Certain New Applications of Faber Polynomial Expansion for a New Class of bi-Univalent Functions Associated with Symmetric q-Calculus

Abstract

In this study, we applied the ideas of subordination and the symmetric q-difference operator and then defined the novel class of bi-univalent functions of complex order $\gamma$. We used the Faber polynomial expansion method to determine the upper bounds for the functions belonging to the newly defined class of complex order $\gamma$. For the functions in the newly specified class, we further obtained coefficient bounds $\left|{\rho }_2\right|$ and the Fekete–Szegő problem $\left|{\rho }_3-{\rho }_2^2\right|$, both of which have been restricted by gap series. We demonstrate many applications of the symmetric Sălăgean q-differential operator using the Faber polynomial expansion technique. The findings in this paper generalize those from previous studies.

1. Introduction and Definitions

The set $\mathfrak{A}$ stands for all $h\left(z\right)$ functions in the open unit disc $E=\{z:\left|z\right|<1\}$ that are analytic, and each $h\in \mathfrak{A}$ is normalized by

$$h\left(0\right)=0\mathrm{and}{h}^{\prime }\left(0\right)=1.$$

This means that the following form may be used to represent any function $h\in \mathfrak{A}$, such that

$$h\left(z\right)=z+\sum _{m=2}^{\infty }{a}_m{z}^m.$$

(1)

In addition, $\mathcal{S}\subset \mathfrak{A}$ and all $h\in \mathcal{S}$ are univalent in E. In the case of $h_1$, $h_2\in \mathfrak{A}$, and $h_1$ is subordinate to $h_2$ in E, represented by

$$h_1\left(z\right)\prec h_2\left(z\right),z\in E.$$

If there exists a Schwarz function $v_0\in \mathfrak{A}$, along with the conditions $v_0\left(0\right)=0$, and $\left|v_0\left(z\right)\right|<1$, then

$$h_1\left(z\right)=h_2\left(v_0\left(z\right)\right),z\in E.$$

The set of all starlike functions is represented by ${\mathcal{S}}^{\ast }$, and for any $h\in {\mathcal{S}}^{\ast }$ if

$$\Re \left(\frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}\right)>0,z\in E.$$

The set of all convex functions is represented by C and for any $h\in C$ if

$$1+\Re \left(\frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}\right)>0,z\in E.$$

These conditions are identical to one another in terms of subordination and are defined as follows:

$${\mathcal{S}}^{\ast }=\left(h\in \mathfrak{A}:\frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}\prec \frac{1+z}{1-z}\right)$$

and

$$\mathcal{C}=\left(h\in \mathfrak{A}:1+\frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}\prec \frac{1+z}{1-z}\right).$$

According to Ma and Minda [1], the aforementioned two classes can be defined as follows:

$${\mathcal{S}}^{\ast }\left(\phi \right)=\left\{h\in \mathfrak{A}:\frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}\prec \phi \left(z\right)\right\}$$

and

$$\mathcal{C}\left(\phi \right)=\left\{h\in \mathfrak{A}:1+\frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}\prec \phi \left(z\right)\right\}.$$

The function $\phi \left(z\right)$ is a positive real part function and satisfies the normalized conditions

$$\phi \left(0\right)=1,{\phi }^{\prime }\left(0\right)>0.$$

The following is the expansion of the two classes mentioned above, as given by Ravichandran et al. [2]:

$${\mathcal{S}}^{\ast }\left(\gamma ,\phi \right)=\left\{h\in \mathfrak{A}:1+\frac{1}{\gamma }\left(\frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}-1\right)\prec \phi \left(z\right);\gamma \in \mathbb{C}\backslash \left\{0\right\}\right\}$$

and

$$\mathcal{C}\left(\gamma ,\phi \right)=\left\{h\in \mathfrak{A}:1+\frac{1}{\gamma }\left(\frac{z{h}^{″}\left(z\right)}{h^{\prime }\left(z\right)}\right)\prec \phi \left(z\right);\gamma \in \mathbb{C}\backslash \left\{0\right\}\right\}.$$

Functions of this kind are known as Ma–Minda starlike and convex functions of order $\gamma$, $\left(\gamma \in \mathbb{C}\backslash \left\{0\right\}\right)$.

The Koebe one-quarter theorem (see [3]) states that the image of E under every $h\in \mathcal{S}$ contains a disk of radius one-quarter centered at the origin. So, there is an inverse function $h^{-1}=g$ for any function $h\in \mathcal{S}$ and

$$g\left(h\right(z\left)\right)=z,z\in E$$

and

$$h\left(g\left(w\right)\right)=w,\left|w\right|<r_0\left(h\right),r_0\left(h\right)\ge \frac{1}{4}.$$

The series of inverse functions g is given by:

$$g\left(w\right)=w-a_2{w}^2+Q_1{w}^3-Q_2{w}^4+\dots ,$$

(2)

where

$$\begin{array}{ccc}\hfill Q_1& =& 2a_2^2-a_3,\hfill \\ \hfill Q_2& =& 5a_2^3-5a_2{a}_3+a_4.\hfill \end{array}$$

Let h be the univalent function and, if its inverse $\left(h^{-1}\right)$ is univalent in E, then the function h is bi-univalent in E, and the collection of all such functions is denoted by the symbol $\Sigma$. Some examples of class $\Sigma$ are:

$$\frac{z}{1-z},log\frac{1}{1-z}\mathrm{and}log\sqrt{\frac{1+z}{1-z}}.$$

Lewin [4] introduced the concept of class $\Sigma$ and demonstrated that $\left|a_2\right|<1.51$ for each $h\in \Sigma$. Styer and Wright [5] proved that there exists a bi-univalent function h such that $\left|a_2\right|<\frac{4}{3}$. Several scholars have been attempting to pin down the connection between the geometric features of $\Sigma$ and the coefficient bounds ever since the class $\Sigma$ was introduced. Lewin [4], Brannan and Taha [6], and Srivastava et al. [7] have contributed significantly to the development of the field of bi-univalent function investigation. All that has come out of these more recent studies are non-sharp estimates of the initial coefficients. In addition, estimates of the coefficients were found in [8] for a wide variety of classes of analytic bi-univalent functions. More recently, in [9], the integral operator based on the Lucas polynomial was used to estimate coefficients for general subclasses of analytic bi-univalent functions, while Oros and Cotirla [10] obtained coefficient estimates for the Fekete–Szegö problem. They achieved this by introducing a new class of m-fold bi-univalent functions. The question of whether or not there exists a sharp coefficient estimate for $|a_m|$, $(m=3,4,5,\dots )$ remains unanswered.

Coefficient bounds $|a_m|$ for $m\ge 3$ have recently been sought by Hamidi and Jahangiri [11,12], who have begun using the Faber polynomials expansion approach. Faber presented the Faber polynomials in [13], and Gong [14] emphasized their significance. By taking into account and including the Faber polynomials expansion approach, a number of novel subclasses of $\Sigma$ have been created and investigated. Using the Faber polynomial approach, Bult [15] defined several new classes of bi-univalent functions and found general coefficient bounds $|a_m|$ for $m\ge 3$ while also discussing the unpredictable behavior of the bounds on the initial coefficients. Recently, the subordination properties and Faber polynomials expansion method were used to obtain the general coefficient bounds $|a_m|$ of class $\Sigma$ in [16], and Altinkaya and Yalcin [17] used the same method to discuss the interesting behavior of coefficient bounds for $\Sigma$. In addition, other authors have used the Faber polynomials method to discover new and exciting facts about the class $\Sigma$; see the following recently published articles on the topic of Faber polynomials [18,19,20,21,22,23].

