Geometric properties of $q$-spiral-like with respect to $(\ell, \jmath)$-symmetric points

1.   Introduction

Let $\mathcal{H}(\Sigma)$ denote the space of all analytic functions in the open unit disk ${\Sigma} = \left\{k\in\mathbb{C}:|k| < 1\right\}$ and let $\mathcal{H}$ denote the class of functions $\hbar\in\mathcal{H}(\Sigma)$ which has the form

and suppose $\mathcal{\widetilde{S}}$ denote the subclass of $\mathcal{H}$ which are univalent in $\Sigma$. Then the convolution or Hadamard product of $\hbar$ of the form (1.1) and $g$ of the form $g(k) = k+\sum_{v = 2}^{\infty}b_{v}k^v$, is defined as:

In order to define new classes of $q$-spiral-like with respect to $(\ell, \jmath)$-symmetric points defined in $\Sigma$, we first recall the concept of quantum calculus (or $q$-calculus), the goal of quantum calculus, often known as $q$-calculus, is to find $q$-analogues without using limits, owing to the fact that it is widely used in a variety of scientific fields, in particular $q$-calculus has a great interest because of its applications in geometric function theory, so this method becomes a crucial component of the subject of our investigation. Jackson was the first participant who introduced fundamental ideas and developed the $q$-calculus theory ^[1,2,3]. In recent years, using quantum calculus approach to studying geometric properties several subclasses of analytic functions by many authors. For example, Naeem et al. ^[4] investigated subclesse of $q$-convex functions. Srivastava et al. ^[5] studied subclasses of $q$-starlike functions. Alsarari et al. ^[6] investigated the convolution conditions of $q$-Janowski symmetrical functions classes. Ovindaraj and Sivasubramanian in ^[7] found subclasses connected with $q$-conic domain. Khan et al. ^[8] used the symmetric $q$-derivative operator. Srivastava ^[9] published survey-cum-expository review paper which is useful for researchers. Many scholars introduced specific classes using $q$-calculus, which helped to advance the theory. See for further information on these contributions ^[10,11,12].

We provide some basic definitions and concept details of $q$-calculus which are used in our work and we will assume that $q$ satisfies the condition $0 < q < 1$ throughout our work. Jackson ^[1] introduced $q$-derivative ${\bf \partial}_q\hbar(k)$ as

Equivalently (1.2), may be written as

where

For $\hbar$ a function defined in a subset of $\mathbb{C}$, provided $\hbar'(0)$ exists, then (1.2) yields

By using (1.2) it can easily be seen that for $n$ and $m$ any real (or complex) constants

As a right inverse Jackson^[2] presented the $q$-integral of a function $\hbar$ as:

provided that $\sum_{v = 0}^{\infty}q^v\hbar(kq^v)$ is converges.

Recent work in the family of analytical functions has shown the use of the idea of $(\ell, \jmath)$-symmetrical functions to take a more general approach. This definition applies concepts of odd, even, and planar Sakaguchi's functions to the $\jmath$-dimensional case. Liczberski and Polubinski ^[13] constructed the concept of $(\ell, \jmath)$-symmetrical functions for the positive integer $\jmath$ and $(\ell = 0, 1, 2, \dots, \jmath-1)$. A non-empty subset $\mathcal{{\bf Q}}$ of the complex plane $\mathbb{C}$ will be called $\jmath$- fold symmetric domain if $\varepsilon \mathcal{{\bf Q}} = \mathcal{{\bf Q}},$ where $\varepsilon = e^{\frac{2\pi i}{\jmath}}$. function $\hbar:\mathcal{{\bf Q}}\rightarrow \mathbb{C}$ is called $(\ell, \jmath)$-symmetrical if for each $k\in \mathcal{{\bf Q}}$, $\hbar(\varepsilon k) = \varepsilon^\ell \hbar(k).$

Theorem 1.1. ^[13] For the $\jmath$-fold symmetric set ${\Sigma}$, then for every function $\hbar:{\Sigma}\mapsto\; \mathbb{C}$, can be written in the form,

Remark 1.1. Equivalently, (1.4) may be written as:

for $I\in \mathbb{N}$.

Recently, many authors have conducted some studies about the concept of $(\ell, \jmath)$-symmetrical functions obtained interesting results for various classes see ^{[14,15,16,17]}.

