Faber polynomial coefficient inequalities for bi-Bazilevič functions associated with the Fibonacci-number series and the square-root functions

Abstract

Two new subclasses of the class of bi-Bazilevič functions, which are related to the Fibonacci-number series and the square-root functions, are introduced and studied in this article. Under a special choice of the parameter involved, these two classes of Bazilevič functions reduce to two new subclasses of star-like biunivalent functions related with the Fibonacci-number series and the square-root functions. Using the Faber polynomial expansion (FPE) technique, we find the general coefficient bounds for the functions belonging to each of these classes. We also find bounds for the initial coefficients for bi-Bazilevič functions and demonstrate how unexpectedly these initial coefficients behave in relation to the square-root functions and the Fibonacci-number series.

1 Introduction and preliminaries

The study of finding bounds on the coefficients is and has remained a major problem in Geometric Function Theory of Complex Analysis. The size of the coefficients of a given analytic function can have an impact on a variety of characteristics, including univalence, rate of growth, and distortion. Formulation of coefficient problems contains the estimation of the general or nth coefficient bounds, the Fekete–Szegö problem, Hankel determinants, and many other entities. A number of researchers have tackled the aforementioned coefficient problems by using different techniques. For instance, Bieberbach [1] provided the estimation on the second coefficient of univalent functions and conjectured a corresponding estimate on the nth coefficient of a univalent function, which was finally settled by de Branges [2]. Another interesting problem, which has a close relationship with the Bieberbach conjecture, was tackled by Littlewood and Paley (see [3]) by deriving the coefficient bounds for odd univalent functions. Further, Fekete and Szegö [4] obtained sharp bounds for the difference of the first two coefficients of univalent functions. Then, the problem of finding bounds on coefficients received much attention for many other subclasses of univalent functions. Additional information on this topic can be found in [5–7] and the references cited therein.

Just like univalent functions, finding coefficient estimates for biunivalent functions has been a subject of substantial interest in recent years. Evidently, researchers working on this topic drew a lot of motivation from the groundbreaking research of Srivastava et al. [8]. Numerous fascinating examples of functions falling under the class of biunivalent functions can be found in the work of Srivastava et al. [8]. Since bounds on these functions can be utilized for predicting their geometry, coefficient problems are also crucial in the study of biunivalent functions. For most of the subclasses of the biunivalent functions, finding the bounds for the general coefficients is still an open problem. However, under some assumptions, the general coefficient estimates for some particular subclasses of biunivalent functions were obtained in recent years (see, for details [9–16]).

In this study, by using the Faber polynomial (see [17, 18]), we obtain coefficient bounds for some classes of biunivalent functions. This polynomial has been extensively studied in the past few years. This polynomial plays an important role in the mathematical sciences, particularly in Geometric Function Theory of Complex Analysis. By employing the Faber polynomial expansion technique, Hamidi and Jahangiri [19, 20] as well as Srivastava et al. [21] developed new subclasses of biunivalent functions and discovered some novel and useful characteristics. Several different subclasses of the analytic and biunivalent function class were introduced and studied analogously by many other authors (see, for example, [10, 22–27]).

Let $\mathcal{A}$ stand for the set of all holomorphic functions ξ in the open unit disk:

$$\begin{aligned} \mathbb{E}=\bigl\{ z:z\in \mathbb{C} \text{ and } \vert z \vert < 1\bigr\} , \end{aligned}$$

which are normalized by

$$\begin{aligned} \xi (0 )=0 \quad\text{and}\quad \xi ^{\prime } ( 0 ) =1. \end{aligned}$$

Each function $\xi \in \mathcal{A}$ can, therefore, be expressed in the series form given by

$$\begin{aligned} \xi (z)=z+\sum_{n=2}^{\infty }a_{n}z^{n}. \end{aligned}$$

(1.1)

Additionally, the members of the class $\mathcal{S}$, which is a subclass of $\mathcal{A}$, are also univalent in $\mathbb{E}$.

Each function $\xi \in \mathcal{S}$ has an inverse function $\xi ^{-1}=g$, which is defined as follows:

$$\begin{aligned} g \bigl(\xi (z) \bigr)=z\quad (z\in \mathbb{E}) \end{aligned}$$

and

$$\begin{aligned} \xi \bigl(g(w) \bigr)=w \quad\biggl( \vert w \vert < r_{0}( \xi ); r_{0}(\xi )\geqq \frac{1}{4} \biggr). \end{aligned}$$

The series expansion of the inverse function is given by

$$\begin{aligned} g(w)=w-a_{2}w^{2}+\bigl(2a_{2}^{2}-a_{3} \bigr)w^{3}-\bigl(5a_{2}^{3} -5a_{2}a_{3}+a_{4} \bigr)w^{4}+ \cdots. \end{aligned}$$

(1.2)

An analytic function ξ is called biunivalent in $\mathbb{E}$ if the function ξ and its inverse $\xi ^{-1}$ are both univalent in $\mathbb{E}$. The class of all biunivalent functions in $\mathbb{E}$ is denoted by Σ. By the same token, a function ξ is said to be bi-Bazilevič in $\mathbb{E}$ if both the function and its inverse are Bazilevič in $\mathbb{E}$ (see [28]). The behavior of these types of functions is unpredictable and, in fact, not much is known about their coefficients.

For $\xi _{1}$, $\xi _{2}\in \mathcal{A}$, if the function $\xi _{1}$ is subordinate to the function $\xi _{2}$ in $\mathbb{E}$, denoted by

$$\begin{aligned} \xi _{1}(z)\prec \xi _{2}(z)\quad (z\in \mathbb{E}), \end{aligned}$$

then we have a function $u_{0}\in \mathcal{A}$ such that $\vert u_{0}(z) \vert <1$, $u_{0}(0)=0$ and

$$\begin{aligned} \xi _{1}(z)=\xi _{2} \bigl(u_{0}(z) \bigr) \quad(z\in \mathbb{E}). \end{aligned}$$

The set of star-like functions of order α in $\mathbb{E}$ is denoted by the symbol $\mathcal{S}^{\ast} ( \alpha )$ and we have

$$\begin{aligned} \mathcal{S}^{\ast }(\alpha )= \biggl\{ \xi:\xi \in \mathcal{S} \text{ and } {\Re} \biggl( \frac{z\xi ^{{\prime }}(z)}{\xi (z)} \biggr) > \alpha\ (0\leqq \alpha < 1) \biggr\} . \end{aligned}$$

