A certain subclass of bi-univalent functions associated with Bell numbers and $q-$Srivastava Attiya operator

1.   Introduction and preliminaries

Let $\mathcal{A}$ be the class of all analytic functions of the form

in the open unit disk $\mathbb{D}$ $= \{z\in \mathbb{C} :|z| < 1\}$ normalized by the conditions $f(0) = 0$ and $f^{\prime }(0) = 1.$ The well-known Koebe one-quarter theorem ^[8] ensures that the image of $\mathbb{D}$ under every univalent function $f\in \mathcal{A}$ contains a disk of radius $1/4$. Thus, every univalent function $f$ has an inverse $f^{-1}$ satisfying $f^{-1}(f(z))$ $= z,$ $(z\in \mathbb{D})$ and

where

A function $f\in \mathcal{A}$ is said to be bi-univalent in $\mathbb{D}$ if both $f$ and $g$ to $\mathbb{D}$ are univalent in $\mathbb{D}$, where $g$ is the analytic continuation of $f^{-1}$ to the unit disk $\mathbb{ D}.$ Let $\Sigma$ denote the class of bi-univalent functions defined in the unit disk $\mathbb{D}$ given by 1.1. For a brief history of functions in the class $\Sigma$, see ^[3,4,16,19]. Later, Srivastava et al.'s ^{[24,26,27,28]} gave very important contributions to this theory. Recently, for coefficient estimates of the functions in some particular subclasses of bi-univalent functions, one may see ^{[6,7,10,15,20,25,29,30]}.

For analytic functions $f$ and $g$ in $\mathbb{D}$, $f$ is said to be subordinate to $g$ if there exists an analytic function $w$ such that $w\left(0\right) = 0,$ $\left\vert w\left(z\right) \right\vert < 1$ and $f\left(z\right) = g\left(w\left(z\right) \right)$. This subordination is denote by $f\left(z\right) \prec g\left(z\right)$. In particular, when $g$ is univalent in $\mathbb{D}$,

The $q-$difference operator, which was introduced by Jackson ^[13], is defined by

for $q\in \left(0, 1\right).$ It is clear that $\underset{q\rightarrow 1^{-} }{\lim }\partial _{q}f\left(z\right) = f^{\prime }\left(z\right)$ and $\partial _{q}f\left(0\right) = f^{\prime }\left(0\right),$ where $f^{\prime }$ is the ordinary derivative of the function. For more properties of $\partial _{q}$ see ^[9,11,12].

Thus, for function $f\in \mathcal{A}$ we have

where $\left[ k\right] _{q}$ is given by

and the $q-$ factorial is defined by

As $q\rightarrow 1^{-}$, then we get $\left[ k\right] _{q}\rightarrow k$. Thus, if we choose the function $g\left(z\right) = z^{k},$ while $q\rightarrow 1,$ then we have

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

In order to derive our main results, we have to recall here the following lemmas.

Lemma 1.1. ( ^[8]) If $p\in \mathcal{P}$ then $\left\vert p_{k}\right\vert \leq 2$ for each $k$, where $\mathcal{P}$ is the family of all functions $p$ analytic in $\mathbb{D}$ for which $\operatorname{Re} p(z) > 0,$

for $z\in \mathbb{D}.$

Lemma 1.2. ( ^[17]) If the function $p\in \mathcal{ P}$ is given by the series 1.8, then

for some $x, z$ with $\left\vert x\right\vert \leq 1$ and $\left\vert z\right\vert \leq 1.$

For a fixed non-negative integer $n,$ the Bell numbers $B_{n}$ count the possible disjoint partitions of a set with $n$ elements into non-empty subsets or, equivalently, the number of equivalence relations on it. The numbers $B_{n}$ are named the Bell numbers after Eric Temple Bell $\left(1883-1960\right)$ (see ^[1,2]) who called then the "exponential numbers". The Bell numbers $B_{n}$ $\left(n\geqq 0\right)$ are generated by the function $e^{e^{z}-1}$ as follows: $e^{e^{z}-1} = \sum\nolimits_{n = 0}^{\infty }B_{n}$ $\left(z^{n}/n!\right)$ $\left(z\in \mathbb{R} \right).$ The Bell numbers $B_{n}$ satisfy the following recurrence relation involving binomial coefficients: $B_{n+1} = \sum\nolimits_{k = 0}^{n} \binom{n}{k}B_{k}.$ Clearly, we have $B_{0} = B_{1} = 1,$ $B_{2} = 2,$ $B_{3} = 5,$ $B_{4} = 15,$ $B_{5} = 52$ and $B_{6} = 203.$ We now consider the function $\varphi \left(z\right) : = e^{e^{z}-1}$ with its domain of definition as the open unit disk $\mathbb{D}$. Recently Srivastava and co-auhors studied geometric properties and coefficients bounds for starlike functions related to the Bell numbers (see ^[5,14]).

