Geometric Properties of Certain Classes of Analytic Functions with Respect to (x,y)-Symmetric Points

Abstract

In this article, the present study employs the utilization of the concepts pertaining to $(x,y)$-symmetrical functions, Janowski type functions, and q-calculus in order to establish a novel subclass within the open unit disk. Specifically, we delve into the examination of convolution properties, which serve as a tool for investigating and inferring adequate and equivalent conditions. Moreover, we also explore specific characteristics of the class ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$, thereby further scrutinizing the convolution properties of these newly defined classes.

1. Introduction

The theory of functions exhibiting $(x,y)$-symmetry has a wide range of intriguing applications. For instance, these functions are useful in exploring the set of fixed points of mappings, estimating the absolute value of certain integrals, and deriving results akin to Cartan’s uniqueness theorem for holomorphic mappings, as demonstrated in [1]. The intrinsic properties of $(x,y)$-symmetrical functions are of great interest in the field of Geometric Function Theory. In this work, we present fundamental definitions and concepts related to $(x,y)$-symmetrical functions. We assume that y is a fixed integer and set $\epsilon =e^{\frac{2\pi i}{y}}$. Furthermore, we definedomain $\mathbf{G}\subset \mathbb{C}$ to be a y-fold symmetric domain if $\mathbf{G}\subset \mathbb{C}$. Function h is referred to as a y-symmetrical function if, for every $\zeta \in \mathbf{G}$, $h\left(\epsilon \zeta \right)=\epsilon h\left(\zeta \right).$

In 1995, Liczberski and Polubinski [1] introduced the notion of $(x,y)$-symmetrical functions for ($y=2,3,\dots$) and $(x=0,1,2,\dots ,y-1)$. Consider a y-fold symmetric domain $\mathbf{G}$ and let x be any integer. Function $h:\mathbf{G}\to \mathbb{C}$ is deemed $(x,y)$-symmetrical if for every $\zeta \in \mathbf{G}$, $h\left(\epsilon \zeta \right)={\epsilon }^{x}h\left(\zeta \right).$ The collection of all $(x,y)$-symmetrical functions is denoted by ${\mathcal{F}}_y^x$. Furthermore, it is worth noting that ${\mathcal{F}}_2^0$, ${\mathcal{F}}_2^1$ and ${\mathcal{F}}_y^1$ represent families of even, odd, and $y$-symmetrical functions, respectively.

Theorem 1 

([1], Page 16). For every mapping $h:k\mapsto \mathbb{C}$, and a y-fold symmetric set k, then

$$h\left(\zeta \right)=\sum _{x=0}^{y-1}{h}_{x,y}\left(\zeta \right),h_{x,y}\left(\zeta \right)=y^{-1}\sum _{r=0}^{y-1}{\epsilon }^{-rx}h\left({\epsilon }^r\zeta \right),\zeta \in k.$$

(1)

Remark 1. 

Alternatively, as expressed in (1) it, can be written as

$$h_{x,y}\left(\zeta \right)=\sum _{v=1}^{\infty }{\delta }_{v,x}{a}_v{\zeta }^v,a_1=1,$$

(2)

where

$${\delta }_{v,x}=\frac{1}{y}\sum _{r=0}^{y-1}{\epsilon }^{(v-x)r}=\left\{\begin{array}{cc}1,\hfill & v=x+ly;\hfill \\ 0,\hfill & v\ne x+ly;\hfill \end{array}\right.$$

(3)

where $l\in \mathbb{N}$ and $y=1,2,\dots ,x=0,1,2,\dots ,y-1.$

Let $\mathcal{F}\left(k\right)$ be the collection of all functions that possess analyticity within the open unit disk $k=\left\{\zeta \in \mathbb{C}:\left|\zeta \right|<1\right\}$ and let $\mathcal{F}$ denote a particular subfamily, which is characterized by class $h\in \mathcal{F}\left(k\right)$, and takes the form of

$$h\left(\zeta \right)=\zeta +\sum _{v=2}^{\infty }{a}_v{\zeta }^v,$$

(4)

consider $\tilde{\mathcal{S}}$ as the set of all functions belonging to $\mathcal{F}$ that are univalent within k. The convolution or Hadamard product of two analytic functions h and g, both belonging to the $\mathcal{F}$ set, can be expressed as

$$(h\ast g)\left(\zeta \right)=\zeta +\sum _{v=2}^{\infty }{a}_v{b}_v{\zeta }^v,$$

where h is defined by Equation (4) and $g\left(\zeta \right)=\zeta +{\sum }_{v=2}^{\infty }{b}_v{\zeta }^v.$ Those interested in the Hadamard product and its generalization are encouraged to refer to sources [2,3]. To introduce new classes of q-Janowski symmetrical functions defined in k, it is necessary to first review the relevant concepts and notations pertaining to Janowski-type functions, $(x,y)$-symmetrical functions, and quantum calculus, also known as q-calculus. In his work [4], Janowski introduced the class $\mathcal{P}[\alpha ,\beta ]$ which pertains to a given $h\in \mathcal{F}\mathrm{and}h\left(0\right)=1$ It is said that h belongs to $\mathcal{P}[\alpha ,\beta ]$ if and only if $p\left(\zeta \right)={\displaystyle \frac{1+\alpha s\left(\zeta \right)}{1+\beta s\left(\zeta \right)}}$, where $-1\le \beta <\alpha \le 1\mathrm{and}s\left(\zeta \right)\in \Delta$. Here, $\Delta$ represents the family of Schwarz functions, that is,

$$\Delta :=\{s\in \mathcal{F},s(0)=0,|s\left(\zeta \right)|<1,\zeta \in k\}.$$

(5)

