Certain new applications of Faber polynomial expansion for some new subclasses of $\upsilon$-fold symmetric bi-univalent functions associated with $q$-calculus

1.   Introduction and definitions

Assume that $\mathfrak{A}$ denotes the set of all analytic functions $\mathfrak{h}(z)$ in the open symmetric unit disk

which are normalized by

Thus, every function $\mathfrak{h} \in \mathfrak{A}$ can be expressed in the form given in (1.1)

Let an analytic function $\mathfrak{h}$ is said to be univalent if it satisfy the following condition:

Furthermore, $\mathcal{S}$ is the subclass of $\mathfrak{A}$ whose members are univalent in $\mathcal{U}$. The idea of subordination was initiated by Lindelof ^[30] and Little-wood and Rogosinski have further improved this idea, see ^[31,35,36]. For $\mathfrak{h}$, $y \in \mathfrak{A}$, and $\mathfrak{h}$ subordinate to $y$ in $\mathcal{U}$, denoted by

if we have a function $u$, such that

and

According to the Koebe one-quarter theorem (see ^[13]), the image of $\mathcal{U}$ under $\mathfrak{h} \in \mathcal{S}$ contains a disk of radius one-quarter centered at origin. Thus, every function $\mathfrak{h} \in \mathcal{S}$ has an inverse $\mathfrak{h}^{-1} = g$, defined as:

and

The power series for the inverse function $g(w)$ is given by

Where

An analytic function $\mathfrak{h}$ is called bi-univalent in $\mathcal{U}$ if $\mathfrak{h}$ and $\mathfrak{h}^{-1}$ are univalent in $\mathcal{U}$ and class of all bi-univalent functions are denoted by $\Sigma$. In $1967$, for $\mathfrak{h} \in \Sigma$, Levin ^[32] showed that $|a_2| < 1.51$ and after twelve years Branan and Clunie ^[8] gave the improvement of $|a_2|$ and proved that $|a_2|\leq \sqrt{2}$. Furthermore, for $\mathfrak{h} \in \Sigma$, Netanyahu ^[34] proved that $\max |a_2| = \frac{4}{3}$ and an intriguing subclass of analytic and bi-univalent functions was proposed and studied by Branan and Taha ^[9], who also discovered estimates for the coefficients of the functions in this subclass. Recently, the investigation of numerous subclasses of the analytic and bi-univalent function class $\Sigma$ was basically revitalized by the pioneering work of Srivastava et al. ^[41]. In 2012, Xu et al. ^[44] defined a general subclass of class $\Sigma$ and investigated coefficient estimates for the functions belonging to the new subclass of class $\Sigma$. Recently, several different subclasses of class $\Sigma$ were introduced and investigated by a number of authors (see for details (^[23,29,38]). In these recent papers only non-sharp estimates on the initial coefficients were obtained.

Faber polynomials was introduced by Faber ^[15] and first time he used it to determine the general coefficient bounds $|a_k|$ for $k \geq 4$. Gong ^[16] interpreted significance of Faber polynomials in mathematical sciences, particularly in Geometric Function Theory. In 1913, Hamidi et al. ^[18] first time used the Faber polynomials expansion technique on meromorphic bi-starlike functions and determined the coefficient estimates. The Faber polynomials expansion method for analytic bi-close-to-convex functions was examined by Hamidi and Jahangiri ^[21,22], who also discovered some new coefficient bounds for new subclasses of close-to-convex functions. Furthermore, many authors ^{[3,4,7,11,12,14,20]} used the same technique and determined some interesting and useful properties for analytic bi-univalent functions. For $\mathfrak{h} \in \Sigma$, by using the Faber polynomial expansions methods, only a few works have been done so far and we recognized very little over the bounds of Maclaurin's series coefficient $|a_k|$ for $k \geq 4$ in the literature. Recently only a few authors, used the Faber polynomials expansion technique and determined the general coefficient bounds $|a_k|$ for $k \geq 4$, (see for detail ^{[6,11,24,39,40,42]}).