On the other hand, Shah and Noor ^[21] introduced the $q-$analogue of the Hurwitz Lerch zeta function by the following series:

where $b\in \mathbb{C} \setminus \mathbb{Z} _{0}^{-},$ $s\in \mathbb{C}$ when $\left\vert z\right\vert < 1,$ and $\operatorname{Re}\left(s\right) > 1$ when $\left\vert z\right\vert = 1.$ The a normalized form of 1.9 as follows:

From 1.10 and 1.1, Shah and Noor ^[21] defined the $q-$Srivastava Attiya operator $J_{q, b}^{s}f\left(z\right) :\mathcal{A} \rightarrow \mathcal{A}$ by

where $\ast$ denotes convolution (or the Hadamard product).

We note that:

$\left(i\right)$ If $q\rightarrow 1^{-},$ then the function $\phi _{q}\left(s, b;z\right)$ reduces to the Hurwitz-Lerch zeta function and the operator $J_{q, b}^{s}$ coincides with the Srivastava-Attiya operator (see ^[22,23]).

$\left(ii\right)$ $J_{q, 0}^{1}f\left(z\right) = \int_{0}^{z}\frac{f\left(t\right) }{t}d_{q}t$ ($q-$Alexander operator).

$\left(iii\right)$ $J_{q, b}^{1}f\left(z\right) = \frac{\left[ 1+b\right] _{q}}{z^{b}}\int_{0}^{z}t^{b-1}f\left(t\right) d_{q}t$ ($q-$Bernardi operator ^[18]).

$\left(iv\right)$ $J_{q, 1}^{1}f\left(z\right) = \frac{\left[ 2\right] _{q} }{z}\int_{0}^{z}t^{b-1}f\left(t\right) d_{q}t$ ($q-$Libera operator ^[18]).

In present paper, we defined a general subclass $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right)$ of bi-univalent functions related to the Bell numbers by using $q-$Srivastava Attiya operator. Using the principles of subordination, the estimates for the coefficients $\left\vert a_{2}\right\vert$, $\left\vert a_{3}\right\vert$ and $\left\vert a_{3}-\delta a_{2}^{2}\right\vert$ of the functions of the form 1.1 in the class $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right)$ have been obtained. For some particular choices of $\tau,$ $\lambda,$ $\mu$ and $s$ the bounds determined.

2.   Coefficient estimates

Let $\Omega$ be the class of analytic functions of the form

in the unit disk $\mathbb{D}$ satisfying the condition $\left\vert w(z)\right\vert < 1.$ There is an important relation between the classes $\Omega$ and $\mathcal{P}$ as follows:

Define the functions $p$ and $s$ in $\mathcal{P}$ given by

and

It follows that

and

Definition 2.1. A function $f\in \Sigma$ is said to be in the class $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right)$ if the following conditions hold true for all $z, w\in \mathbb{D}$:

and

where $\varphi (z) = e^{e^{z-1}},$ $g(w) = f^{-1}(w),$ $\tau \in \mathbb{C} \backslash \left\{ 0\right\},$ $\mu > 0,$ $0 < q < 1$ and $\lambda \geq 0.$

Remark 2.1. We note that, for suitable choices parameters, the class $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right)$ reduces to the following classes.

1) Let $\lambda = 1$ in $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right).$ Then a function $f\in \Sigma$ is said to be in the class $\Sigma H_{q, b}^{s}\left(\tau, \mu \right)$ if the following subordinations hold for all $z, w\in \mathbb{D}$:

and

2) Let $\lambda = 1$ and $\tau = 1$ in $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right).$ Then a function $f\in \Sigma$ is said to be in the class $\Sigma H_{q, b}^{s}\left(\mu \right)$ if the following subordinations hold for all $z, w\in \mathbb{D}$:

and

3) Let $\mu = 1$ in $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right).$ Then a function $f\in \Sigma$ is said to be in the class $\Sigma H_{q, b}^{s}\left(\tau, \lambda \right)$ if the following subordinations hold for all $z, w\in \mathbb{D}$:

and

4) Let $\mu = 1$ and $\tau = 1$ in $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right).$ Then a function $f\in \Sigma$ is said to be in the class $\Sigma H_{q, b}^{s}\left(\lambda \right)$ if the following subordinations hold for all $z, w\in \mathbb{D}$:

and

5) Let $\mu = 1$, $\tau = 1$ and $\lambda = 0$ in $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right).$ Then a function $f\in \Sigma$ is said to be in the class $\Sigma H_{q, b}^{s}$ if the following subordinations hold for all $z, w\in \mathbb{D}$:

and

6) Let $s = 0$ in $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right).$ Then a function $f\in \Sigma$ is said to be in the class $\Sigma H\left(\tau, \lambda, \mu \right)$ if the following subordinations hold for all $z, w\in \mathbb{D}$:

and

The following theorem derives the estimates for the coefficients $\left\vert a_{2}\right\vert$ and $\left\vert a_{3}\right\vert$ for the functions given by 1.1 that belong to the class $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right)$.

Theorem 2.1. Let $\mathit{f}$ given by (1.1) be in the class $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right)$. Then

and

Proof. Let $\mathit{f\in }\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right)$ and $g = f^{-1}.$ Then, there are analytic functions $u, v\in \Omega$ satisfying

and

In other words, by using 2.1 in 2.6 and 2.7 we write

and

From 2.8 and 2.9, we have

and

Comparing the coefficients on the both sides of above last equalities, we have the relations

and

Therefore, from the Eqs 2.10 and 2.12, we find that

and

which upon applying Lemma 1.1, yields

On the other hand, by using 2.11 and 2.13, we obtain

which yields

We now, investigate the upper bound of $\left\vert a_{3}\right\vert.$ For this, by using 2.11 and 2.13, we have

Therefore for substituting 2.15 in 2.17, we have

or

On the other hand, according to the Lemma 1.2 and 2.14, we write

and so, from 2.19 and 2.20, we have

If we apply triangle inequality to equation 2.21, we obtain

Since the function $p(e^{i\theta }z)$ $(\theta \in \mathbb{R})$ is in the class $\mathcal{P}$ for any $p\in \mathcal{P}$, there is no loss of generality in assuming $p_{1} > 0$. Write $p_{1} = \mathfrak{p},$ $\mathfrak{p}\in \lbrack 0, 2].$ Thus, for $\left\vert x\right\vert \leq 1$ and $\left\vert y\right\vert \leq 1$ we obtain

which upon applying Lemma 1.1, yields upper bound of $\left\vert a_{3}\right\vert.$

Theorem 2.2. If $f(z)$ given by (1.1) be in the class $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right)$ and $\delta \in \mathbb{C},$ then

where

Proof. From the Eqs 2.16 and 2.18 we obtain

and

Therefore, by using the equalities 2.23 and 2.24 for $\delta \in \mathbb{C},$ we have

After the necessary arrangements, we rewrite the above last equality as

where $K$ and $L$ are given by 2.22. Taking the absolute value of 2.25, from Lemma 1.1 we obtain the desired inequality.

Theorem 2.3. If $f(z)$ given by (1.1) be in the class $\Sigma H_{q, b}^{s}\left(\tau, \lambda, \mu \right)$ and $\delta \in \mathbb{C},$ then

where

Proof. From Eq 2.17, we write

By substituting 2.16 in 2.26, we have

where

Therefore, we conclude that

which evidently complete the proof of the theorem.

Corollary 2.1. Let $\mathit{f}$ given by (1.1) be in the class $\Sigma H_{q, b}^{s}\left(\tau, \mu \right)$. Then

and

where

Corollary 2.2. Let $\mathit{f}$ given by (1.1) be in the class $\Sigma H_{q, b}^{s}\left(\tau, \lambda \right)$. Then

and

where

Corollary 2.3. Let $\mathit{f}$ given by (1.1) be in the class $\Sigma H\left(\tau, \lambda, \mu \right)$. Then

and

where

3.   Conclusions

In this paper, we defined a general subclass of bi-univalent functions related with $q-$Srivastava Attiya operator by using the Bell numbers and subordination. For the functions belonging to this class, we obtained non-sharp bounds for the initial coefficients and the Fekete-Szegö functional. Some interesting corollaries and applications of the results are also discussed.

Acknowledgement

The research was supported by the Commission for the Scientific Research Projects of Kyrgyz-Turkish Manas University, project number KTMU-BAP-2020.FB.04.