Numerical analysis, fractional calculus, special polynomials, analytic number theory, and quantum group theory are just a few of the many areas of study that make use of the q-calculus. Quantum calculus and fractional calculus is a vast area that is attracting the attention of mathematicians and physicists. There is now a unified framework between the theory of analytic functions and that of q-calculus. The q-difference equations are employed in a wide variety of mathematical models, and they may be solved with the help of q-differential operators. In reality, nonlinear q-differential equations vie with fractional differential equations as a theoretical framework (see, for example, [24,25,26,27,28]).

Although Jackson [29] introduced the concept of the q-calculus operator and developed the $D_q$, the first to utilize the q-difference operator ($D_q$) to create a class of q-starlike functions was Ismail et al. [30]. Subclasses of analytic functions in q-calculus were subsequently developed by other scholars (for more information, see [31,32,33,34,35,36,37]).

Fractional calculus and quantum physics are only two examples of fields where the symmetric q-calculus has been shown to be important [38,39]. Fractional q-symmetric integrals and symmetric q-derivatives were introduced and some of their features were studied by Sun et al. in [40]. They investigated problems associated with non-local boundary conditions using fractional difference operators and symmetric q-fractional integrals. Some of the potential uses of the symmetric q-derivative operator in the conic domain were investigated and a new class of analytic functions was established by Kanas et al. [41]. Using the concepts of symmetric q-calculus and conic regions, Khan et al. [42] recently created an updated version of the generalized symmetric conic domains. It was also utilized to establish new findings and create a new class of q-starlike functions in E. The work of Khan et al. [43] includes applications of the symmetric q-operator, a generalization of the conic domain, and the analysis of subclasses of q-starlike and q-convex functions. The symmetric q-difference operator for m-fold symmetric functions was recently built by Khan et al. [44], allowing them to investigate certain significant results for m-fold symmetric bi-univalent functions. For multivalent functions, the idea of a symmetric q-derivative operator was developed further by Khan et. al in article [45], and many novel applications of this operator were discussed.

After defining the symmetric q-derivative (q-difference) operator, we utilized it to construct a new family of class $\Sigma$ in E.

Definition 1

([46]). The symmetric q-difference operator ${\widehat{D}}_{q}f\left(z\right)$ for $f\in \mathfrak{A}$ is defined by

$$\begin{array}{ccc}\hfill {\widehat{D}}_{q}f\left(z\right)& =& \frac{1}{z}\left(\frac{f\left(q^{-1}z\right)-f\left(qz\right)}{q^{-1}-q}\right),z\in U,\hfill \\ & & \\ & =& 1+\sum _{m=2}^{\infty }\widehat{{\left[m\right]}_q}{a}_m{z}^{m-1},\left(z\ne 0,q\ne 1\right)\hfill \end{array}$$

and

$${\widehat{D}}_q{z}^m=\widehat{{\left[m\right]}_q}{z}^{m-1},{\widehat{D}}_q\left\{\sum _{m=1}^{\infty }{a}_m{z}^m\right\}=\sum _{m=1}^{\infty }\widehat{{\left[m\right]}_q}{a}_m{z}^{m-1}.$$

We can observe that

$$\underset{q\to 1-}{lim}{\widehat{D}}_{q}f\left(z\right)=f^{\prime }\left(z\right),$$

where $f^{\prime }\left(z\right)$ is the ordinary derivative and the $\widehat{{\left[m\right]}_q}$ number is defined by

$${\widehat{\left[m\right]}}_q=\frac{q^{-m}-q^m}{q^{-1}-q},\mathrm{and}\widehat{{\left[0\right]}_q}=0.$$

The symmetric q-factorial ${\left[m\right]}_q!$ is defined by:

$$\begin{array}{ccc}\hfill {\widehat{\left[m\right]}}_q!& =& \prod _{k=1}^m\widehat{{\left[k\right]}_q},\mathrm{if}(m\in \mathbb{N})\hfill \\ & =& 1\mathrm{if}m=0.\hfill \end{array}$$

(3)

Suppose that $\phi$ is an analytic function with a positive real part in the unit disk E satisfying

$$\phi \left(0\right)=1\mathrm{and}{\phi }^{\prime }\left(0\right)>0$$

and that $\phi \left(E\right)$ is symmetric with respect to the real axis and has the series

$$\phi \left(z\right)=1+L_{1}z+L_2{z}^2+L_3{z}^3+\dots \mathrm{and}(L_1>0).$$

(4)

Based on the work given in [47] and the formulation of the symmetric q-difference operator (${\widehat{D}}_q$), a new class $\widetilde{{\mathfrak{R}}_{\gamma }^{\alpha ,q}}\left(\phi \right)$ of generalized bi-univalent functions is presented. In the next part, we will use the Faber polynomial method and two lemmas to prove the original findings.

Definition 2.

Let h be of the type (1); then, $h\in \widetilde{{\mathfrak{R}}_{\gamma }^{\alpha ,q}}\left(\phi \right)$ if

$$1+\frac{1}{\gamma }\left(\frac{z{\widehat{D}}_{q}h\left(z\right)+\alpha z^2{\widehat{D}}_q^2\left(h\left(z\right)\right)}{\left(1-\alpha \right)h\left(z\right)+\alpha z{\widehat{D}}_{q}h\left(z\right)}-1\right)\prec \phi \left(z\right)$$

and

$$1+\frac{1}{\gamma }\left(\frac{w{\widehat{D}}_{q}g\left(w\right)+\alpha w^2{\widehat{D}}_q^2\left(g\left(w\right)\right)}{\left(1-\alpha \right)g\left(w\right)+\alpha w{\widehat{D}}_{q}g\left(w\right)}-1\right)\prec \phi \left(w\right),$$

where $0\le \alpha \le 1$, $\gamma \in \mathbb{C}\backslash \left\{0\right\}$, $g=h^{-1}$, and $z,w\in E$.

Note: the function h is said to be of complex order $\gamma$ if and only if $g=h^{-1}$ also exists in $\widetilde{{\mathfrak{R}}_{\gamma }^{\alpha ,q}}\left(\phi \right)$.

Remark 1.

Taking $q\to 1-$, then $\widetilde{{\mathfrak{R}}_{\gamma }^{\alpha ,q}}\left(\phi \right)={\mathfrak{R}}_{\gamma }^{\alpha }\left(\phi \right)$, introduced by Deniz in [48].