A class $\mathcal{B}(\gamma,\rho )$ of analytic functions ξ was first studied by Bazilevič [28] in 1955. Its definition in an open unit disk $\mathbb{E}$ is as follows:

$$\begin{aligned} \xi (z)= \biggl(\frac{\rho }{1+\gamma ^{2}} \int _{0}^{z} \bigl( p(t)-i \gamma \bigr) t^{ (-\frac{\gamma \rho i}{1+\gamma ^{2}}-1 ) } \bigl[h(t) \bigr]^{\frac{\rho}{1+\gamma ^{2}}} \mathrm{d}t \biggr) ^{ \frac{ 1+i\gamma }{\rho }}, \end{aligned}$$

(1.3)

where $h\in \mathcal{S}^{\ast}$, $p\in \mathcal{P}$, $\rho \in \mathbb{R}^{+}$ and $\gamma \in \mathbb{R}$. Bazilevič [28] also showed that $\mathcal{B}(\gamma,\rho )$ is a subclass of $\mathcal{S}$. The following inequality results upon taking $\gamma =0$ in (1.3) and then differentiating each side:

$$\begin{aligned} {\Re} \biggl( \frac{z\xi ^{{\prime }}(z)}{\xi ^{1-\rho } ( \xi (z) ) ^{\rho }} \biggr) >0, \end{aligned}$$

(1.4)

wherein all the powers are considered to be principal values. Thomas [29] gave the name “Bazilevič functions of type ρ” for such type of functions that satisfy (1.4). Despite the fact that the class $\mathcal{B}(\rho )$ is the largest subclass of univalent functions and contains a large number of known subclasses of $\mathcal{S}$, little is known about the class $\mathbb{B}(\gamma,\rho )$ of functions defined by (1.3) or for the class $\mathbb{B}(\rho )$ in general. The class of Bazilevič functions and the associated counterparts have undergone substantial research in a variety of areas. For further information, see [30–33].

The idea of subordination was used in order to define many subclasses of analytic functions. For instance, the Carathéodory class $\mathcal{P}$ of functions with positive real part can be defined as follows:

$$\begin{aligned} \mathcal{P}= \biggl\{ p:p (0 )=1 \text{ and } p(z) \prec \frac{1+z}{1-z} (z\in \mathbb{E}) \biggr\} . \end{aligned}$$

Geometric Function Theory of Complex Analysis is especially intriguing in the geometric structure of the image domain. On an understanding of the ranges of these functions, other classes of analytic functions have been developed and explored. Several well-known subclasses of the class $\mathcal{P}$ can be created by substituting the function $\frac{1+z}{1-z}$ with appropriate functions. For instance, we cite the following special cases:

1. By taking

$$\begin{aligned} \varphi _{1} ( z )= \frac{1+ (1-2\alpha )z }{1-z}\quad (0\leqq \alpha < 1), \end{aligned}$$

we obtain the plane to the right of the vertical line $u=\alpha$ (see [34]).

2. By taking

$$\begin{aligned} \varphi _{1} (z )=\frac{1+Az}{1+Bz}\quad (-1\leqq B< A \leqq 1), \end{aligned}$$

we have the circular domain centered at $\frac{1-AB}{1-B^{2}}$ and with radius $\frac{A-B}{1-B^{2}}$ (see [35]).

3. If we take

$$\begin{aligned} \varphi _{2} ( z )=z+\sqrt{1+z^{2}}, \end{aligned}$$

we obtain the crescent-shaped region that was studied in [36].

4. For

$$\begin{aligned} \varphi _{3} ( z )=1+z-\frac{z^{3}}{3}, \end{aligned}$$

we have the nephroid domain that was investigated in [37].

5. If we set

$$\begin{aligned} \varphi _{4} (z )= \biggl( \frac{1+z}{1+\frac{1-\beta}{\beta}z } \biggr)^{\alpha} \quad\biggl(\alpha \geqq 1; \beta \geqq \frac{1}{2} \biggr), \end{aligned}$$

we obtain the leaf-like domain (see [38]).

6. By taking $\phi _{5} (z )=1+\sin z$, we obtain the eight-shaped region (see [39]).

7. For

$$\begin{aligned} \phi _{6} (z )= \frac{1+\tau ^{2}z^{2}}{1-\tau z-\tau ^{2}z^{2}} \quad\biggl(\tau:= \frac{1-\sqrt{5}}{2} \biggr), \end{aligned}$$

we have the shell-like domains studied in [40, 41]. The function

$$\begin{aligned} \phi _{6} (z )= \frac{1+\tau ^{2}z^{2}}{1-\tau z-\tau ^{2}z^{2}} \end{aligned}$$

generates the shell-like curve. In a more detailed way, commonly known as a shell-like curve, is produced by mapping the unit circle through the function $\phi _{6} (z )$ given by:

$$\begin{aligned} \phi _{6} \bigl(e^{i\theta} \bigr)= \frac{\sqrt{5}}{2 ( 3-2\cos \theta _{1} ) }+i \frac{\sin \theta _{1} ( 4\cos \theta _{1}-1 )}{ 2 ( 3-2\cos \theta _{1} ) 2 ( 1+\cos \theta _{1} ) } \quad(0\leqq \theta _{1}< 2\pi ). \end{aligned}$$

The series representation for this significant function is as follows:

$$\begin{aligned} \varphi _{6}(z)&=1+\sum_{n=1}^{\infty } ( u_{n-1}+u_{n+1} ) \tau ^{n}z^{n} \\ &=1+\sum_{n=1}^{\infty}T_{n}z^{n}, \end{aligned}$$

where

$$\begin{aligned} T_{n}= (u_{n-1}+u_{n+1} ) \tau ^{n} \end{aligned}$$

(1.5)

and

$$\begin{aligned} u_{n}=\frac{ ( 1-\tau ) ^{n}-\tau ^{n}}{\sqrt{5}} \quad\biggl(\tau =\frac{1-\sqrt{5}}{2} \biggr), \end{aligned}$$

which brings it closer to the Fibonacci numbers by producing a series of constant coefficients. For indepth research on the aforementioned functions and a large number of other similar functions, see ([42–48]) and the references cited therein.