A domain $\mathcal{U}$ is said to be the $\upsilon$-fold symmetric if

and every $\mathfrak{h}_\upsilon$ has the series of the form

The class $\mathcal{S}^\upsilon$ represents the set of all $\upsilon$-fold symmetric univalent functions. For $\upsilon = 1$, then $\mathcal{S}^\upsilon$ reduce to the class $\mathcal{S}$ of univalent functions. If the inverse $g_\upsilon$ of univalent $\mathfrak{h}$ is univalent then $\mathfrak{h}$ is called $\upsilon$-fold symmetric bi-univalent functions in $\mathcal{U}$ and denoted by $\Sigma_\upsilon$. The series expansion of inverse function $g_\upsilon$ investigated by Srivastava et al. in ^[43]:

For $\upsilon = 1$, the series in (1.4) reduces to the (1.2) of the class $\Sigma$. In ^[43] Srivastava et al. defined a subclass of $\upsilon$-fold symmetric bi-univalent functions and investigated coeffiients problem for $\upsilon$-fold symmetric bi-univalent functions. Hamidi and Jahangiri ^[19] defined $\upsilon$-fold symmetric bi-starlike functions and discussed the unpredictability of the coefficients of $\upsilon$-fold symmetric bi-starlike functions.

Many researchers have used the $q$-calculus and fractional $q$-calculus in the field of Geometric Function Theory (GFT) and they defined and studied several new subclasses of analytic, univalent and bi-univalent functions. In 1909, Jackson (^[26,27]), gave the idea of $q$-calculus operator and defined the q-difference operator ($D_q$) while in ^[25], Ismail et al. was the first who used $D_q$ in order to define a class of q-starlike functions in open unit disk $\mathcal{U}$. The most signifcant usages of $q$-calculus in the perspective of GFT was basically furnished and the basic $(or\; q-)$ hypergeometric functions were first used in GFT in a book chapter by Srivastava (see, for details, ^[37]). For more study about $q$-calculus operator theory in GFT, see the following articles ^[5,28,33].

Now we recall, some basic definitions and concepts of the $q$-calculus which will be used to define some new subclasses of the this paper.

For a non-negative integer $t$, the $q$-number $[t, q],$ $(0 < q < 1),$ is defined by

and the $q$-number shift factorial is given by

For $q \rightarrow1-$, then $[t, q]!$ reduces to $t!$.

The q-generalized Pochhammer symbol is defined by

Remark 1.1. For $q \rightarrow1-$, then $[t, q]_k$ reduces to $(t)_k = \frac{\Gamma(t+k)}{\Gamma(t)}.$

Definition 1.2. Jackson ^[27] defined the $q$-integral of function $\mathfrak{h}(z)$ as follows:

Jackson ^[26] introduced the $q$-difference operator for analytic functions as follows:

Definition 1.3. ^[26]. For $\mathfrak{h} \in \mathcal{A}$, the $q$-difference operator is defined as:

Note that, for $k \in \mathbb{N}$ and $z \in \mathcal{U}$ and

Here, we introduce the $q$-difference operator for $\upsilon$-fold symmetric functions related to the $q$-calculus as follows:

Definition 1.4. Let $\mathfrak{h}_{\upsilon} \in \Sigma_\upsilon$, of the form (1.3). Then $q$-difference operator will be defined as

and

Now we define Salagean $q$-differential operator for $\upsilon$-fold symmetric functions as follows:

Definition 1.5. For $m \in \mathbb{N}$, the Salagean $q$-differential operator for $\mathfrak{h}_{\upsilon} \in \Sigma_\upsilon$ is defined by

Remark 1.6. For $\upsilon = 1$, we have Salagean $q$-differential operator for analytic functions proved in [17].