2. Some Uses of the Faber Polynomial Expansion Technique

The Faber polynomial was used by Airault and Bouali ([49], page 184) to prove that, for the function $h\in \mathfrak{A}$,

$$\frac{z{h}^{\prime }\left(z\right)}{h\left(z\right)}=1-\sum _{m=2}^{\infty }\left[T_{m-1}(a_2,a_3,\dots a_m)\right]z^{m-1},$$

(5)

where

$$T_{m-1}(a_2,a_3,\dots a_m)=\sum _{i_1+2i_2+\dots \left(m-1\right)i_{m-1}=m-1}^{\infty }A\left(i_1,i_2,i_3,\dots ,i_{m-1}\right)\left(a_2^{i_1}{a}_3^{i_2}\dots a_m^{i_{m-1}}\right)$$

and

$$\begin{array}{ccc}\hfill A\left(i_1,i_2,i_3,\dots ,i_{m-1}\right)& =& {\left(-1\right)}^{\left(m-1\right)+2i_1+\dots +m{i}_{m-1}}\hfill \\ & & \times \left(\frac{\left(i_1+i_2+i_3,\dots +i_{m-1}-1\right)!\left(m-1\right)}{\left(i_1!\right)\left(i_2!\right)\left(i_3!\right),\dots \left(i_{m-1}!\right)}\right).\hfill \end{array}$$

The first terms of the Faber polynomial $T_{m-1}$, $m\ge 2$ are (see page 52, see [50])

$$\begin{array}{ccc}\hfill T_1& =& -a_2,T_2=a_2^2-2a_3,\hfill \\ \hfill T_3& =& -a_2^3+3a_2{a}_3-3a_4,\hfill \\ \hfill T_3& =& a_2^4-4a_2^2{a}_3+4a_2{a}_4+2a_3^2-4a_5.\hfill \end{array}$$

The coefficients of the inverse map g can be constructed using the Faber polynomial method applied to the analytic functions h (see [49], page 185):

$$h^{-1}\left(w\right)=g\left(w\right)=w+\sum _{m=2}^{\infty }\frac{1}{m}{T}_{m-1}^m(a_2,a_3,\dots ,a_m)w^m,$$

where the coefficients of m parametric function $T_m^t(a_2,a_3,\dots ,a_m)$ are given by

$$\begin{array}{ccc}\hfill T_1^t& =& t{a}_2,\hfill \\ \hfill T_2^t& =& \frac{t\left(t-1\right)}{2}{a}_2^2+t{a}_{3,}\hfill \\ \hfill T_3^t& =& t(t-1)a_2{a}_3+t{a}_4+\frac{t\left(t-1\right)\left(t-2\right)}{3!}{a}_2^3,\hfill \\ \hfill T_4^t& =& t(t-1)a_2{a}_4+t{a}_5+\frac{t\left(t-1\right)}{2}{a}_3^2+\frac{t\left(t-1\right)\left(t-2\right)}{2}{a}_2^2{a}_3+\frac{t!}{(t-4)!4!}{a}_2^4,\hfill \\ & & ⋮\hfill \end{array}$$

$$\begin{array}{ccc}\hfill T_m^t& =& \frac{t!}{(t-m)!m!}{a}_2^m++\frac{t!}{(t-m+1)!(m-2)!}{a}_2^{m-2}{a}_3+\frac{t!}{(t-n+2)!(m-3)!}{a}_2^{m-3}{a}_4\hfill \\ & & \\ & & +\frac{t!}{(t-n+3)!(m-4)!}{a}_2^{m-4}\left(a_5+\frac{t-n+3}{2}{a}_3^2\right)\hfill \\ & & \\ & & +\frac{t!}{(t-n+4)!(m-5)!}{a}_2^{m-4}\left(a_6+(t-m+3)a_3{a}_4\right)+\sum _{\mathfrak{i}\ge 6}{a}_2^{m-\mathfrak{i}}{Q}_{\mathfrak{i}}.\hfill \end{array}$$

where, for $6\le \mathfrak{i}\le m$, $Q_{\mathfrak{i}}$ represents the homogeneous polynomial in the variables $a_2,a_3,\dots a_m$; see (page 349, [51]) and (pages 183 and 205, [49]). In particular, the first three terms of $T_m^t$ are

$$\frac{1}{2}{T}_1^1=-a_2,\frac{1}{3}{T}_2^{-3}=2a_2^2-a_3$$

and

$$\frac{1}{4}{T}_3^3=-(5a_2^3-5a_2{a}_3+a_4).$$

In most cases, and every $r\in \mathbb{Z}$, ($m\ge 2)$, an expansion of $T_m^r$ has the form:

$$T_m^r=r{a}_m+\frac{r(r-1)}{2}{\mathcal{V}}_m^2+\frac{r!}{(r-3)!3!}{\mathcal{V}}_m^3+\dots +\frac{r!}{(r-m)!\left(m\right)!}{\mathcal{V}}_m^m,$$

where

$${\mathcal{V}}_m^r={\mathcal{V}}_m^r(a_2,a_3\dots ).$$

By [52], we have

$${\mathcal{V}}_m^v(a_2,\dots ,a_m)=\sum _{m=1}^{\infty }\frac{v!{\left(a_2\right)}^{{\mu }_1}\dots {\left(a_m\right)}^{{\mu }_m}}{{\mu }_{1!},\dots ,{\mu }_m!},\mathrm{for}{a}_1=1\mathrm{and}v\le m$$

and nonnegative integer ${\mu }_1,\dots ,{\mu }_m$, which satisfies

$${\mu }_1+{\mu }_2+\dots +{\mu }_m=v$$

and

$${\mu }_1+2{\mu }_2+\dots +m{\mu }_m=m.$$

Clearly,

$${\mathcal{V}}_m^m(a_1,\dots ,a_m)={\mathcal{V}}_1^m$$

and the first and last polynomials are

$${\mathcal{V}}_m^m=a_1^m,\mathrm{and}{\mathcal{V}}_m^1=a_m.$$

Set of Lemmas

In order to prove our main theorems, we need the following known lemmas:

Lemma 1

([3]). Let $p\left(z\right)=1+{\displaystyle \sum _{m=1}^{\infty }}{p}_m{z}^m$ and

$$\left\{Re\right\}\left(p\right(z\left)\right)>0\mathrm{for}z\in E;$$

then, for $-\infty <\lambda <\infty$,

$$\left|p_2-\lambda p_1^2\right|\le \left\{\begin{array}{cc}2-\lambda {\left|p_1\right|}^2& \mathrm{if}\lambda <\frac{1}{2}\\ 2-\left(1-\lambda \right){\left|p_1\right|}^2& \mathrm{if}\lambda \ge \frac{1}{2}.\end{array}\right.$$

Lemma 2

([48]). Let the Schwarz function $\varphi \left(z\right)={\displaystyle \sum _{m=1}^{\infty }}{\varphi }_m{z}^m$ and $\left|\varphi \left(z\right)\right|<1$ for $z\in E$; then,

$$\left|{\varphi }_2+\beta {\varphi }_1^2\right|\le \left\{\begin{array}{cc}1-\left(1-\beta \right){\left|{\varphi }_1\right|}^2& \mathrm{if}\beta >0\\ 1-\left(1+\beta \right){\left|{\varphi }_1\right|}^2& \mathrm{if}\beta \le 0.\end{array}\right.$$

(6)

Here, we estimate the coefficients $\left|{\rho }_m\right|$ for the family $\widetilde{{\mathfrak{R}}_{\gamma }^{\alpha ,q}}\left(\phi \right)$ by considering a symmetric q-difference operator and using the Faber polynomial approach. Subject to gap series conditions, we examine the unexpected behavior of initial coefficient bounds $\left|{\rho }_2\right|$ and the Fekete–Szegö problem $\left|{\rho }_3-{\rho }_2^2\right|$ in this family, and we achieve some known findings by inserting the specific value of parameters. In Section 4, by using the symmetric Sălăgean q-differential operator, we define a new class $\widetilde{{\mathfrak{R}}_{\gamma ,m}^{\alpha ,q}}\left(\phi \right)$ of analytic functions and discuss some of its applications in the form of some new results. In Section 5, we provide some concluding thoughts.