8. The function $\varphi _{7} (z )=\sqrt{1+z}$ yields the right-half of the lemniscate of Bernoulli, which was introduced and studied in [49, 50]. Moreover, $\phi _{7} ( z )=\sqrt{1+z}$ is an analytic function with positive real part in the unit disk $\mathbb{E}$, which satisfies the following conditions:

$$\begin{aligned} \varphi _{7}(0)=1\quad \text{and}\quad \varphi _{7}^{{\prime}}(0)>0 \end{aligned}$$

such that the series expansion of the form $\varphi _{7}(z)=\sqrt{1+z}$ is given as follows:

$$\begin{aligned} \varphi _{7}(z) &=1+\sum_{n=1}^{\infty } \frac{ ( 2n-2 ) ! ( -1 ) ^{n-1}}{ (n-1 )! n! 2^{2n-1}}z^{n} \\ &=1+\sum_{n=1}^{\infty }Q_{n}z^{n} \\ &=1+Q_{1}z+Q_{2}z^{2}+\cdots, \end{aligned}$$

where

$$\begin{aligned} Q_{n}= \frac{ (2n-2 )! (-1 )^{n-1}}{ (n-1 ) ! n! 2^{2n-1}}. \end{aligned}$$

(1.6)

In our work, we shall use the function:

$$\begin{aligned} \phi _{6} ( z ) = \frac{ 1+\tau ^{2}z^{2}}{1-\tau z-\tau ^{2}z^{2}} \quad\biggl(\tau = \frac{1-\sqrt{5}}{2} \biggr) \end{aligned}$$

and the function

$$\begin{aligned} \phi _{7} (z )=\sqrt{1+z}. \end{aligned}$$

Definition 1.1

Let ξ be an analytic function and be of the form (1.1). Then, $\xi \in \mathcal{B}_{\Sigma}^{\beta} (\varphi _{6}(z) )$ if and only if

$$\begin{aligned} \biggl( \frac{z}{\xi (z)} \biggr)^{1-\beta} \xi ^{{\prime }}(z) \prec \varphi _{6}(z) \end{aligned}$$

and

$$\begin{aligned} \biggl(\frac{w}{g(w)} \biggr)^{1-\beta} g^{{\prime}}(w)\prec \varphi _{6}(w), \end{aligned}$$

where $0\leqq \beta <1$ and $z,w\in \mathbb{E}$.

Definition 1.2

Let ξ be an analytic function and be of the form (1.1). Then, $\xi \in \mathcal{B}_{\Sigma}^{\beta} (\varphi _{7}(z) )$ if and only if

$$\begin{aligned} \biggl(\frac{z}{\xi (z)} \biggr)^{1-\beta} \xi ^{{\prime}}(z)\prec \varphi _{7}(z) \end{aligned}$$

and

$$\begin{aligned} \biggl( \frac{w}{g(w)} \biggr)^{1-\beta} g^{{\prime}}(w)\prec \varphi _{7}(w), \end{aligned}$$

where $0\leqq \beta <1$ and $z,w\in \mathbb{E}$.

For $\beta =0$, the above two classes of Bazilevič functions reduce to two new subclasses of star-like biunivalent functions related with the Fibonacci-number series and the square-root function.

Definition 1.3

Assume that ξ is an analytic function and has the form given by (1.1). Then, $\xi \in \mathcal{S}_{\Sigma} (\varphi _{6}(z) )$ if and only if

$$\begin{aligned} \frac{z\xi ^{{\prime}}(z)}{\xi (z)}\prec \varphi _{6}(z) \end{aligned}$$

and

$$\begin{aligned} \frac{wg^{{\prime}}(w)}{g(w)}\prec \varphi _{6}(w), \end{aligned}$$

where $z,w\in \mathbb{E}$.

Definition 1.4

Suppose that ξ is an analytic function and has the form (1.1). Then, $\xi \in \mathcal{S}_{\Sigma} (\varphi _{7}(z) )$ if and only if

$$\begin{aligned} \frac{z\xi ^{{\prime}}(z)}{\xi (z)}\prec \varphi _{7}(z) \end{aligned}$$

and

$$\begin{aligned} \frac{wg^{{\prime}}(w)}{g(w)}\prec \varphi _{7}(w), \end{aligned}$$

where $z,w\in \mathbb{E}$.

Throughout this article, we will be presuming that $0\leqq \beta <1$.

Next, with a view to introducing the Faber polynomial expansion (FPE) method and its applications, we assume that the coefficients of the inverse map g of the analytic function ξ can be expressed as follows by using the FPE method (see [51, 52]):

$$\begin{aligned} g(w)=\xi ^{-1}(w)=w+\sum_{n=2}^{\infty} \frac{1}{n} \mathfrak{S} _{n-1}^{n}(a_{2},a_{3}, \ldots,a_{n})w^{n}, \end{aligned}$$

where

$$\begin{aligned} \mathfrak{S}_{n-1}^{-n} ={}&\mathfrak{S}_{n-1}^{-n} ( a_{2},a_{3}, \ldots,a_{n} ) \\ ={}&\frac{(-n)!}{(-2n+1)! (n-1)!} a_{2}^{n-1}+ \frac{(-n)!}{ (2(-n+1) )! (n-3)!} a_{2}^{n-3}a_{3} \\ &{} +\frac{(-n)!}{(-2n+3)!(n-4)!} a_{2}^{n-4}a_{4} \\ &{} +\frac{(-n)!}{ (2(-n+2) )! (n-5)!} a_{2}^{n-5} \bigl[ a_{5}+(-n+2)a_{3}^{2} \bigr] \\ &{} +\frac{(-n)!}{(-2n+5)! (n-6)!} a_{2}^{n-6} \bigl[ a_{6}+(-2n+5)a_{3}a_{4} \bigr] \\ &{} +\sum_{\mathfrak{j}\geqq 7}a_{2}^{j-\mathfrak{n}} \mathcal{Q}_{\mathfrak{n}}, \end{aligned}$$

in which $\mathcal{Q}_{\mathfrak{n}}$ is a homogeneous polynomial in the variables $a_{2},a_{3},\ldots,a_{n}$ for $7\leqq \mathfrak{j}\leqq n$ such as (for example) $(-n)!$ are symbolically interpreted as follows:

$$\begin{aligned} ( -n )!\equiv \Gamma (1-n )= ( -n ) ( -n-1 ) ( -n-2 ) \cdots\quad \bigl(n\in \mathbb{N}_{0}:=\mathbb{N}\cup \{0\}=\{0,1,2,3,\ldots \} \bigr). \end{aligned}$$

In particular, the first three terms of $\mathfrak{S}_{n-1}^{-n}$ are given by:

$$\begin{aligned} &\frac{1}{2}\mathfrak{S}_{1}^{-2}=-a_{2},\\ &\frac{1}{3}\mathfrak{S}_{2}^{-3}=2a_{2}^{2}-a_{3} \end{aligned}$$

and

$$\begin{aligned} \frac{1}{4}\mathfrak{S}_{3}^{-4}=- \bigl(5a_{2}^{3}-5a_{2}a_{3}+a_{4} \bigr). \end{aligned}$$