Motivated by the following articles ^[1,10,25] and using the $q$-analysis in order to define new subclasses of class $\Sigma_\upsilon$, we apply Faber polynomial expansions technique in order to determine the estimates for the general coefficient bounds $|a_{\upsilon k+1}|$. We also derive initial coefficients $|a_{\upsilon +1}|$ and $|a_{2\upsilon +1}|$ and obtain Feketo-Sezego coefficient bounds for the functions belonging to the new subclasses of $\Sigma_\upsilon$.

Definition 1.7. A function $\mathfrak{h}_\upsilon \in \Sigma_\upsilon$ is in the class $\mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi)$ if and only if

and

where, $\varphi \in \mathcal{P}$, $\gamma \geq 0$, $b \in \mathbb{C} \setminus \{0\}$, $z$, $w \in \mathcal{U}$, and $g_\upsilon(w)$ is defined by (1.4).

Remark 1.8. For $q \rightarrow 1-$, $\upsilon = 1$, and $\gamma = 0$, then $\mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi) = \mathcal{R}_b(\varphi)$ introduced in ^[22].

Definition 1.9. A function $\mathfrak{h}_\upsilon \in \Sigma_\upsilon$, is in the class $\mathcal{R}^\upsilon_b(b, \alpha, \gamma)$ if and only if

and

Or equivalently by using subordination, we can write the above conditions as:

and

where, $0\leq \alpha < 1$, $\gamma \geq 0$, $b \in \mathbb{C} \setminus \left\{0\right\}$, $z$, $w \in \mathcal{U}$, $g_\upsilon(w)$ is defined by (1.4).

Remark 1.10. For $q \rightarrow1-$, $\upsilon = 1$, $\alpha = 0$ and $\gamma = 0$, then $\mathcal{R}^\upsilon_b(b, \alpha, \gamma) = \mathcal{R}_b(\varphi)$ introduced in [22].

Definition 1.11. A function $\mathfrak{h}_\upsilon \in \Sigma_\upsilon$, is in the class $\mathcal{R}^{\upsilon, \gamma, m}_{b, q}(\varphi)$ if and only if

and

where, $\varphi\in \mathcal{P}$, $\gamma \geq 0, m \in \mathbb{N}$, $b \in \mathbb{C} \setminus \left\{0\right\}$, $z$, $w \in \mathcal{U}$, $g_\upsilon(w)$ is defined by (1.4).

2.   The faber polynomial expansion method and application

Using the Faber polynomial technique for the analytic function $\mathfrak{h},$ then the coefficient of its inverse map $g$ can be written as follows (see ^[2,4]):

where

and $\mathcal{Q}_{i}$ is a homogeneous polynomial in the variables $a_{2}, a_{3}, ...a_{k},$ for $7 \leq i \leq k.$ Particularly, the first three term of $\Re^{-k}_{k-1}$ are

In general, for $r\in N \ \ and \ \ k\geq 2,$ an expansion of $\Re^{r}_{k}$ of the form:

where,

and by ^[2], we have

The sum is taken over all non negative integer $\mu_{1}, ..., \mu_{k}$ which is satisfying

Clearly,

and

are first and last polynomials.

Now, using the Faber polynomial expansion for $\mathfrak{h}_{\upsilon}$ of the form (1.3) we have

The coefficient of inverse map $g_{\upsilon}$ can be expressed of the form:

Theorem 2.1. For $b \in \mathbb{C}\setminus \{0\}.$ Let $\mathfrak{h}_{\upsilon} \in \mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi)$ by given by (1.3). If $a_{\upsilon i+1} = 0, \ \ 1\leq i \leq k-1$, then

Proof. For $\mathfrak{h}_{\upsilon} \in \mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi)$ we have

and

where,

Since $\mathfrak{h}_{\upsilon} \in \mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi)$ and $g_{\upsilon} \in \mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi)$ by definition, we have

and

where

Equating the coefficient of (2.1) and (2.5) we obtain

Similarly, corresponding coefficient of (2.2) and (2.6), we have

Since, $1\leq i \leq k-1, \ \ and \ \ a_{\upsilon i+1} = 0$; we have

and

Taking the modulus on both sides of (2.9) and (2.10), we have

Now using the fact $|\varphi_{1}|\leq 2, |c_{k}|\leq 1,$ and $|d_{k}|\leq 1,$ we have

Hence, Theorem 2.1 is completed.