3. Main Results

Theorem 1.

Let $0\le \alpha \le 1$, $\gamma \in \mathbb{C}\backslash \left\{0\right\}$. If both functions $h\left(z\right)=z+{\displaystyle \sum _{m=2}^{\infty }}{\rho }_m{z}^m$ and $g=h^{-1}\in \widetilde{{\mathfrak{R}}_{\gamma }^{\alpha ,q}}\left(\phi \right)$ and ${\rho }_k=0$, $2\le k\le m-1$. Then,

$$\left|{\rho }_m\right|\le \frac{\left|\gamma \right|L_1}{\left(\widehat{{\left[m\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[m\right]}_q}-1\right)\right)},\left(L_1>0\right).$$

Proof. 

If we set

$$\Lambda \left(h\left(z\right)\right)=\left(1-\alpha \right)h\left(z\right)+\alpha z{\widehat{D}}_{q}h\left(z\right),$$

then

$$h\in \widetilde{{\mathfrak{R}}_{\gamma }^{\alpha ,q}}\left(\phi \right)\iff 1+\frac{1}{\gamma }\left(\frac{z{\widehat{D}}_q\left(\Lambda \left(h\left(z\right)\right)\right)}{\Lambda \left(h\left(z\right)\right)}-1\right)\prec \phi \left(z\right)$$

and

$$g=h^{-1}\in \widetilde{{\mathfrak{R}}_{\gamma }^{\alpha ,q}}\left(\phi \right)\iff 1+\frac{1}{\gamma }\left(\frac{w{\widehat{D}}_q\left(\Lambda \left(g\left(w\right)\right)\right)}{\Lambda \left(g\left(w\right)\right)}-1\right)\prec \phi \left(w\right).$$

We notice that

$$a_m=1+\alpha \left(\widehat{{\left[m\right]}_q}-1\right){\rho }_n$$

for

$$\Lambda \left(h\left(z\right)\right)=z+\sum _{m=2}^{\infty }{a}_m{z}^m.$$

The power series $\widetilde{{\mathfrak{R}}_{\gamma }^{\alpha ,q}}\left(\phi \right)$ can be expanded using the Faber polynomial (5) in this way:

$$1+\frac{1}{\gamma }\left(\frac{z{D}_q\left(\Lambda \left(h\left(z\right)\right)\right)}{\Lambda \left(h\left(z\right)\right)}-1\right)=1-\frac{1}{\gamma }\sum _{m=2}^{\infty }\left[T_{m-1}(a_2,a_3,\dots a_m)\right]z^{m-1}$$

(7)

and $g=h^{-1}$; then, obviously, we obtain

$$1+\frac{1}{\gamma }\left(\frac{w{D}_q\left(\Lambda \left(h\left(w\right)\right)\right)}{\Lambda \left(h\left(w\right)\right)}-1\right)=1-\frac{1}{\gamma }\sum _{m=2}^{\infty }{T}_{m-1}(b_2,b_3,\dots b_m)w^{m-1},$$

(8)

where

$$b_m=1+\alpha \left(\widehat{{\left[m\right]}_q}-1\right)\frac{1}{m}{T}_{m-1}^m(a_2,a_3,\dots ,a_m).$$

Applying the definition of subordination, we have two Schwarz functions:

$$u\left(z\right)=\sum _{m=1}^{\infty }{c}_m{z}^m$$

and

$$v\left(w\right)=\sum _{m=1}^{\infty }{d}_m{w}^m.$$

Moreover, we also have

$$1-L_1\sum _{m=1}^{\infty }{T}_m^{-1}(c_1,-c_2,\dots {\left(-1\right)}^{m+1}{c}_m,L_1,L_2,\dots L_m)z^m=\phi \left(u\left(z\right)\right)$$

(9)

and

$$1-L_1\sum _{m=1}^{\infty }{T}_m^{-1}(d_1,-d_2,\dots ,{\left(-1\right)}^{m+1}{d}_m,L_1,L_2,\dots L_m)w^m=\phi \left(v\left(w\right)\right).$$

(10)

In general [48], the coefficients

$$T_m^t=T_m^t\left(k_1,k_2,\dots ,k_m,L_1,L_2,\dots L_m\right)$$

are given by

$$\begin{array}{ccc}\hfill T_m^t& =& \frac{t!}{(t-m)!\left(m\right)!}{k}_1^m\frac{L_m}{L_1}+\frac{t!}{\left(t-m+1\right)!(m-2)!}{k}_1^{m-2}{k}_2\frac{L_{m-1}}{L_1}\hfill \\ & & +\frac{t!}{\left(t-m+2\right)!(m-4)!}{k}_1^{m-3}{k}_3\left(\frac{L_{m-2}}{L_1}\right)\hfill \\ & & \\ & & +\frac{t!}{(t-m+3)!(m-4)!}{k}_1^{m-4}\left[k_4\left(\frac{L_{m-3}}{L_1}\right)+\left(\frac{t-m+3}{2}\right)k_2^2\left(\frac{L_{m-2}}{L_1}\right)\right]\hfill \\ & & \\ & & +\frac{t!}{(t-m+4)!(m-5)!}{k}_1^{m-5}\left[k_5\left(\frac{L_{m-4}}{L_1}\right)+\left(t-m+4\right)k_2{k}_3\left(\frac{L_{m-3}}{L_1}\right)\right]\hfill \\ & & \\ & & +\sum _{j\ge 6}{k}_1^{m-j}{Q}_j,\hfill \end{array}$$

where $Q_j$ in the variables $k_2,k_3,\dots k_m$ is a homogeneous polynomial of degree j.

As a result of comparing the coefficients in Equations (7) and (9), we obtain

$$\frac{1}{\gamma }{T}_{m-1}(a_2,a_3,\dots a_m)=L_1{T}_m^{-1}(c_1,-c_2,\dots {\left(-1\right)}^m{c}_m,L_1,L_2,\dots L_m).$$

(11)

But, using the fact that $\left|c_m\right|\le 1$ and $\left|d_m\right|\le 1$, ([3]) and under the assumption that $2\le k\le m-1$ and $a_k=0$, respectively, we have

$$\frac{1}{\gamma }\left(\widehat{{\left[m\right]}_q}-1\right)a_m=\frac{1}{\gamma }\left(\widehat{{\left[m\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[m\right]}_q}-1\right)\right){\rho }_m=-L_1{c}_{m-1}.$$

(12)

Evaluating the coefficients of Equations (8) and (10) yields

$$\frac{1}{\gamma }{T}_{m-1}(b_2,b_3,\dots b_m)=L_1{T}_m^{-1}(d_1,-d_2,\dots {\left(-1\right)}^m{d}_m,L_1,L_2,\dots L_m).$$

(13)

This leads us to

$$-\frac{1}{\gamma }\left(\widehat{{\left[m\right]}_q}-1\right)b_m=-L_1{d}_{m-1}.$$

Note that, for $2\le k\le m-1$, $b_m=-a_m$ and $a_k=0$; thus,

$$\frac{1}{\gamma }\left(\widehat{{\left[m\right]}_q}-1\right)a_m=\frac{1}{\gamma }\left(\widehat{{\left[m\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[m\right]}_q}-1\right)\right){\rho }_m=-L_1{d}_{m-1}.$$

(14)

By solving either (12) or (14) at their absolute values, we obtain the required bound.