In general, for $\mathfrak{r}\in \mathbb{Z}:=\{0,\pm 1,\pm 2,\ldots \}$ and $n\geqq 2$, an expansion of $\mathfrak{S}_{n}^{\mathfrak{r}}$ is of the following form:

$$\begin{aligned} \mathfrak{S}_{n}^{\mathfrak{p}}=\mathfrak{r}a_{n}+ \frac{\mathfrak{r}(\mathfrak{r}-1)}{2}\mathcal{D}_{n}^{2}+ \frac{\mathfrak{r}!}{(\mathfrak{r}-3)! 3!} \mathcal{D}_{n}^{3}+\cdots + \frac{\mathfrak{r}!}{(\mathfrak{r}-n)! n!} \mathcal{D}_{n}^{n}, \end{aligned}$$

where

$$\begin{aligned} \mathcal{D}_{n}^{\mathfrak{r}}=\mathcal{D}_{n}^{\mathfrak{r}}(a_{2},a_{3}, \ldots ). \end{aligned}$$

Also, by [51, 52], we have

$$\begin{aligned} \mathcal{D}_{n}^{\nu }(a_{2},\ldots,a_{n+1})= \sum_{n=1}^{\infty } \frac{\nu !(a_{2})^{\mu _{1}}\cdots (a_{n})^{\mu _{n}}}{\mu _{1!},\ldots,\mu _{n}!}\quad ( a_{1}=1; \nu \leqq n ). \end{aligned}$$

The sum is taken over all nonnegative integers $\mu _{1},\ldots,\mu _{n}$ that satisfy the following conditions:

$$\begin{aligned} \mu _{1}+\mu _{2}+\cdots +\mu _{n}=v \end{aligned}$$

and

$$\begin{aligned} \mu _{1}+2\mu _{2}+\cdots +n\mu _{n}=n. \end{aligned}$$

Clearly, we have

$$\begin{aligned} \mathcal{D}_{n}^{n}(a_{1},\ldots,a_{n})=a_{2}^{n}. \end{aligned}$$

In this paper, we introduce new subclasses of analytic biunivalent functions connected with the Fibonacci-number series and the square-root functions. We also obtain the estimates for the initial Taylor–Maclaurin coefficients for functions in each of these subclasses. Moreover, we investigate estimates for the general coefficients $a_{n}$ for functions in these subclasses of biunivalent functions by using the Faber polynomial expansion (FBE) method.

2 Main results

To prove our main results, we shall need the following known results (see [1]).

Lemma 2.1

Let

$$\begin{aligned} \psi (z)=\sum_{n=1}^{\infty }c_{n}z^{n} \end{aligned}$$

be a Schwarz function so that

$$\begin{aligned} \bigl\vert \psi (z) \bigr\vert < 1 \quad\bigl( \vert z \vert < 1\bigr). \end{aligned}$$

If $\gamma \geqq 0$, then

$$\begin{aligned} \bigl\vert c_{2}+\gamma c_{1}^{2} \bigr\vert \leqq 1+ ( \gamma -1 ) \vert c_{1} \vert ^{2}. \end{aligned}$$

In the following theorem, we use the Faber polynomial expansion method and determine the unpredictable behavior of the coefficients’ estimates of the functions class ${\mathcal{B}}_{\Sigma}^{\beta} (\varphi _{6}(z) )$ defined by the Fibonacci-number series.

Theorem 2.2

Let $\xi \in {\mathcal{B}}_{\Sigma }^{\beta } (\varphi _{6}(z) )$ be given by (1.1). Also, let $a_{k}=0$ $(2\leqq k\leqq n-1)$. Then,

$$\begin{aligned} \vert a_{n} \vert \leqq \frac{ \vert T_{1} \vert }{n+\beta } \quad\bigl(n\in \mathbb{N} \setminus \{1,2\}\bigr), \end{aligned}$$

where $T_{1}$ is given in (1.5) for all values of n.

Proof

For $\xi \in {\mathcal{B}}_{\Sigma}^{\beta} (\varphi _{6}(z) )$, then the FPE for $(\frac{z}{\xi (z)} ) ^{1-\beta }\xi ^{{\prime }}(z)$ is given by

$$\begin{aligned} \biggl( \frac{z}{\xi (z)} \biggr) ^{1-\beta }\xi ^{{\prime }}(z)=1+ \sum _{n=1}^{\infty } \biggl( 1+\frac{n}{\beta } \biggr) K_{k}^{ \beta } ( a_{2},a_{3}, \ldots,a_{n+1} ) z^{n}. \end{aligned}$$

(2.1)

For the inverse mapping $g=\xi ^{-1}$, the FPE for $( \frac{w}{g(w)} ) ^{1-\beta }g^{{\prime }}(w)$ is

$$\begin{aligned} \biggl( \frac{w}{g(w)} \biggr) ^{1-\beta }g^{{\prime }}(w)=1+\sum _{n=1}^{\infty } \biggl( 1+\frac{n}{\beta } \biggr) K_{k}^{ \beta } ( b_{2},b_{3}, \ldots,b_{n+1} ) w^{n}, \end{aligned}$$

(2.2)

where

$$\begin{aligned} b_{n+1}=\frac{1}{n+1}K_{k}^{-(n+1)}(a_{2},a_{3}, \ldots,a_{n+1}) \quad(k\geqq 1). \end{aligned}$$

Furthermore, since $\xi \in {\mathcal{B}}_{\Sigma }^{\beta } (\varphi _{6}(z) )$, by the definition of subordination, there exists a Schwarz function $\psi (z)$ given by

$$\begin{aligned} \psi (z)=\sum_{n=1}^{\infty }c_{n}z^{n}\quad (z\in \mathbb{E}) \end{aligned}$$

such that

$$\begin{aligned} \biggl( \frac{z}{\xi (z)} \biggr) ^{1-\beta }\xi ^{{\prime }}(z)& = \varphi _{6} \bigl(\psi (z) \bigr) \\ & =1+\sum_{n=1}^{\infty }\sum _{k=1}^{n}T_{k} \mathfrak{S}_{n}^{k} ( c_{1},c_{2},\ldots,c_{n} ) z^{n}. \end{aligned}$$

(2.3)

Similarly, we have

$$\begin{aligned} \biggl( \frac{w}{g(w)} \biggr) ^{1-\beta }g^{{\prime }}(w)\quad (w \in \mathbb{E}). \end{aligned}$$