For $\upsilon = 0, \gamma = 0, q \rightarrow 1-, k = n-1,$ in Theorem 2.1, we obtain known corollary proved in ^[22].

Corollary 2.2. For $b\in \mathbb{C}\setminus\{0\},$ Let $\mathfrak{h}_{\upsilon} \in \mathcal{R}_{b}(\varphi)$, If $a_{\upsilon i+1} = 0, 1\leq i \leq n.$ Then

Theorem 2.3. For $b \in \mathbb{C}\setminus\{0\}.$ Let $\mathfrak{h}_{\upsilon} \in \mathcal{R}^{\upsilon, \gamma}_{b, q}(\varphi)$ be given by (1.3). Then

where,

Proof. Taking $k = 1$ and $k = 2$ in (2.7) and (2.8), then, we have

From (2.11) and (2.13) and using the fact $|\varphi_{1}|\leq 2, |c_{k}|\leq 1$ and $|d_{k}|\leq 1,$ we have

Adding (2.12) and (2.14) we have

Taking absolute value of (2.16), we have

Now the bounds given for $|a_{\upsilon+1}|$ can be justified since

for

From (2.12), we get

Subtract (2.14) from (2.12), we have

or

Taking the absolute, we have

Using the assertion (2.15) on (2.20), we have

Follows from (2.17) and (2.21) upon nothing that

Now, rewrite (2.14) as follows:

Using the fact $\mid \varphi_1 \mid \leq 2$, $\mid c_k\mid \leq 1$ and $\mid d_k \mid \leq 1$, we have

From (2.18), we have

Again using the fact $\mid \varphi_1 \mid \leq 2$, $\mid c_k \mid \leq 1$ and $\mid d_k \mid \leq 1$, we have

Take $q\rightarrow 1-, \gamma = 0, \upsilon = 1,$ and $k = n-1$ in the Theorem 2.3, we get known corollary.

Corollary 2.4. ^[22]. For $b \in \mathbb{C} \setminus \{0\},$ let $\mathfrak{h} \in \mathcal{R}_b (\varphi)$ be given by (1.1), then

Theorem 2.5. For $b \in \mathbb{C} \setminus\{0\}.$ Let $\mathfrak{h}_\upsilon \in \mathcal{R}^{\upsilon }_{q}(b, \alpha, \gamma)$ by given by (1.3). If $a_{\upsilon i+1} = 0, 1\leq i \leq k-1.$ Then

where, $\mathfrak{B}_0 = 1 - \alpha(1+q)$ and $\mathfrak{B}_1 = -q$.

Proof. Let $\mathfrak{h}_\upsilon \in \mathcal{R}^{\upsilon }_{q}(b, \alpha, \gamma)$. Then

and

where,

Since $h_\upsilon \in \mathcal{R}^{\upsilon }_{q}(b, \alpha, \gamma)$ and $g_\upsilon \in \mathcal{R}^{\upsilon }_{q}(b, \alpha, \gamma)$ by definition, there exist two positive real functions $p(z)$ and $r(w)$ given in (2.3) and (2.4), then we have

Equating the corresponding coefficients of (2.22) and (2.24), we have

Similarly, corresponding coefficient of (2.23)and (2.25), we have

For $a_{\upsilon i+1} = 0;\; 1 \leq i \leq k-1,$ we get

and we have

and

Taking modulus on (2.28) and (2.29), we have

Since

we have

which complete the proof of Theorem.

For $b = 1, k = 1, \upsilon = n-1, q \rightarrow 1-, and \gamma \geqq 0$ in the above Theorem 2.5, we obtain the following result given in ^[40].