The function $\hbar$ is called $\lambda$-spirallike if $\Re\left\{e^{i\lambda} \frac{k\hbar'(k)}{\hbar(k)}\right\} > 0, \; \lambda$ is real and $|\lambda| < \frac{\pi}{2}.$ Furthermore, let $\mathcal{P}$ the Carath$\check{e}$odory class of functions form $p(k) = 1+\sum\limits_{v = 1}^\infty c_v k^v$ defined on $\Sigma$ and satisfying $p(0) = 1$, $\operatorname{Re}\{p(k)\} > 0$, $k\in \Sigma$ and $p\in \mathcal{P}\Leftrightarrow p(k) = \dfrac{1+s(k)}{1-s(k)}$, where $s\in\Delta$ denote for the family of Schwarz functions, that is

We amalgamate the notion of $(\ell, \jmath)$-symmetrical functions and $q$-derivative to originate new classes of $q$-spirallike functions with respect to $(\ell, \jmath)$-symmetric points $\mathcal{\widetilde{S}}^{\ell, \jmath}_q(\lambda)$.

Definition 1.1. For arbitrary fixed numbers $\lambda$ and $q$, $\; |\lambda| < \frac{\pi}{2}, \; 0 < q < 1,$ let $\mathcal{\widetilde{S}}^{\ell, \jmath}_q(\lambda)$ denote the family of functions $\hbar\in \mathcal{H}$ which satisfies

where $\hbar_{\ell, \jmath}$ is defined in (1.4).

For special cases for the parameters $q, \lambda, \ell$ and $\jmath$ the class $\mathcal{\widetilde{S}}^{\ell, \jmath}_q(\lambda)$ yield several known subclasses of $\mathcal{H}$, namely $\mathcal{{\widetilde{S}}}^{1, \jmath}_1(0): = \mathcal{{\widetilde{S}}}_{\jmath}$ by defined by Sakaguchi ^[18], $\mathcal{\widetilde{S}}^{1, 1}_q(1)$ = $\mathcal{\widetilde{S}}_q$ which was first introduced by Ismail et al. ^[3], etc.

We denote by $\mathcal{{T}}^{\ell, \jmath}_q(\lambda)$ consisting all functions $\hbar$, satisfying

The following neighborhood principle was first proposed by Goodman ^[19] and generalized by Ruscheweyh ^[20].

Definition 1.2. ^[19,20] For $\rho\geq0$ and any $\hbar\in \mathcal{H}$, $\rho$-neighborhood of function $\hbar$ defined as:

For the identity function $e(k) = k$, defined as:

For all $\eta\in \mathbb{C}$, with $|\eta| < \rho$, Ruscheweyh ^[20] proved

Lemma 1.1. ^[19] Let $P(k) = 1+\sum_{v = 1}^{\infty}p_{v}k^v, \; \; \; \; (k\in \Sigma)$, with the condition $\Re\{p(k)\} > 0,$ then

The goal of this research to investigate a convolution conditions and coefficient estimates for a function $\hbar$ to be in the classes $\mathcal{\widetilde{S}}^{\ell, \jmath}_q(\lambda)$ and $\hbar\in \mathcal{T}^{\ell, \jmath}_q(\lambda)$, which will be used as a assisting result for to discuss a sufficient prerequisites and associated neighborhood results.

2.   Main results

Theorem 2.1. A function $\hbar\in \mathcal{T}^{\ell, \jmath}_q(\lambda)$ if and only if

where, $0\leq\phi < 2\pi, 0 < q < 1$ and $\alpha_\ell$ are defined by (2.3).

Proof. We have, $\hbar\in \mathcal{T}^{\ell, \jmath}_q(\lambda)$ if and only if

which implies

Setting $\hbar(k) = k+\sum_{v = 2}^{\infty}a_{v}k^v$, we have

where

The left hand side of (2.1) is equivalent to

simplifying (2.4), we get

since $k\partial_q\hbar*g = \hbar*k\partial_qg$, then the above equation can be written as:

Remark 2.1. As $q\rightarrow 1^-$ and particular values of $\ell, \jmath$ and $\lambda$ Theorem 2.1 yields to the results found in ^[21,22].

