Based on a family of bi-univalent functions introduced through the Faber polynomial expansions and Noor integral operator

1.   Introduction

Conformal mapping is used in electromagnetic theory as well as in heat transfer theory. Univalent functions have wide application in heat transfer problems (see ^[16]). Special functions contain a very old branch of mathematics. In addition, in recent years, special functions and inequalities are widely used for solving some problems in physics, integer-order differential equations and systems, electromagnetizm, heat-transfer problems, mathematical models, etc ^[24]. Specially harmonic, analytical functions and inequalities of coefficients are widely used in thermodynamics, electricity and magnetism and quantum physics. In electricity, current and impedance equations can be expressed in a complex plane, and basic electrical relations become complex functions. However, in this study, we consider only upper coefficient bounds and some important inequalities for analytic and bi-univalent functions family by using special functions.

Let's denote by $\mathbb{C}$ which is the complex plane in the open unit disk $\mathfrak{D} = \left\{ z\, :\, \, z\in\mathbb{C}\, {\rm{and}}\, |z| < 1\right\}.$ Additionaly, $\mathcal{A}$ denotes the family of functions $s(z)$ which are analytic in the open unit disk and normalized by $s(0) = s^{\prime }(0)-1 = 0$ and having the style:

Let $\mathcal{S}$ be a subfamily of $\mathcal{A}$ which is univalent in $\mathfrak{D}$ (for details, see ^[11]). Furthermore, $\mathcal{P}$ be the family of functions, formed:

$\mathfrak{D}$ and hold the necessity $\Re(\varphi(z)) > 0$ in $\mathfrak{D}$. By the Carathéodory's Lemma (e.g., see ^[11]), we get $|\varphi_{k}|\leq2$.

In accordance with the Koebe Theorem (e.g., see ^[11]), each univalent function $s(z)\in \mathcal{A}$ has an inverse $s^{-1}$ fulfilling

and

Actually, the inverse function $s^{-1}$ is denoted by

If both $s(z)$ and $s^{-1}(z)$ are univalent, we can say that, $s \in \mathcal{A}$ is be bi-univalent in $\mathfrak{D}.$ All families of bi-univalent functions in $\mathfrak{D}$ with Taylor-Maclaurin series expansion (1.1) are presented by $\Sigma$.

For both of some knowledges and different examples for functions belong to $\Sigma$, see the following references ^{[9,17,18,22,23,25]}. Also, see references by Ali et al. ^[5], Jahangiri and Hamidi ^[15], and other studies such as ^{[6,7,10,13,21]}.

Definition 1. For analytic functions $s$ and $r \;, \; s$ is subordinate to $r$, presented by

if there is an analytic function $w$ such that

The following definition gives us the knowledge about fractional q-calculus operators (see, ^[22]).

Definition 2. ^[22] For $q\in (0, 1)$, the q-derivative of $s\in \mathcal{A}$ is given by

and

Thus we have

where $\left[ k, q\right]$ is presented by

and define the q-fractional by

Furthermore, Pochhammer symbol which is $q-$generalized for $\mathfrak{p}\geq 0$ is denoted by

In addition, as $q\rightarrow 1^{-}, \; [k, q]\rightarrow k,$ if we select $r(z) = z^{k},$ then we obtain

where $r^{\prime }$ is the ordinary derivative.

Recently, $F_{q, \mu +1}^{-1}(z)$, given with the following relation, was defined by Arif et al. (see^[8])

where

Due to the fact that series given in (1.10) is convergent absolutely in $z\in \mathfrak{D}$, by taking advantage of the characterization of $q$ -derivative via convolution, one can define the integral operator $\zeta _{q}^{\mu }:\mathfrak{D\rightarrow D}$ by

where

We note that

and

Equation (1.14) means that the operator denoted by (1.11) reduces to the known Noor integral operator by getting $q\rightarrow 1,$ which is presented in (see^[19,20]). For further informations on the $q$-analogue of differential-integral operators, see the study of Aldweby and Darus (see^[4]).

This work was motivated by Akgül and Sakar's study ^[3]. The basic purpose of this study is to give a new subfamily, which is in $\Sigma$ and provide general coefficient bound $|a_{n}|$ by using Faber polynomial technics for this subfamily. Additionaly, we derive bounds of the $|a_{2}|$ and $|a_{3}|$ which are the first two coefficients of this subfamily.

2.   The family $W_{\Sigma }^{\mu, q}(\alpha, \tau; \varphi)$

In this part, firstly we will introduce the class $W_{\Sigma }^{\mu, q}(\alpha, \tau; \varphi)$ and then give the knowledgements about Faber polynomial expansions.

Definition 3. A function $s \; \in \Sigma$ is known in the class $W_{\Sigma }^{\mu, q}(\alpha, \tau; \varphi)$ if the requirements given below hold:

and

where $(\mu > -1, \; 0 < q < 1$, $\tau > 0, \alpha \geq 0)$ and $s = r^{-1}(w)$ is given by (1.2).