Corollary 2.6. Let $\mathfrak{h}_\upsilon \in \mathcal{R}(n, \alpha, \gamma)$ be given by (1.3). If $a_{n-1} = 0$, and $1 \leq i \leq k-1$, then

Theorem 2.7. For $b \in \mathbb{C} \setminus \{0\},$ let $\mathfrak{h}_\upsilon \in \mathcal{R}_q^\upsilon (b, \alpha, \gamma)$ be given by (1.3), then

and

where

Proof. Take $k = 1$ and $k = 2$ in (2.26) and (2.27). Then we have

From (2.30) and (2.32) and using the fact $|\varphi_1 | \leq 2, | c_k|\leq 1$ and $| d_k|\leq 1,$ we have

Adding (2.31) and (2.33) we have

and

Taking the square-root of (2.35), we have

Now the bounds given for $|a_{\upsilon +1} |$ can be justified since

From (2.31), we have

Next we subtract (2.33) from (2.31), we get

or

Taking the absolute values yield

Using the assertion (2.34) on (2.39), we have

It follows from (2.36) and (2.40) upon noting that

Now, we rewrite (2.33) as follows:

Taking the modulus and using $|\varphi_1| \leq 2, \ |c_k| \leq 1$ and $|d_k| \leq 1,$ we have

Finally, from (2.37), we have

Taking the modulus and using $|c_k| \leq 1$ and $|d_k| \leq 1,$ we have

For $\upsilon = 1, \gamma = 0, q \rightarrow 1-, k = n-1$ in Theorem 2.7, then we obtain result proved in ^[22].

Corollary 2.8. ^[22]. For $b \in \mathbb{C} \setminus \{0\},$ let $\mathfrak{h}_\upsilon \in \mathcal{R}_b (\varphi)$ be given by (1.1), then

2.1.   Applications of our main results

Here, in this section, we consider the newly defined Salagean $q-$differential operator for subclass of $R^{\upsilon, \gamma, m}_{b, q}(\varphi)$ of class of $\sum_{\upsilon }$ and investigate some new application in the form of results

Theorem 2.9. For $b \in \mathbb{C} \setminus\{0\}.$ Let $\mathfrak{h}_\upsilon \in R^{\upsilon, \gamma, m}_{b, q}(\varphi)$ by given by (1.3). If $a_{\upsilon i+1} = 0,$ and $1\leq i \leq k-1,$ then

Proof. We can prove Theorem 2.9 by using the similar method of Theorem 2.1.

Theorem 2.10. For $b \in \mathbb{C} \setminus \{0\}.$ Let $\mathfrak{h}_\upsilon \in R^{\upsilon, \gamma, m}_{b, q}(\varphi)$ by given by (1.3). Then

where

Proof. We can prove Theorem 2.10 by using the similar method of Theorem 2.3.

3.   Conclusions

In this article, first of all, we used the q-difference operator for $\upsilon$-fold symmetric functions in order to define some new subclasses of the $\upsilon$-fold symmetric bi-univalent functions in the open symmetric unit disk $\mathcal{U}$. We also used the basic concepts of q-calculus and defined the Salagean q-differential operator for $\upsilon$-fold symmetric functions. We considered this operator and investigated a new subclass of $\upsilon$-fold symmetric bi-univalent functions. Faber Polynomial expansion method and q-analysis are used in order to determined general coefficient bounds $|a_{\upsilon+1}|$ for functions in each of these newly defined $\upsilon$-fold symmetric bi-univalent functions classes. Feketo-Sezego problems and initial coefficient bounds $|a_{\upsilon+1}|$ and $|a_{2\upsilon+1}|$ for the function belonging to the subclasses of $\upsilon$-fold symmetric bi-univalent functions in open symmetric unit disk $\mathcal{U}$ are also investigated.

Acknowledgments

I would like to thank to the editor and referees for their valuable comments and suggestions.

Conflict of interest

The author declares no conflict of interest.