In [5], Jackson presented and examined the notion of the q-derivative operator ${\partial }_{q}h\left(\zeta \right)$ where q satisfies the condition $0<q<1$. The operator is defined as follows

$${\partial }_{q}h\left(\zeta \right)=\left\{\begin{array}{c}\frac{h\left(\zeta \right)-h\left(q\zeta \right)}{\zeta (1-q)},\zeta \ne 0,\hfill \\ h^{\prime }\left(0\right),\zeta =0.\hfill \end{array}\right.$$

(6)

Alternatively, as expressed in (6), it can be written as

$${\partial }_{q}h\left(\zeta \right)=1+\sum _{v=2}^{\infty }{\left[v\right]}_q{a}_v{\zeta }^{v-1}\zeta \ne 0,$$

where

$${\left[v\right]}_q=\frac{1-q^v}{1-q}=1+q+q^2+\dots +q^{v-1}.$$

(7)

Note that as $q\to 1^{-}$, ${\left[v\right]}_q\to v$. For a function $h\left(\zeta \right)={\zeta }^v,$ we can note that

$${\partial }_{q}h\left(\zeta \right)={\partial }_q\left({\zeta }^v\right)=\frac{1-q^v}{1-q}{\zeta }^{v-1}={\left[v\right]}_q{\zeta }^{v-1}.$$

Then,

$$\underset{q\to 1^{-}}{lim}{\partial }_{q}h\left(\zeta \right)=\underset{q\to 1^{-}}{lim}{\left[v\right]}_q{\zeta }^{v-1}=v{\zeta }^{v-1}=h^{\prime }\left(\zeta \right),$$

where $h^{\prime }\left(\zeta \right)$ is the ordinary derivative.

The q-integral of a function h, which was presented by Jackson [6], is regarded as a right inverse given the condition that

$${\int }_0^{\zeta }h\left(z\right)d_{q}z=\zeta (1-q)\sum _{v=0}^{\infty }{q}^{v}h\left(\zeta q^v\right),$$

provided that the series ${\sum }_{v=0}^{\infty }{q}^{v}h\left(\zeta q^v\right)$ converges. Ismail et al. [7] introduced the initial link between quantum calculus and geometric function theory. In recent years, numerous authors have utilized the quantum calculus approach to investigate the geometric properties of several subclasses of analytic functions. For instance, Naeem et al. [8] delved into the subclass of q-convex functions, while Srivastava et al. [9] examined subclasses of q-starlike functions. Additionally, Alsarari and Alzahrani [10] explored the convolution conditions of q-Janowski symmetrical function classes. Moreover, Ovindaraj and Sivasubramanian in [11] discovered subclasses related to q-conic domains, and several recent studies can be found in [12,13,14]. Khan et al. [15] employed the symmetric q-derivative operator, while Srivastava [16] published a survey-cum-expository review paper that is highly beneficial to researchers.

Proposition 1. 

For n and m any real (or complex) constants and $\zeta \in k,$ we have

1 

${\partial }_q(nh\left(\zeta \right)\pm mg\left(\zeta \right))=n{\partial }_{q}h\left(\zeta \right)\pm m{\partial }_{q}g\left(\zeta \right)$,

2 

${\partial }_q\left(h\left(\zeta \right)g\left(\zeta \right)\right)=h\left(q\zeta \right){\partial }_{q}g\left(\zeta \right)+{\partial }_{q}h\left(\zeta \right)g\left(\zeta \right)=h\left(\zeta \right){\partial }_{q}g\left(\zeta \right)+{\partial }_{q}h\left(\zeta \right)g\left(q\zeta \right),$

3 

${\partial }_q\left(\frac{h\left(\zeta \right)}{g\left(\zeta \right)}\right)=\frac{g\left(\zeta \right){\partial }_{q}h\left(\zeta \right)-h\left(\zeta \right){\partial }_{q}g\left(\zeta \right)}{g\left(q\zeta \right)g\left(\zeta \right)}.$

The function h is denoted as $\lambda$-spiral-like if the real part of $\left\{e^{i\lambda }\frac{\zeta h^{\prime }\left(\zeta \right)}{h\left(\zeta \right)}\right\}$ is greater than zero. Here, $\lambda$ represents a real number such that its absolute value is less than $\frac{\pi }{2}.$ In recent research conducted by Al sarari et al. [17,18], numerous intriguing findings were obtained for various classes by utilizing the concept of $(x,y)$-symmetrical functions and the q-derivative. Consequently, we combine the notion of $(x,y)$-symmetrical functions, the q-derivative, and Janowski-type functions to establish the ensuing classes.