There exists a Schwarz function $q(z)$ given by

$$\begin{aligned} q(w)=\sum_{n=1}^{\infty }d_{n}w^{n} \end{aligned}$$

such that

$$\begin{aligned} \biggl( \frac{w}{g(w)} \biggr) ^{1-\beta }g^{{\prime }}(w)& = \varphi _{1} \bigl(q(w) \bigr) \\ & =1+\sum_{n=1}^{\infty }\sum _{k=1}^{n}T_{k} \mathfrak{S}_{n}^{k} ( d_{1},d_{2},\ldots,d_{n} ) z^{n}. \end{aligned}$$

(2.4)

Evaluating the coefficients of equations (2.1) and (2.3), for any $n\geqq 2$, we obtain

$$\begin{aligned} \biggl( 1+\frac{n}{\beta } \biggr) \mathfrak{S}_{n}^{\beta } ( a_{2},a_{3}, \ldots,a_{n+1} ) a_{n}=\sum _{k=1}^{n-1}T_{k}\mathfrak{S}_{n}^{k} ( c_{1},c_{2}, \ldots,c_{n-1} ). \end{aligned}$$

(2.5)

Evaluating the coefficients of equations (2.2) and (2.4), for any $n\geqq 2$, we have

$$\begin{aligned} \biggl( 1+\frac{n}{\beta } \biggr) \mathfrak{S}_{k}^{\beta } ( b_{2},b_{3}, \ldots,b_{n} ) =\sum _{k=1}^{n-1}T_{k}\mathfrak{S}_{n}^{k} ( d_{1},d_{2},\ldots,d_{n} ). \end{aligned}$$

(2.6)

Solving for $a_{2}$ and taking the moduli of the coefficients of the Schwarz functions ψ and q, we obtain (see [1])

$$\begin{aligned} \vert c_{n} \vert \leqq 1\quad \text{and} \quad \vert d_{n} \vert \leqq 1. \end{aligned}$$

Also, from the assumptions that $2\leqq k\leqq n-1$ and $a_{k}=0$, respectively, we obtain

$$\begin{aligned} b_{n}=-a_{n} \end{aligned}$$

as well as

$$\begin{aligned} \biggl( 1+\frac{n}{\beta } \biggr) \beta a_{n}=T_{1}c_{n-1} \end{aligned}$$

(2.7)

and

$$\begin{aligned} - \biggl( 1+\frac{n}{\beta } \biggr) \beta a_{n}=T_{1}d_{n-1}. \end{aligned}$$

(2.8)

By solving equations (2.7) and (2.8) for $a_{n}$ and determining the moduli, and by the coefficients of the Schwarz functions $\psi (z)$ and $q(w)$, we have $\vert c_{n} \vert \leq 1$and $\vert d_{n} \vert \leq 1$, (see [1]):

$$\begin{aligned} \vert a_{n} \vert \leqq \frac{ \vert T_{1} \vert }{n+\beta }. \end{aligned}$$

This completes the proof of Theorem 2.2. □

Theorem 2.3

If an analytic function ξ given by (1.1) belongs to the class ${\mathcal{B}}_{\Sigma }^{\beta } (\varphi _{6}(z) )$, and if

$$\begin{aligned} T_{2}=\alpha T_{1} \quad(\alpha >0), \end{aligned}$$

then

$$\begin{aligned} &\vert a_{2} \vert \leqq \textstyle\begin{cases} \sqrt{\frac{2T_{1}}{(1+\beta )(\beta +2)}} & ( 0< T_{1} \leqq \frac{2 ( \beta +1 ) }{(\beta +2)} ), \\ \frac{T_{1}}{\beta +1} & ( T_{1}\geqq \frac{2 ( \beta +1 ) }{(\beta +2)} ),\end{cases}\displaystyle \\ &\vert a_{3} \vert \leqq \frac{T_{1}}{\beta +2} \end{aligned}$$

and

$$\begin{aligned} \bigl\vert a_{3}-a_{2}^{2} \bigr\vert \leqq \frac{T_{1}}{(\beta +1)(\beta +2)}, \end{aligned}$$

where $T_{1}$ and $T_{2}$ are given by (1.5).

Proof

Taking $n=2$ in (2.5) and (2.6), we have

$$\begin{aligned} (\beta +1)a_{2}=T_{1}c_{1} \end{aligned}$$

(2.9)

and

$$\begin{aligned} -(\beta +1)a_{2}=T_{1}d_{1}. \end{aligned}$$

(2.10)

From (2.9) and (2.10), we find that

$$\begin{aligned} c_{1}=-d_{1}. \end{aligned}$$

(2.11)

If we take the moduli of both sides of any of these two equations, for the coefficients of the Schwarz functions ψ and q given by (see [1])

$$\begin{aligned} \vert c_{n} \vert \leqq 1\quad \text{and} \quad \vert d_{n} \vert \leqq 1, \end{aligned}$$

we have

$$\begin{aligned} \vert a_{2} \vert \leqq \frac{T_{1}}{\beta +1}. \end{aligned}$$

(2.12)

For $n=3$, equations (2.5) and (2.6) yield

$$\begin{aligned} \frac{(\beta -1)(\beta +2)}{2}a_{2}^{2}+(\beta +2)a_{3}=T_{1}c_{2}+ \alpha T_{1}c_{1}^{2} \end{aligned}$$

(2.13)

and

$$\begin{aligned} \frac{(3+\beta )(\beta +2)}{2}a_{2}^{2}-(\beta +2)a_{3}=T_{1}d_{2}+ \alpha T_{1}d_{1}^{2}, \end{aligned}$$

(2.14)

respectively. Thus, upon adding (2.13) and (2.14), we obtain

$$\begin{aligned} (1+\beta ) (\beta +2)a_{2}^{2}=T_{1} \bigl[ \bigl( c_{2}+\alpha c_{1}^{2} \bigr) + \bigl( d_{2}+\alpha d_{1}^{2} \bigr) \bigr]. \end{aligned}$$

Again, for the coefficients of the Schwarz functions ψ and q, we have

$$\begin{aligned} (1+\beta ) (\beta +2) \vert a_{2} \vert ^{2}\leqq T_{1} \bigl[ \bigl\vert c_{2}+\alpha c_{1}^{2} \bigr\vert + \bigl\vert d_{2}+ \alpha d_{1}^{2} \bigr\vert \bigr]. \end{aligned}$$