Thus, Theorem 1 is finished. □

Here, we provide a new corollary to Theorem 1 that is obtained by setting $\alpha =0$.

Corollary 1.

Let $\gamma \in \mathbb{C}\backslash \left\{0\right\}$. If $h\left(z\right)$ and inverse $g\left(z\right)$ are both in $\widetilde{{\mathfrak{R}}_{\gamma }^{0,q}}\left(\phi \right)$ and ${\rho }_k=0$, $2\le k\le m-1$. Then,

$$\left|{\rho }_m\right|\le \frac{\left|\gamma \right|L_1}{\widehat{{\left[m\right]}_q}-1},L_1>0.$$

Theorem 1 has a novel corollary that we obtain when $\alpha =1$.

Corollary 2.

Let $\gamma \in \mathbb{C}\backslash \left\{0\right\}$. If $h\left(z\right)$ and inverse $g\left(z\right)$ are both in $\widetilde{{\mathfrak{R}}_{\gamma }^{1,q}}\left(\phi \right)$ and ${\rho }_k=0$, $2\le k\le m-1$. Then,

$$\left|{\rho }_m\right|\le \frac{\left|\gamma \right|L_1}{\widehat{{\left[m\right]}_q}\left(\widehat{{\left[m\right]}_q}-1\right)},L_1>0.$$

Using Theorem 1, we obtain a well-established corollary that is proved in [48] for $q\to 1-$.

Corollary 3

([48]). Let $0\le \alpha \le 1$, $\gamma \in \mathbb{C}\backslash \left\{0\right\}$. If both functions $h\left(z\right)$ and $g=h^{-1}$ belong to ${\mathfrak{R}}_{\gamma }^{\alpha }\left(\phi \right)$ and ${\rho }_k=0$, $2\le k\le m-1$. Then,

$$\left|{\rho }_m\right|\le \frac{\left|\gamma \right|L_1}{\left(m-1\right)\left(1+\alpha \left(m-1\right)\right)},L_1>0.$$

Using Theorem 1, we gain a well-established corollary that is proved in [48] for $q\to 1-$ and $\alpha =0$.

Corollary 4

([48]). Let $0\le \alpha \le 1$, $\gamma \in \mathbb{C}\backslash \left\{0\right\}$. If both functions $h\left(z\right)$ and $g=h^{-1}$ are in ${\mathfrak{R}}_{\gamma }\left(\phi \right)$ and, for $2\le k\le m-1$, ${\rho }_k=0$, then

$$\left|{\rho }_m\right|\le \frac{\left|\gamma \right|L_1}{m-1},L_1>0.$$

Example 1.

If we take $\phi \left(z\right)={\left(\frac{1+z}{1-z}\right)}^{\eta }$, which gives $L_1=2\eta$ in Theorem 1, then we obtain

$$\left|{\rho }_m\right|\le \frac{2\left|\gamma \right|\eta }{\left(\widehat{{\left[m\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[m\right]}_q}-1\right)\right)}.$$

Example 2.

If we take $\phi \left(z\right)=\frac{1+\left(1-2\eta \right)z}{1-z}$, which gives $L_1=2\left(1-\eta \right)$ in Theorem 1, then we obtain

$$\left|{\rho }_m\right|\le \frac{2\left(1-\eta \right)\left|\gamma \right|}{\left(\widehat{{\left[m\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[m\right]}_q}-1\right)\right)}.$$

Theorem 2.

Let $0\le \alpha \le 1$, $\gamma \in \mathbb{C}\backslash \left\{0\right\}$. If both functions $h\left(z\right)=z+{\displaystyle \sum _{m=2}^{\infty }}{\rho }_m{z}^m$ and $g=h^{-1}$ are in $\widetilde{{\mathfrak{R}}_{\gamma }^{\alpha ,q}}\left(\phi \right)$, then

$$\left|{\rho }_2\right|\le \left\{\begin{array}{cc}\sqrt{\frac{\left|\gamma \right|L_1}{\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\right\}}}& \mathrm{if}{L}_1\ge \left|L_2\right|\\ \sqrt{\frac{\left|\gamma \right|L_2}{\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\right\}}}& \mathrm{if}{L}_1<\left|L_2\right|\end{array}\right\}$$

and

$$\left|{\rho }_3-{\rho }_2^2\right|\le \left\{\begin{array}{cc}\frac{\left|\gamma \right|\left|L_1\right|}{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)}& \mathrm{if}{L}_1\ge \left|L_2\right|\\ \frac{\left|\gamma \right|\left|L_2\right|}{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)}& \mathrm{if}{L}_1<\left|L_2\right|\end{array}\right\}.$$

Proof. 

For $m=2$, Equations (11) and (13), respectively, yield

$${\rho }_2=\frac{\gamma L_1{c}_1}{\left(\widehat{{\left[2\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}\mathrm{and}{\rho }_2=\frac{-\gamma L_1{d}_1.}{\left(\widehat{{\left[2\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}.$$

(15)

By substituting $|c_m|\le 1$ and $|d_m|\le 1$ into any of these two equations (for an example, see [3]), we obtain

$$\left|{\rho }_2\right|\le \frac{\left|\gamma L_1\right|}{\left(\widehat{{\left[2\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}.$$

For $m=3$, Equations (11) and (13), respectively, yield

$$\frac{1}{\gamma }\left(\begin{array}{c}\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right){\rho }_3\\ \\ -\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2{\rho }_2^2\end{array}\right)=L_1{c}_2+L_2{c}_1^2$$

(16)

and

$$\frac{1}{\gamma }\left(\begin{array}{c}-\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right){\rho }_3\\ \\ +\left\{\begin{array}{c}2\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)\\ -\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\end{array}\right\}{\rho }_2^2\end{array}\right)=L_1{d}_2+L_2{d}_1^2.$$

(17)

By combining the two equations mentioned above and finding $\left|{\rho }_2\right|$, we arrive at

$${\rho }_2^2=\frac{\gamma \left(L_1{c}_2+L_2{c}_1^2+L_1{d}_2+L_2{d}_1^2\right)}{2\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\right\}}.$$

Or,

$${\left|{\rho }_2\right|}^2\le \frac{\left|\gamma \right|L_1\left(\left|c_2+\frac{L_2}{L_1}{c}_1^2\right|+\left|d_2+\frac{L_2}{L_1}{d}_1^2\right|\right)}{2\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\right\}}.$$

(18)

If $L_2\le 0$ and $\delta =\frac{L_2}{L_1}$, then, by using the Lemma 2 on (18), we obtain

$${\left|{\rho }_2\right|}^2\le \frac{\left|\gamma \right|L_1\left[1-\left(\frac{L_1+L_2}{L_1}\right){\left|c_1\right|}^2\right]+\left[1-\left(\frac{L_1+L_2}{L_1}\right){\left|d_1\right|}^2\right]}{2\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\right\}}.$$

(19)

If $L_1+L_2>0$, then (19) yields

$$\left|{\rho }_2\right|\le \sqrt{\frac{\left|\gamma \right|L_1}{\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\right\}}}.$$