Theorem 2.2. A function $\hbar\in \mathcal{\widetilde{S}}^{\ell, \jmath}_q(\lambda)$ if and only if

where $0 < q < 1,$ $0\leq\phi < 2\pi$ and $\alpha_\ell$ are defined by (2.3).

Proof. Since $\hbar\in \mathcal{\widetilde{S}}^{\ell, \jmath}_q(\lambda)$ if and only if $g(k) = \int_{0}^{k}\frac{\hbar(\zeta)}{\zeta}d_q\zeta \in \mathcal{T}^{\ell, \jmath}_q(\lambda)$, we have

Thus the result follows from Theorem 2.2.

Remark 2.2. Note that from Theorem 2.2, we can easily get

where $g(k)$ has the form

Theorem 2.3. Let $\hbar(k)\in \mathcal{H}$, for $|\lambda| < \frac{\pi}{2}$ and $0 < q < 1$, if

then $\hbar(k)\in\mathcal{{\widetilde{S}}}_q^{\ell, \jmath}(\lambda)$.

Proof. Theorem 2.3 can be proved by demonstrating Remark 2.2 by showing $\frac{(\hbar*g)(k)}{k}\neq 0$. For $\hbar$ and $g$ given by (1.1) and (2.7) respectfully.

It is known from Remark 2.2 that $\hbar(k)\in\mathcal{{\widetilde{S}}}_q^{\ell, \jmath}(\lambda)\Leftrightarrow\frac{(\hbar*g)(k)}{k}\neq0$. Using (2.7) and (2.8), we get

Thus, $\hbar(k)\in\mathcal{{\widetilde{S}}}_q^{\ell, \jmath}(\lambda)$.

Theorem 2.4. If $\hbar(k)\in\mathcal{{\widetilde{S}}}_q^{\ell, \jmath}(\lambda)$, then

where $\delta_{m, \jmath}$ is given by (1.5).

Proof. Let $\hbar(k)\in\mathcal{{\widetilde{S}}}_q^{\ell, \jmath}(\lambda)$ from Definition 1.1, we have

where $p(k)$ is Carath$\acute{e}$odory function.

Since

we have

where $\delta_{v, \jmath}$ is given by (1.5), $\delta_{1, \jmath} = 1$.

By equating coefficients of $k^v$ in (2.10) both sides we have

By Lemma1.1 and using the fact that $|e^{i\lambda}| = 1$, we get

It now suffices the prove that

For this, we use the induction method. (2.12) is true for $v = 2$ and $3$.

Let us suppose (2.12)holds for all $v\leq m$.

From (2.11), we have

From (2.9), we have

By the induction hypothesis, we have

Multiplying both sides by

we have

Hence

The inequality (2.12) holds for $v = m+1$, thus proving the result.

Theorem 2.5. If $\hbar\in \mathcal{T}^{\ell, \jmath}_q(\lambda)$ then

where $\delta_{r, \jmath}$ is given by (1.5).

The proof follows by using Theorem 2.4 and (1.8).

3.   $(\rho, q)$-neighborhoods for functions in the classes $\mathcal{{\widetilde{S}}}_q^{\ell, \jmath}(\lambda)$ and $\mathcal{T}^{\ell, \jmath}_q(\lambda)$

In the order to find some neighborhood results, we assume that $v = [v]_q$ in Definition 1.2 to get definition of neighborhood with $q$-derivative $\mathcal{N}^\lambda_{q, \rho}(h)$ and $\mathcal{N}^\lambda_{q, \rho}(e)$, where $[v]_q$ is given by Equation (1.3). In particular. For $v = \frac{([v]_q-\delta_{v, \ell})+\left|[v]_q+\delta_{v, \ell}e^{-2i\lambda}\right|} {|1+e^{-2i\lambda}|}$ in Definition 1.2 to get definition of neighborhood for the classes $\mathcal{{\widetilde{S}}}_q^{\ell, \jmath}(\lambda)$ and $\mathcal{T}^{\ell, \jmath}_q(\lambda)$ which is $\mathcal{N}_{q, \rho}^{\ell, \jmath}(\lambda; \hbar)$.

Theorem 3.1. Let $\hbar\in \mathcal{N}_{q, 1}(e)$, and defined by the form (1.1), then

where $\hbar_{\ell, \jmath}$ is defined by (1.4).