Thus, by Lemma 2.1, we have

$$\begin{aligned} (1+\beta ) (\beta +2) \vert a_{2} \vert ^{2}\leqq T_{1} \bigl[ 1+ ( \alpha -1 ) \vert c_{1} \vert ^{2}+1+ ( \alpha -1 ) \vert d_{1} \vert ^{2} \bigr]. \end{aligned}$$

Now, by using the moduli of the coefficients of the Schwarz functions, we obtain

$$\begin{aligned} \vert a_{2} \vert \leqq \sqrt{\frac{2T_{1}}{(1+\beta )(\beta +2)}}. \end{aligned}$$

(2.15)

Consequently, we note that

$$\begin{aligned} \sqrt{\frac{2T_{1}}{(1+\beta )(\beta +2)}}< \frac{T_{1}}{\beta +1}. \end{aligned}$$

Multiplying equation (2.13) by $(3+\beta )$ and equation (2.14) by $(\beta -1)$, we obtain

$$\begin{aligned} 2 ( \beta +1 ) ( \beta +2 ) a_{3}=T_{1} \bigl[ (3+ \beta ) \bigl( c_{2}+ ( \alpha ) c_{1}^{2} \bigr) -( \beta -1) \bigl( d_{2}+ ( -B ) d_{1}^{2} \bigr) \bigr]. \end{aligned}$$

From Lemma 2.1, we have

$$\begin{aligned} \vert a_{3} \vert \leqq \frac{T_{1}}{2 ( \beta +1 ) ( \beta +2 ) } \bigl[ (3+\beta ) \bigl( 1+ ( \alpha -1 ) \bigr) \vert c_{1} \vert ^{2}+( \beta -1) \bigl( 1+ ( \alpha -1 ) \bigr) \vert d_{1} \vert ^{2} \bigr]. \end{aligned}$$

Applying the moduli of the coefficients of the Schwarz functions to the above equation, we have

$$\begin{aligned} \vert a_{3} \vert \leqq \frac{T_{1}}{ (\beta +2 ) }. \end{aligned}$$

Lastly, upon subtracting equation (2.13) from equation (2.14), if we use equation (2.11), we find that

$$\begin{aligned} (\beta +1) (\beta +2) \bigl( a_{3}-a_{2}^{2} \bigr) =T_{1} \vert c_{2}-d_{2} \vert , \end{aligned}$$

which can be rewritten as follows:

$$\begin{aligned} \bigl\vert a_{3}-a_{2}^{2} \bigr\vert \leqq \frac{T_{1}}{(\beta +1)(\beta +2)}. \end{aligned}$$

This completes the proof. □

Taking $\beta =0$ in Theorem 2.2, we obtain Theorem 2.4 below for the class ${\mathcal{S}}_{\Sigma} (\varphi _{6}(z) )$.

Theorem 2.4

Let $\xi \in {\mathcal{S}}_{\Sigma} (\varphi _{6}(z) )$ be an analytic function. Also, let $a_{k}=0$ $2 (\leqq k\leqq n-1)$. Then,

$$\begin{aligned} \vert a_{n} \vert \leqq \frac{ \vert T_{1} \vert }{n} \quad\bigl(n\in \mathbb{N} \setminus \{1,2\}\bigr). \end{aligned}$$

(2.16)

Taking $\beta =0$ in Theorem 2.3, we obtain the following Theorem 2.5 for the class ${\mathcal{S}}_{\Sigma} (\varphi _{6}(z) )$.

Theorem 2.5

If an analytic function ξ given by (1.1) belongs to the class ${\mathcal{S}}_{\Sigma} (\varphi _{6}(z) )$, and if

$$\begin{aligned} T_{2}=\alpha T_{1} \quad(\alpha >0), \end{aligned}$$

then

$$\begin{aligned} &\vert a_{2} \vert \leqq \textstyle\begin{cases} \sqrt{T_{1}} & (0< T_{1}\leqq 1), \\ \frac{T_{1}}{2} & (T_{1}\geqq 1),\end{cases}\displaystyle \end{aligned}$$

(2.17)

$$\begin{aligned} &\vert a_{3} \vert \leqq \frac{T_{1}}{2} \end{aligned}$$

(2.18)

and

$$\begin{aligned} \bigl\vert a_{3}-a_{2}^{2} \bigr\vert \leqq \frac{T_{1}}{2}. \end{aligned}$$

(2.19)

In the following theorem, we use the Faber polynomial expansion method and determine the unpredictable behavior of coefficient estimates of the functions class ${\mathcal{B}}_{\Sigma}^{\beta} (\varphi _{7}(z) )$ defined by the series of the square-root function.

Theorem 2.6

Let $\xi \in {\mathcal{B}}_{\Sigma}^{\beta} (\varphi _{7}(z) )$ be an analytic function. Also, let $a_{k}=0$ $(2\leqq k\leqq n-1)$. Then,

$$\begin{aligned} \vert a_{n} \vert \leqq \frac{ \vert Q_{1} \vert }{ n+\beta} \quad\bigl(n\in \mathbb{N} \setminus \{1,2\}\bigr), \end{aligned}$$

(2.20)

where the value of $Q_{1}$ is given by (1.6) for all values of n.

Proof

For $\xi \in {\mathcal{B}}_{\Sigma}^{\beta} (\varphi _{7}(z) )$, the FPE for $( \frac{z}{\xi (z)} ) ^{1-\beta }\xi ^{{\prime }}(z)$ is given by

$$\begin{aligned} \biggl( \frac{z}{\xi (z)} \biggr) ^{1-\beta }\xi ^{{\prime }}(z)=1+ \sum _{n=1}^{\infty } \biggl( 1+\frac{n}{\beta } \biggr) K_{k}^{ \beta } ( a_{2},a_{3}, \ldots,a_{n+1} ) z^{n}. \end{aligned}$$

(2.21)

For the inverse mappings $g=\xi ^{-1}$, the FPE for $( \frac{w}{g(w)} ) ^{1-\beta }g^{{\prime }}(w)$ is:

$$\begin{aligned} \biggl( \frac{w}{g(w)} \biggr) ^{1-\beta }g^{{\prime }}(w)=1+\sum _{n=1}^{\infty } \biggl( 1+\frac{n}{\beta } \biggr) K_{k}^{ \beta } ( b_{2},b_{3}, \ldots,b_{n+1} ) w^{n}, \end{aligned}$$

(2.22)

where

$$\begin{aligned} b_{n+1}=\frac{1}{n+1}K_{k}^{-(n+1)}(a_{2},a_{3}, \ldots,a_{n+1}) \quad(k\geqq 1). \end{aligned}$$