If $L_1+L_2<0$, then, for the maximum values of $\left|c_1\right|=\left|d_1\right|$,

$$\begin{array}{ccc}\hfill {\left|{\rho }_2\right|}^2& \le & \frac{2\left|\gamma \right|L_1\left[1-\left(\frac{L_1+L_2}{L_1}\right)\right]}{2\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\right\}}\hfill \\ & =& \frac{-\left|\gamma \right|\left|L_2\right|}{\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\right\}}.\hfill \end{array}$$

If $L_2>0$ and $\delta =\frac{L_2}{L_1}$, then, by using the Lemma 2 on (18), we obtain

$${\left|{\rho }_2\right|}^2\le \frac{\left|\gamma \right|L_1\left\{\left[1-\left(\frac{L_1-L_2}{L_1}\right){\left|c_1\right|}^2\right]+\left[1-\left(\frac{L_1-L_2}{L_1}\right){\left|d_1\right|}^2\right]\right\}}{2\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\right\}}.$$

(20)

If $L_1-L_2>0$, then (20) yields

$$\left|{\rho }_2\right|\le \sqrt{\frac{\left|\gamma \right|L_1}{\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left({\left[2\right]}_q-1\right)\right)}^2\right\}}.}$$

If $L_1-L_2<0$, then, for the maximum values of $\left|c_1\right|=\left|d_1\right|$, we have

$$\begin{array}{ccc}\hfill {\left|{\rho }_2\right|}^2& \le & \frac{2\left|\gamma \right|L_1\left[1-\left(\frac{L_1+L_2}{L_1}\right)\right]}{2\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\right\}}\hfill \\ & & \\ & =& \frac{\left|\gamma \right|\left|L_2\right|}{\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\right\}}.\hfill \end{array}$$

Therefore,

$$\left|{\rho }_2\right|\le \sqrt{\frac{\left|\gamma \right|\left|L_2\right|}{\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2\right\}}}.$$

Now, we subtract (16) and (17), and $L_1>0$. We have

$${\rho }_3-{\rho }_2^2=\frac{\gamma L_1}{2\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)}\left\{\left(c_2+\frac{L_2}{L_1}{c}_1^2\right)-\left(d_2+\frac{L_2}{L_1}{d}_1^2\right)\right\}.$$

(21)

Adding the absolute values of the numbers on each side of (21) yields

$$\begin{array}{ccc}\hfill \left|{\rho }_3-{\rho }_2^2\right|& \le & \frac{\left|\gamma \right|L_1\left\{\left|c_2+\frac{L_2}{L_1}{c}_1^2\right|+\left|d_2+\frac{L_2}{L_1}{d}_1^2\right|\right\}}{2\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)}.\hfill \end{array}$$

(22)

If $L_2\le 0$ and $\delta =\frac{L_2}{L_1}$, then, by using the Lemma 2 on (22), we obtain

$$\left|{\rho }_3-{\rho }_2^2\right|\le \frac{\left|\gamma \right|L_1\left\{\left[1-\left(\frac{L_1+L_2}{L_1}\right){\left|c_1\right|}^2\right]+\left[1-\left(\frac{L_1+L_2}{L_1}\right){\left|d_1\right|}^2\right]\right\}}{2\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)}.$$

(23)

If $L_1+L_2>0$, then (23) yields

$$\left|{\rho }_3-{\rho }_2^2\right|\le \frac{\left|\gamma \right|L_1}{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)}.$$

If $L_1+L_2<0$, then, for the maximum values of $\left|c_1\right|=\left|d_1\right|$, the inequality (23) yields

$$\begin{array}{ccc}\hfill \left|{\rho }_3-{\rho }_2^2\right|& \le & \frac{2\left|\gamma \right|L_1\left[1-\left(\frac{L_1+L_2}{L_1}\right)\right]}{2\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)}\hfill \\ & =& \frac{-\left|\gamma \right|\left|L_2\right|}{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)}.\hfill \end{array}$$

If $L_2>0$ and $\delta =\frac{L_2}{L_1}$, then, by using the Lemma 2 on (22), we obtain

$$\left|{\rho }_3-{\rho }_2^2\right|\le \frac{\left|\gamma \right|L_1\left\{\left[1-\left(\frac{L_1-L_2}{L_1}\right){\left|c_1\right|}^2\right]+\left[1-\left(\frac{L_1-L_2}{L_1}\right){\left|d_1\right|}^2\right]\right\}}{2\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)}.$$

(24)

If $L_1-L_2>0$, then (24) yields

$$\left|{\rho }_3-{\rho }_2^2\right|\le \frac{\left|\gamma \right|L_1}{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)}.$$

If $L_1-L_2<0$, then, for the maximum values of $\left|c_1\right|=\left|d_1\right|$, the inequality (24) yields

$$\begin{array}{ccc}\hfill \left|{\rho }_3-{\rho }_2^2\right|& \le & \frac{2\left|\gamma \right|L_1\left[1-\left(\frac{L_1+L_2}{L_1}\right)\right]}{2\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)}\hfill \\ & =& \frac{\left|\gamma \right|\left|L_2\right|}{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right)}.\hfill \end{array}$$

Theorem 2 has now been proved. □

By substituting $\alpha =0$ in Theorem 2, a new corollary follows.

Corollary 5.

Let $0\le \alpha \le 1$, $\gamma \in \mathbb{C}\backslash \left\{0\right\}$. If both functions $h\left(z\right)=z+{\displaystyle \sum _{m=2}^{\infty }}{\rho }_m{z}^m$ and $g=h^{-1}$ are in $\widetilde{{\mathfrak{R}}_{\gamma }^{0,q}}\left(\phi \right)$, then

$$\left|{\rho }_2\right|\le \left\{\begin{array}{cc}\sqrt{\frac{\left|\gamma \right|\left|L_1\right|}{\widehat{{\left[3\right]}_q}-\widehat{{\left[2\right]}_q}}}& \mathrm{if}{L}_1\ge \left|L_2\right|\\ \sqrt{\frac{\left|\gamma \right|\left|L_2\right|}{\widehat{{\left[3\right]}_q}-{\widehat{{\left[2\right]}_q}}_q}}& \mathrm{if}{L}_1<\left|L_2\right|\end{array}\right\}$$

and

$$\left|{\rho }_3-{\rho }_2^2\right|\le \left\{\begin{array}{cc}\frac{\left|\gamma \right|\left|L_1\right|}{\widehat{{\left[3\right]}_q}-1}& \mathrm{if}{L}_1\ge \left|L_2\right|\\ \frac{\left|\gamma \right|\left|L_2\right|}{\widehat{{\left[3\right]}_q}-1}& \mathrm{if}{L}_1<\left|L_2\right|\end{array}\right\}.$$

By substituting $\alpha =0$ in Theorem 2, a new corollary follows.

Corollary 6.