Proof. Let $\hbar\in \mathcal{H}$, and $\partial_q\hbar(k) = k+\sum_{v = 2}^\infty [v]_qa_vk^v, \hbar_{\ell, \jmath}(k) = k+\sum_{v = 2}^\infty \delta_{v, \ell}a_vk^v,$ where $\delta_{v, \ell}$ is given by (1.5). Consider

This gives us the required result.

Theorem 3.2. Let $\hbar\in \mathcal{H}$, and for all complex number $\eta$, with $|\eta| < \rho$, if

Then

Proof. Let $f\in \mathcal{N}_{q, \frac{\rho|1+e^{-2i\lambda}\;\;|}{4}}^{\ell, \jmath}\;\;\;(\lambda; \hbar)$ and defined by $f(k) = k+\sum_{v = 2}^\infty b_vk^v$. It is sufficient to prove that $f\in\mathcal{\widetilde{S}}_q^{\ell, \jmath}(\lambda)$ to to prove the Theorem 3.2. We would prove this claim in next three steps.

We first note that Theorem 2.2 is equivalent to

where

and $0\leq\phi < 2\pi$. We can write $t_\phi(k) = k-\sum_{v = 2}^\infty t_vk^v$,

where

so that $|t_v|\leq \frac{4[v]_q}{|1+e^{-2i\lambda}\;\;|}.$ Secondly we obtain that (3.2) is equivalent to

because, if $\hbar(k) = k+\sum_{v = 2}^\infty a_vk^v \in \mathcal{H}$ and satisfy (3.2), then (3.3) is equivalent to

Thirdly letting $f(k) = k+\sum_{v = 2}^\infty b_vk^v$ we notice that

this prove that

In view of our observations (3.3), it follows that $f\in \mathcal{\widetilde{S}}_q^{\ell, \jmath}(\lambda)$. The theorem's proof is now complete.

Theorem 3.3. Let $\hbar \in \mathcal{\widetilde{S}}_q^{\ell, \jmath}(\lambda)$, for $\rho_1 < c$. Then

where $c\neq0$ with $\left|\frac{(\hbar*t_\phi)(k)}{k}\right|\geq c$, $\rho_1 = \frac{\rho|1+e^{-2i\lambda}|}{4}$ and $t_\phi$ is defined by (3.4).

Proof. Let the function $m = k+\sum_{v = 2}^\infty b_vk^v\in \mathcal{N}_{q, \rho_1}^{\ell, \jmath} (\lambda; \hbar)$. It is enough to demonstrate that $\frac{(m*t_\phi)(k)}{k}\neq 0$ for Theorem 3.3's proof, where $t_\phi$ is given by (3.4). Consider

Since $\hbar \in \mathcal{\widetilde{S}}_q^{\ell, \jmath}(\lambda)$, therefore applying Theorem 2.2, we obtain

where $c$ is a real value that is not zero and $k\in \Sigma$. Now

using (3.8) and (3.9) in (3.7), we obtain

where $\rho_1 < c$. This completes the proof.

4.   Conclusions

The centered $(\ell, \jmath)$-symmetrical functions in geometric function theory were the subject of this work. As a new topic emerged to take a more general approach and there are numerous uses for $(\ell, \jmath)$-symmetrical functions, including investigating fixed points, estimating the absolute values of particular integrals, and deriving conclusions of the Cartan's uniqueness theorem variety and motivated by the recent applications of the $q$-calculus, we have applied the two concepts for classes of $\lambda$-spirallike functions to introduce and study the classes $\mathcal{{\widetilde{S}}}_q^{\ell, \jmath}(\lambda)$ and $\mathcal{T}^{\ell, \jmath}_q(\lambda).$ We investigate a convolution conditions and coefficient estimates. Furthermore, these results are used to find sufficient condition, coefficient estimates investigate related neighborhood results. The idea used in this paper can easily be implemented to define several classes with different image domains. The opportunities for research using symmetric $q$-calculus, Janowski class or the basic $q$-hypergeometric functions in several diverse areas.

Acknowledgments

This research received funding from Taif University, Researchers Supporting and Project number (TURSP-2020/207), Taif University, Taif, Saudi Arabia.

Conflict of interest

The authors declare no conflict of interest.