Furthermore, since $\xi \in {\mathcal{B}}_{\Sigma }^{\beta } (\varphi _{7}(z) )$, by the definition of subordination, there exists a Schwarz function $\psi (z)$ given by

$$\begin{aligned} \psi (z)=\sum_{n=1}^{\infty }c_{n}z^{n},\quad z\in E \end{aligned}$$

such that

$$\begin{aligned} \biggl( \frac{z}{\xi (z)} \biggr) ^{1-\beta }\xi ^{{\prime }}(z)& = \varphi _{7} \bigl(\psi (z) \bigr) \\ & =1+\sum_{n=1}^{\infty }\sum _{k=1}^{n}Q_{k} \mathfrak{S}_{n}^{k} ( c_{1},c_{2},\ldots,c_{n} ) z^{n}. \end{aligned}$$

(2.23)

Similarly, for $( \frac{w}{g(w)} ) ^{1-\beta }g^{{\prime }}(w)$, there exists another Schwarz function $q(w)$ given by

$$\begin{aligned} q(w)=\sum_{n=1}^{\infty }d_{n}w^{n} \end{aligned}$$

such that

$$\begin{aligned} \biggl( \frac{w}{g(w)} \biggr) ^{1-\beta }g^{{\prime }}(w)& = \varphi _{2} \bigl(q(w) \bigr) \\ & =1+\sum_{n=1}^{\infty }\sum _{k=1}^{n}Q_{k} \mathfrak{S}_{n}^{k} ( d_{1},d_{2},\ldots,d_{n} ) z^{n}. \end{aligned}$$

(2.24)

Evaluating the coefficients of equations (2.21) and (2.23), for any $n\geqq 2$, we obtain

$$\begin{aligned} \biggl( 1+\frac{n}{\beta } \biggr) \mathfrak{S}_{n}^{\beta } ( a_{2},a_{3}, \ldots,a_{n+1} ) a_{n}=\sum _{k=1}^{n-1}Q_{k}\mathfrak{S}_{n}^{k} ( c_{1},c_{2}, \ldots,c_{n-1} ). \end{aligned}$$

(2.25)

Furthermore, by evaluating the coefficients of equations (2.22) and (2.24), for any $n\geqq 2$, we obtain

$$\begin{aligned} \biggl( 1+\frac{n}{\beta } \biggr) \mathfrak{S}_{k}^{\beta } ( b_{2},b_{3}, \ldots,b_{n} ) =\sum _{k=1}^{n-1}Q_{k}\mathfrak{S}_{n}^{k} ( d_{1},d_{2},\ldots,d_{n} ). \end{aligned}$$

(2.26)

Solving for $a_{2}$ and taking the moduli of the coefficients of the Schwarz functions ψ and q, we have (see [1])

$$\begin{aligned} \vert c_{n} \vert \leqq 1 \quad\text{and} \quad \vert d_{n} \vert \leqq 1. \end{aligned}$$

Also, from the assumptions that $2\leqq k\leqq n-1$ and $a_{k}=0$; respectively, we obtain

$$\begin{aligned} b_{n}=-a_{n} \end{aligned}$$

as well as

$$\begin{aligned} \biggl( 1+\frac{n}{\beta } \biggr) \beta a_{n}=Q_{1}c_{n-1} \end{aligned}$$

(2.27)

and

$$\begin{aligned} - \biggl( 1+\frac{n}{\beta } \biggr) \beta a_{n}=Q_{1}d_{n-1}. \end{aligned}$$

(2.28)

Thus, by solving equations (2.27) and (2.28) for $a_{n}$ and determining the moduli with the coefficients of the Schwarz functions $\psi (z)$ and $q(w)$, which are $\vert c_{n} \vert \leq 1$ and $\vert d_{n} \vert \leq 1$, respectively (see [1]), we obtain

$$\begin{aligned} \vert a_{n} \vert \leqq \frac{ \vert Q_{1} \vert }{n+\beta }. \end{aligned}$$

This completes the proof of Theorem 2.6. □

Theorem 2.7

If an analytic function ξ given by (1.1) belongs to the class ${\mathcal{B}}_{\Sigma }^{\beta } (\varphi _{7}(z) )$, and if

$$\begin{aligned} Q_{2}=\alpha Q_{1} \quad(\alpha >0), \end{aligned}$$

then

$$\begin{aligned} &\vert a_{2} \vert \leqq \textstyle\begin{cases} \sqrt{\frac{2Q_{1}}{(1+\beta )(\beta +2)}} & ( 0< Q_{1} \leqq \frac{2 ( \beta +1 ) }{(\beta +2)} ), \\ \frac{Q_{1}}{\beta +1} & ( Q_{1}\geqq \frac{2 ( \beta +1 ) }{(\beta +2)} ),\end{cases}\displaystyle \\ &\vert a_{3} \vert \leqq \frac{Q_{1}}{\beta +2} \end{aligned}$$

(2.29)

and

$$\begin{aligned} \bigl\vert a_{3}-a_{2}^{2} \bigr\vert \leqq \frac{Q_{1}}{(\beta +1)(\beta +2)}, \end{aligned}$$

where $Q_{1}$ and $Q_{2}$ are given by (1.6).

Proof

Taking $n=2$ in (2.25) and (2.26), we have

$$\begin{aligned} (\beta +1)a_{2}=Q_{1}c_{1} \end{aligned}$$

(2.30)

and

$$\begin{aligned} -(\beta +1)a_{2}=Q_{1}d_{1}. \end{aligned}$$

(2.31)

From (2.30) and (2.31), we obtain

$$\begin{aligned} c_{1}=-d_{1}. \end{aligned}$$

(2.32)

If we take the moduli of both sides of any of these two equations, for the coefficients of the Schwarz functions ψ and q given by

$$\begin{aligned} \vert c_{n} \vert \leqq 1 \quad\text{and}\quad \vert d_{n} \vert \leqq 1, \end{aligned}$$

we obtain

$$\begin{aligned} \vert a_{2} \vert \leqq \frac{Q_{1}}{\beta +1}. \end{aligned}$$

(2.33)

For $n=3$, equations (2.5) and (2.6) yield

$$\begin{aligned} \frac{(\beta -1)(\beta +2)}{2}a_{2}^{2}+(\beta +2)a_{3}=Q_{1}c_{2}+ \alpha Q_{1}c_{1}^{2} \end{aligned}$$

(2.34)

and

$$\begin{aligned} \frac{(3+\beta )(\beta +2)}{2}a_{2}^{2}-(\beta +2)a_{3}=Q_{1}d_{2}+ \alpha Q_{1}d_{1}^{2}, \end{aligned}$$

