose that $f\in\mathfrak{R}_{\Sigma}\left(\widetilde{\mathfrak{p}}\right)$. If $a_{m}=0~{}\left(2\leq m\leq n-1\right)$, then

[TABLE]

Theorem 10

Let $f\in\mathfrak{R}_{\Sigma,\gamma}^{\mu,\rho}\left(\widetilde{\mathfrak{p}}\right)\ (\gamma\in\mathbb{C}\backslash\{0\}).$Then

[TABLE]

and

[TABLE]

Proof. Substituting $n$ by $2$ and $3$ in (8) and (9), respectively, we find that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Obviously, we obtain

[TABLE]

If we add the equation (13) to (11) and use (14), we get

[TABLE]

Using the value of $t_{1}^{2}$ from (10), we get

[TABLE]

Combining (16) and (4), we obtain

[TABLE]

It follows from (10) that

[TABLE]

Additionally, by (4) and (15)

[TABLE]

Since $\left|\widetilde{\mathfrak{p}}_{2}\right|>\left|\widetilde{\mathfrak{p}}_{1}\right|$, we get

[TABLE]

Next, in order to derive the bounds on $\left|a_{3}\right|,$ by subtracting (13) from (11), we may obtain

[TABLE]

Evidently, from (15), we state that

[TABLE]

and consequently

[TABLE]

Since $\left|\widetilde{\mathfrak{p}}_{2}\right|>\left|\widetilde{\mathfrak{p}}_{1}\right|$, we must write

[TABLE]

On the other hand, by (4) and (18), we have

[TABLE]

Then, with the help of (10), we have

[TABLE]

By considering (17), we deduce that

[TABLE]

Corollary 11

Let $f\in\mathfrak{R}_{\Sigma,\gamma}\left(\widetilde{\mathfrak{p}}\right)\ (\gamma\in\mathbb{C}\backslash\{0\}).$Then

[TABLE]

and

[TABLE]

Corollary 12

Let $f\in\mathfrak{R}_{\Sigma}\left(\widetilde{\mathfrak{p}}\right).$Then

[TABLE]

and

[TABLE]