Let $0\le \alpha \le 1$, $\gamma \in \mathbb{C}\backslash \left\{0\right\}$. If both functions $h\left(z\right)=z+{\displaystyle \sum _{m=2}^{\infty }}{\rho }_m{z}^m$ and $g=h^{-1}$ are in $\widetilde{{\mathfrak{R}}_{\gamma }^{1,q}}\left(\phi \right)$, then

$$\left|{\rho }_2\right|\le \left\{\begin{array}{cc}\sqrt{\frac{\left|\gamma \right|L_1}{\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\widehat{{\left[3\right]}_q}-\left(\widehat{{\left[2\right]}_q}-1\right){\widehat{\left[2\right]}}_q^2\right\}}}& \mathrm{if}{L}_1\ge \left|L_2\right|\\ \sqrt{\frac{\left|\gamma \right|\left|L_2\right|}{\left\{\widehat{{\left[3\right]}_q}\left(\widehat{{\left[3\right]}_q}-1\right)-\left(\widehat{{\left[2\right]}_q}-1\right){\widehat{\left[2\right]}}_q^2\right\}}}& \mathrm{if}{L}_1<\left|L_2\right|\end{array}\right\}$$

and

$$\left|{\rho }_3-{\rho }_2^2\right|\le \left\{\begin{array}{cc}\frac{\left|\gamma \right|L_1}{\widehat{{\left[3\right]}_q}\left(\widehat{{\left[3\right]}_q}-1\right)}& \mathrm{if}{L}_1\ge \left|L_2\right|\\ \frac{\left|\gamma \right|\left|L_2\right|}{\widehat{{\left[3\right]}_q}\left(\widehat{{\left[3\right]}_q}-1\right)}& \mathrm{if}{L}_1<\left|L_2\right|\end{array}\right\}.$$

Taking the $q\to 1-$ in Theorem 2, we obtain the known corollary proved in [48].

Corollary 7.

Let $0\le \alpha \le 1$, $\gamma \in \mathbb{C}\backslash \left\{0\right\}$. If both functions $h\left(z\right)=z+{\displaystyle \sum _{m=2}^{\infty }}{\rho }_m{z}^m$ and its inverse map $g=h^{-1}$ are in $\widetilde{{\mathfrak{R}}_{\gamma }^{\alpha }}\left(\phi \right)$, then

$$\left|{\rho }_3\right|\le \left\{\begin{array}{cc}\sqrt{\frac{\left|\gamma \right|L_1}{2\left(1+2\alpha \right)-{\left(1+\alpha \right)}^2}}& \mathrm{if}{L}_1\ge \left|L_2\right|\\ \sqrt{\frac{\left|\gamma \right|\left|L_2\right|}{2\left(1+2\alpha \right)-{\left(1+\alpha \right)}^2}}& \mathrm{if}{L}_1<\left|L_2\right|\end{array}\right\}$$

and

$$\left|{\rho }_3-{\rho }_2^2\right|\le \left\{\begin{array}{cc}\frac{\left|\gamma \right|L_1}{2\left(1+2\alpha \right)}& \mathrm{if}{L}_1\ge \left|L_2\right|\\ \frac{\left|\gamma \right|\left|L_2\right|}{2\left(1+2\alpha \right)}& \mathrm{if}{L}_1<\left|L_2\right|\end{array}\right\}.$$

4. Applications of Symmetric Sălăgean q-Differential Operator

Many mathematicians have used the Sălăgean differential operator to create new types of convex and star-shaped functions. In this study, a novel subclass of class $\Sigma$ was defined, and its application to the symmetric Sălăgean q-differential operator was explored. Using the Faber polynomial approach, we obtained the $\left|{\rho }_m\right|$ bounds, the initial coefficients, and the Fekete–Szegő problem for this class. In terms of the uses of the symmetric q-difference operator stated in (16), the symmetric Sălăgean q-differential operator is explained as follows:

Definition 3

([53]). For the positive integer n, the symmetric Sălăgean q-differential operator for $f\in \mathfrak{A}$ is defined by

$$\begin{array}{ccc}\hfill {\widehat{S}}_q^{0}f\left(z\right)& =& f\left(z\right),{\widehat{S}}_q^{1}f\left(z\right)=z{\widehat{D}}_{q}f\left(z\right)=\frac{f\left(q^{-1}z\right)-f\left(qz\right)}{q^{-1}-q},\dots ,\hfill \\ \hfill {\widehat{S}}_q^{n}f\left(z\right)& =& {\widehat{S}}_q\left({\widehat{S}}_q^{n-1}f\left(z\right)\right)\hfill \\ \hfill {\widehat{S}}_q^{n}f\left(z\right)& =& z+\sum _{m=2}^{\infty }{\widehat{\left[m\right]}}_q^n{a}_m{z}^m.\hfill \end{array}$$

We observe that

$${\widehat{D}}_q^{n}f\left(z\right)=f\left(z\right)\ast {\widehat{D}}_q\left(\frac{z}{1-z}\right)$$

(25)

and

$$\begin{array}{ccc}\hfill {\widehat{D}}_q\left(\frac{z}{1-z}\right)& =& z+\sum _{m=2}^{\infty }{\widehat{\left[m\right]}}_q^n{a}_m{z}^m\hfill \\ & =& \frac{z}{\left(1-q^{-1}z\right)\left(1-qz\right)}.\hfill \end{array}$$

(26)

It is obvious that

$$\underset{q\to 1-}{lim}{\widehat{D}}_q^{n}f\left(z\right)=z+\sum _{m=2}^{\infty }{m}^n{a}_m{z}^m,$$

This refers to the known Sălăgean differential operator as stated in [54].

Definition 4.

Let h satisfy the Equation (1) and $h\in \widetilde{{\mathfrak{R}}_{\gamma ,n}^{\alpha ,q}}\left(\phi \right)$ if

$$1+\frac{1}{\gamma }\left(\frac{z{D}_q\left({\mathcal{S}}_q^{n}h\left(z\right)\right)+\alpha z^2{D}_q^2\left({\mathcal{S}}_q^{n}h\left(z\right)\right)}{\left(1-\alpha \right){\mathcal{S}}_q^{n}h\left(z\right)+\alpha z{D}_q\left({\mathcal{S}}_q^{n}h\left(z\right)\right)}-1\right)\prec \phi \left(z\right)$$

and

$$1+\frac{1}{\gamma }\left(\frac{z{D}_q\left({\mathcal{S}}_q^{n}g\left(z\right)\right)+\alpha w^2{D}_q^2\left({\mathcal{S}}_q^{n}g\left(w\right)\right)}{\left(1-\alpha \right)\left({\mathcal{S}}_q^{n}g\left(w\right)\right)+\alpha w{D}_q\left({\mathcal{S}}_q^{n}g\left(w\right)\right)}-1\right)\prec \phi \left(w\right),$$

where $0\le \alpha \le 1$, $n\in \mathbb{N}$, $\gamma \in \mathbb{C}\backslash \left\{0\right\}$, $z,w\in E$, and $g=h^{-1}$.

Theorem 3.

Let $0\le \alpha \le 1$, $\gamma \in \mathbb{C}\backslash \left\{0\right\}$. If both functions $h\left(z\right)=z+{\displaystyle \sum _{m=2}^{\infty }}{\rho }_m{z}^m$ and $g=h^{-1}$ are in $\widetilde{{\mathfrak{R}}_{\gamma ,n}^{\alpha ,q}}\left(\phi \right)$ and ${\rho }_k=0$, $2\le k\le m-1$. Then,

$$\left|{\rho }_m\right|\le \frac{\left|\gamma \right|L_1}{{\widetilde{\left[m\right]}}_q^n\left(\widehat{{\left[m\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[m\right]}_q}-1\right)\right)}.$$

Proof. 