(2.35)

respectively. Thus, upon adding (2.34) and (2.35), we obtain

$$\begin{aligned} (1+\beta ) (\beta +2)a_{2}^{2}=Q_{1} \bigl[ \bigl( c_{2}+\alpha c_{1}^{2} \bigr) + \bigl( d_{2}+\alpha d_{1}^{2} \bigr) \bigr]. \end{aligned}$$

Using the moduli of the coefficients of the Schwarz functions, we have

$$\begin{aligned} (1+\beta ) (\beta +2) \vert a_{2} \vert ^{2}\leqq Q_{1} \bigl[ \bigl\vert c_{2}+\alpha c_{1}^{2} \bigr\vert + \bigl\vert d_{2}+ \alpha d_{1}^{2} \bigr\vert \bigr]. \end{aligned}$$

Thus, by Lemma 2.1, we have

$$\begin{aligned} (1+\beta ) (\beta +2) \vert a_{2} \vert ^{2}\leqq Q_{1} \bigl[ 1+ ( \alpha -1 ) \vert c_{1} \vert ^{2}+1+ ( \alpha -1 ) \vert d_{1} \vert ^{2} \bigr]. \end{aligned}$$

Taking the moduli on both sides and using the fact that

$$\begin{aligned} \vert c_{n} \vert \leqq 1 \quad\text{and}\quad \vert d_{n} \vert \leqq 1, \end{aligned}$$

we obtain

$$\begin{aligned} \vert a_{2} \vert \leqq \sqrt{\frac{2Q_{1}}{(1+\beta )(\beta +2)}}. \end{aligned}$$

Consequently, we note that

$$\begin{aligned} \sqrt{\frac{2Q_{1}}{(1+\beta )(\beta +2)}}< \frac{Q_{1}}{\beta +1}. \end{aligned}$$

Multiplying equation (2.34) by $(3+\beta )$ and equation (2.35) by $(\beta -1)$, we obtain

$$\begin{aligned} 2 ( \beta +1 ) ( \beta +2 ) a_{3}=Q_{1} \bigl[ (3+ \beta ) \bigl( c_{2}+ ( \alpha ) c_{1}^{2} \bigr) -( \beta -1) \bigl( d_{2}+ ( -B ) d_{1}^{2} \bigr) \bigr]. \end{aligned}$$

Thus, by Lemma 2.1, we have

$$\begin{aligned} \vert a_{3} \vert \leqq \frac{Q_{1}}{2 ( \beta +1 ) ( \beta +2 ) } \bigl[ (3+\beta ) \bigl( 1+ ( \alpha -1 ) \bigr) \vert c_{1} \vert ^{2}+( \beta -1) \bigl(1+ ( \alpha -1 ) \bigr) \vert d_{1} \vert ^{2} \bigr]. \end{aligned}$$

Also, for the coefficients of the Schwarz functions, we obtain

$$\begin{aligned} \vert a_{3} \vert \leqq \frac{Q_{1}}{\beta +2}. \end{aligned}$$

Lastly, upon subtracting equation (2.34) from equation (2.35), if we use relation (2.32), we obtain

$$\begin{aligned} (\beta +1) (\beta +2) \bigl( a_{3}-a_{2}^{2} \bigr) =Q_{1} \vert c_{2}-d_{2} \vert . \end{aligned}$$

For the coefficients of the Schwarz functions $\psi (z)$ and $q(w)$, we have $\vert c_{n} \vert \leq 1$ and $\vert d_{n} \vert \leq 1$, respectively (see [1]), we obtain

$$ \bigl\vert a_{3}-a_{2}^{2} \bigr\vert \leqq \frac{Q_{1}}{(\beta +1)(\beta +2)}. $$

 □

Taking $\beta =0$ in Theorem 2.6, we obtain the following result for the class ${\mathcal{S}}_{\Sigma} (\varphi _{7}(z) )$.

Theorem 2.8

Let $\xi \in {\mathcal{S}}_{\Sigma } (\varphi _{7}(z) )$ be an analytic function given by (1.1). Also, let $a_{k}=0$ $(2\leqq k\leqq n-1)$. Then,

$$\begin{aligned} \vert a_{n} \vert \leqq \frac{ \vert Q_{1} \vert }{n}\quad \bigl(n\in \mathbb{N}\setminus \{1,2\}\bigr). \end{aligned}$$

Taking $\beta =0$ in Theorem 2.7, we deduce Theorem 2.9 below for the class ${\mathcal{S}}_{\Sigma} (\varphi _{7}(z) )$.

Theorem 2.9

If an analytic function ξ given by (1.1) belongs to the class ${\mathcal{S}}_{\Sigma } (\varphi _{7}(z) )$, and if

$$\begin{aligned} Q_{2}=\alpha Q_{1} \quad(\alpha >0), \end{aligned}$$

then

$$\begin{aligned} &\vert a_{2} \vert \leqq \textstyle\begin{cases} \sqrt{Q_{1}} & (0< Q_{1}\leqq 1), \\ \frac{Q_{1}}{2} & (Q_{1}\geqq 1),\end{cases}\displaystyle \\ &\vert a_{3} \vert \leqq \frac{Q_{1}}{2} \end{aligned}$$

and

$$\begin{aligned} \bigl\vert a_{3}-a_{2}^{2} \bigr\vert \leqq \frac{Q_{1}}{2}. \end{aligned}$$

3 Conclusion

In our present investigation, we have introduced and studied the properties and characteristics of functions belonging to two new subclasses of the class of bi-Bazilevič functions, which are related to the Fibonacci-number series and the square-root functions. Under a special choice of the parameters involved, these two classes of Bazilevič functions are shown to reduce to two new subclasses of star-like biunivalent functions related with the Fibonacci-number series and the square-root functions. We have applied the Faber polynomial expansion (FPE) technique in order to find the general coefficient bounds for the functions belonging to each of these functions’ classes. We have also established bounds for the initial coefficients for bi-Bazilevič functions and have demonstrated the unexpected behavior of these initial coefficients in relation to the square-root functions and the Fibonacci-number series. The technique described in this article allows us to define several new subclasses of analytic functions and meromorphic functions connected with Fibonacci numbers and square-root functions. For these classes, we can determine the nth coefficient bounds by using the technique of the Faber polynomial. Furthermore, we can expand this work by incorporating Chebyshev polynomials, quantum calculus, symmetric quantum calculus, sigmoid activation functions, linear operators, and differential operators.