It is clear from Definition $3$ that upon setting $q\rightarrow 1^{-}$, for $\tau = 1$, $\alpha = 1$ and $\mu = 1,$ one can easily see that $s \; \in \Sigma$ is in $\ $

if the conditions given below hold true:

and

where $r = s^{-1}$ is given by (1.2). The class $\mathcal{H} _{\sigma}(\varphi)$ was investigated by Ali et al. ^[5].

The Faber polynomials act effective role in several fields of mathematical sciences, specially, in the Theory of Geometric Function ^[12]. Also, Grunsky ^[14] gave some sufficient conditions for the univalency.

To obtain our main results, we need to following knowledgements owing to Airault and Bouali ^[1].

Using the Faber polynomial expansion of function $s\in \mathcal{A}$ given in (1.1), $s^{-1} = g$ may be given as

where

symbolically such term $(-k!)\equiv \Gamma(1-k): = (-k)(-k-1)(-k-2)\cdots(k\in \mathbb{N}_{0}, \, \mathbb{N}: = \{1, 2, 3, \cdots\})$ and $V_{j}$ with $7\leq j\leq k$ is a homologous polynomial in $a_{2}, a_{3}, \ldots a_{k}$, ^[2]. Particularly, some initial terms of $K_{k-1}^{-k}$ are

Generally, for any $p\in \mathbb{N}: = \{1, 2, 3\ldots \}$, an expansion of $K_{k}^{p}$ is given, ^[1],

where $D_{k}^{p} = D_{k}^{p}(a_{1}, a_{2}, a_{3}, \ldots, a_{k})$, and by ^[26],

while $a_{1} = 1$, and non-negative integers $i_{1}, \ldots, i_{k}$ satisfying

It is obvious that $D_{k}^{k}(a_{1}, a_{2}, ...a_{k}) = a_{1}^{k}$.

As a result, for $s\in W_{\Sigma }^{\mu, q}(\alpha, \tau; \varphi)$ given by (1.1), we can write

where

Theorem 4. For $\alpha \geq 1, \mu > -1, \; 0 < q < 1$, $\tau > 0$, let the function given by (1.1) $s\in W_{\Sigma }^{\mu, q}(\alpha, \tau; \varphi)$. If $a_{m} = 0$ for $2\leq m\leq k-1$, then

Proof. For analytic functions $s$ given by (1.1), we get

and

Moreover, the correlations (2.1) and (2.2) refer to the presence of Schwartz functions

so that

where

Thus, from (2.4), (2.6) and (2.9) we have

Similarly, by using (2.5), (2.6) and (2.10) we find that

For $a_{n} = 0 \; (2\leq n\leq k-1),$ we get

and so

When we take the absolute values of either of the above two equalities and using $\left\vert \varphi _{1}\right\vert \leq 2, \left\vert c_{k-1}\right\vert \leq 1$ and $\left\vert d_{k-1}\right\vert \leq 1,$ we obtain

which evidently completes the proof of theorem.

We have the Corollary 5, when we choose $\tau = 1$ in Theorem $4$.

Corollary 5. For $\alpha \geq 1, \, \, \mu > -1 \; 0 < q < 1$, let $s$ in the form (1.1) be in $W_{\Sigma }^{\mu, q}(\alpha; \varphi)$. If $a_{m} = 0$ for $2\leq m\leq k-1$, then

Comforting the coeefficient restricts produced in Theorem 4, we get coefficients given early of $s\in W_{\Sigma }^{\mu, q}(\alpha, \tau; \varphi)$ given below.

Theorem 6. Let $s\in W_{\Sigma }^{\mu, q}(\alpha, \tau; \varphi)$ and for $\alpha \geq 1, \, \, \mu > -1$, $0 < q < 1$, $\tau > 0$. Then

and

Proof. we obtain following equalities by replacing $k$ by 2 and 3 in (2.11) and (2.12), respectively,

From (2.13) and (2.15), we obtain,

and taking their absolute values,

Now, by adding (2.14) and (2.16), implies that

or equivalently, (by taking the square roots and using Caratheodary Lemma)

Next, in order to obtain the coefficient estimate of $|a_{3}|$, we subtract (2.16) from (2.14). Thus we get

or equivalently,

By replacing $|a_{2}^{2}|$ from (2.17) and (2.18) into (2.19), we get,

and

Finally, from (2.16), we deduce that (by Caratheodary Lemma)

So, the proof is over.

By letting $q\rightarrow1^{-}$ in Theorem $6$, we get the Corollary 7.

Corollary 7. Let $s$ presented by (1.1) be in the family $W_{\Sigma }^{\mu }(\alpha, \tau; \varphi)$ if $a_{m} = 0$ for $2\leq m\leq k-1$, then

and

By letting $\tau = 1$ in Corrollary $7$, we obtain Corollary 8.

Corollary 8. Let $s$ indicated by (1.1) be in the family $W_{\Sigma }^{\mu }(\alpha; \varphi)$ if $a_{m} = 0$ for $2\leq m\leq k-1$, then

and

Conflict of interest

The authors declare no conflict of interest.