Let

$$\Lambda \left(h\left(z\right)\right)=\left(1-\alpha \right){\mathcal{S}}_q^{n}h\left(z\right)+\alpha z{D}_q\left({\mathcal{S}}_q^{n}h\left(z\right)\right);$$

then,

$$h\in \widetilde{{\mathfrak{R}}_{\gamma ,n}^{\alpha ,q}}\left(\phi \right)\iff 1+\frac{1}{\gamma }\left(\frac{z{D}_q\left(\Lambda \left({\mathcal{S}}_q^{n}h\left(z\right)\right)\right)}{\Lambda \left({\mathcal{S}}_q^{n}h\left(z\right)\right)}-1\right)\prec \phi \left(z\right)$$

and

$$g=h^{-1}\in \widetilde{{\mathfrak{R}}_{\gamma ,n}^{\alpha ,q}}\left(\phi \right)\iff 1+\frac{1}{\gamma }\left(\frac{w{D}_q\left(\Lambda {\mathcal{S}}_q^{n}g\left(w\right)\right)}{\Lambda \left(g\left(w\right)\right)}-1\right)\prec \phi \left(w\right).$$

We see that

$$a_m={\widehat{\left[m\right]}}_q^n\left(1+\alpha \left(\widehat{{\left[m\right]}_q}-1\right)\right)$$

for

$$\Lambda \left(h\left(z\right)\right)=z+\sum _{m=2}^{\infty }{a}_m{z}^m.$$

Extending the power series of $\widetilde{{\mathfrak{R}}_{\gamma ,n}^{\alpha ,q}}\left(\phi \right)$ with the Faber polynomial yields

$$1+\frac{1}{\gamma }\left(\frac{z{D}_q\left(\Lambda \left(h\left(z\right)\right)\right)}{\Lambda \left(h\left(z\right)\right)}-1\right)=1-\frac{1}{\gamma }\sum _{m=2}^{\infty }\left[T_{m-1}(a_2,a_3,\dots a_m)\right]z^{m-1}.$$

Then, we may obtain Theorem 3 by using the same steps as we carried out for Theorem 1. □

Theorem 4.

Let $0\le \alpha \le 1$, $\gamma \in \mathbb{C}\backslash \left\{0\right\}$. If both functions $h\left(z\right)=z+{\displaystyle \sum _{m=2}^{\infty }}{\rho }_m{z}^m$ and $g=h^{-1}$ are in $\mathfrak{J}\left(\alpha ,\gamma ,\psi ,q;\phi \right)$, then

$$\left|{\rho }_2\right|\le \left\{\begin{array}{cc}\sqrt{\frac{\left|\gamma \right|L_1}{\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right){\widehat{\left[3\right]}}_q^n-\left({\left[2\right]}_q-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2{\left({\widehat{\left[2\right]}}_q^n\right)}^2\right\}}}& \mathrm{if}{L}_1\ge \left|L_2\right|\\ \sqrt{\frac{\left|\gamma \right|L_2}{\left\{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right){\widehat{\left[3\right]}}_q^n-\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2{\left({\widehat{\left[2\right]}}_q^n\right)}^2\right\}}}& \mathrm{if}{L}_1<\left|L_2\right|\end{array}\right\}$$

and

$$\left|{\rho }_3-{\rho }_2^2\right|\le \left\{\begin{array}{cc}\frac{\left|\gamma \right|L_1}{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right){\widehat{\left[3\right]}}_q^n}& \mathrm{if}{L}_1\ge \left|L_2\right|\\ \frac{\left|\gamma \right|L_2}{\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right){\widehat{\left[3\right]}}_q^n}& \mathrm{if}{L}_1<\left|L_2\right|\end{array}\right\}.$$

Proof. 

For $m=2$, Equations (11) and (13), respectively, yield

$${\rho }_2=\frac{\gamma L_1{c}_1}{{\widehat{\left[2\right]}}_q^n\left(\widehat{{\left[2\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}\mathrm{and}{\rho }_2=\frac{-L_1{d}_1.}{{\widehat{\left[2\right]}}_q^n\left(\widehat{{\left[2\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}.$$

If we take the absolute values of any of these two equations and apply $|c_m|\le 1$ and $|d_m|\le 1$ (e.g, see [3]), we obtain

$$\left|{\rho }_2\right|\le \frac{\left|\gamma L_1\right|}{{\widehat{\left[2\right]}}_q^n\left(\widehat{{\left[2\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}.$$

For $m=3$, Equations (11) and (13), respectively, yield

$$\frac{1}{\gamma }\left(\begin{array}{c}\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right){\widehat{\left[3\right]}}_q^n{\rho }_3\\ -\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2{\left({\widehat{\left[2\right]}}_q^n\right)}^2{\rho }_2^2\end{array}\right)=L_1{c}_2+L_2{c}_1^2$$

and

$$\frac{1}{\gamma }\left(\begin{array}{c}-\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right){\widehat{\left[3\right]}}_q^n{\rho }_3\\ +\left\{\begin{array}{c}2\left(\widehat{{\left[3\right]}_q}-1\right)\left(1+\alpha \left(\widehat{{\left[3\right]}_q}-1\right)\right){\widehat{\left[3\right]}}_q^n\\ -\left(\widehat{{\left[2\right]}_q}-1\right){\left(1+\alpha \left(\widehat{{\left[2\right]}_q}-1\right)\right)}^2{\left({\widehat{\left[2\right]}}_q^n\right)}^2\end{array}\right\}{\rho }_2^2\end{array}\right)=L_1{d}_2+L_2{d}_1^2.$$

We can obtain the required outcome of Theorem 4 by using a similar procedure to that of Theorem 2. □

5. Conclusions

In this study, we utilized the idea of convolution and symmetric q-calculus, and we then introduced a new family of bi-univalent functions $\widetilde{{\mathfrak{R}}_{\gamma ,n}^{\alpha ,q}}\left(\phi \right)$ on the open unit disc E. The Faber polynomial expansion method serves as the backbone of our strategy, and we used it to investigate the estimates for the general coefficients of Taylor–Maclaurin series expansions in the open unit disc E for the function $h\in \widetilde{{\mathfrak{R}}_{\gamma ,n}^{\alpha ,q}}\left(\phi \right)$. New applications of the symmetric Sălăgean q-differential operator were also identified in the form of some results. In addition to our primary findings, we also discussed their numerous consequences.

There are four parts of this article. In Section 1, we briefly reviewed some elementary concepts from the theory of geometric functions since they were important to our primary conclusion. These elements are all common fare, and we have appropriately referenced them. We presented the Faber polynomial method, several related applications, and some preliminary lemmas in Section 2. We summarized our results in Section 3 and determined some applications of the Sălăgean q-differential operator in Section 4.

Future study recommendations that make use of these newly developed operators may be connected to the application of the idea of the Faber polynomial method and subordination. It is possible that the approach provided in this study may be used to create further subclasses of multivalent, meromorphic, and harmonic functions, each of which can then be studied for extra characteristics. Only by being inspired by these results will researchers be able to come up with new approaches to the various types of studies that may be undertaken in these contexts. Due to differential operators, functional analysis and operator theory can be used for the study of differential equations. Differential operator characteristics are employed in the operator approach for solving differential equations.