Definition 1. 

For arbitrary fixed numbers $q,\alpha ,\beta$ and λ, $0<q<1,\left|\lambda \right|<\frac{\pi }{2},$ $-1\le \beta <\alpha \le 1,$ let ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ denote the family of functions $h\in \mathcal{F}$ which satisfies

$$\Re \left\{e^{i\lambda }\frac{\zeta {\partial }_{q}h\left(\zeta \right)}{h_{x,y}\left(\zeta \right)}\right\}\in \mathcal{P}[\alpha ,\beta ],\mathrm{for}\mathrm{all}\zeta \in k,$$

(8)

where $h_{x,y}$ is defined in (1).

The aforementioned classes can be reduced to well-established classes that were originally defined by Janowski [4]. The class ${\tilde{\mathcal{S}}}_q^{1,1}(1-2\kappa ,-1,0)$ =${\tilde{\mathcal{S}}}_q\left(\kappa \right)$, which was introduced and extensively studied by Agrawal and Sahoo [19], can be associated with the class Se1,${\tilde{\mathcal{S}}}_q^{1,1}(1,-1,0)$ = ${\tilde{\mathcal{S}}}_q$, first introduced by Ismail et al. [7]. The class ${\tilde{\mathcal{S}}}_1^{1,1}(1-2\kappa ,-1,0)$ = $\tilde{\mathcal{S}}\left(\kappa \right)$ represents the widely recognized class of starlike functions of order ${\tilde{\mathcal{S}}}_1^{1,1}(1-2\kappa ,-1,0)$ = $\tilde{\mathcal{S}}\left(\kappa \right)$, as introduced by Robertson [20]. Lastly, the class ${\mathcal{S}}_1^{1,1}(1,-1,0)={\mathcal{S}}^{\ast }$ can be attributed to the work of Nevanlinna [21]. ${\tilde{\mathcal{S}}}_1^{1,y}(1,-1,0):={\tilde{\mathcal{S}}}_y$ We denote by ${\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ the subclass of $\mathcal{F}$ consisting of all functions h such that

$$w{\partial }_{q}h\left(\zeta \right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda ).$$

(9)

We must call to mind the subsequent concept of a neighborhood

Definition 2. 

For any $h\in \mathcal{F}$, the ρ-neighborhood of function h can be defined as:

$${\mathcal{N}}_{\mu ,\rho }\left(h\right)=\left\{g\in \mathcal{F}:g\left(\zeta \right)=\zeta +\sum _{v=2}^{\infty }{b}_v{\zeta }^v,\sum _{v=2}^{\infty }{\mu }_v|a_v-b_v|\le \rho ,(\rho \ge 0)\right\}.$$

(10)

For $e\left(\zeta \right)=\zeta$, we can see that

$${\mathcal{N}}_{\mu ,\rho }\left(e\right)=\left\{g\in \mathcal{F}:g\left(\zeta \right)=\zeta +\sum _{v=2}^{\infty }{b}_v{\zeta }^v,\sum _{v=2}^{\infty }{\mu }_v\left|b_v\right|\le \rho ,(\rho \ge 0)\right\}.$$

(11)

Remark 2. 

• If we substitute ${\mu }_v=v$ in Definition 2 we obtain the concept initially introduced by Goodman [22] and later generalized by Ruscheweyh [23].
• If we substitute ${\mu }_v={\left[v\right]}_q$ in Definition 2, we arrive at the definition of a neighborhood with q-derivative ${\mathcal{N}}_{q,\rho }^{\mu }\left(h\right),{\mathcal{N}}_{q,\rho }^{\mu }\left(e\right)$, where ${\left[v\right]}_q$ is defined by Equation (7).

Ruscheweyh [23] demonstrated, among other findings, that for all $\eta \in \mathbb{C}$, with $\left|\eta \right|<\rho$,

$$\frac{h\left(\zeta \right)+\eta \zeta }{1+\eta }\in {\tilde{\mathcal{S}}}^{\ast }\Rightarrow {\mathcal{N}}_{\rho }\left(h\right)\subset {\tilde{\mathcal{S}}}^{\ast }.$$

2. Main Results

In this section, we investigate the convolution conditions for the class of convex functions in Theorem 2, which will be used to get the convolution conditions for the class of star-like functions in Theorem 3, which is the equivalent of Remark 4. Theorem 5, with its application of Corollary 2, will be used as a supporting result to deduce the sufficient condition of Theorem 4 and Corollary 1, which will be used to get the neighborhood results of Theorems 6 and 7.

Theorem 2. 

A function $h\in {\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ if and only if

$$\frac{1}{\zeta }\left[h\ast \left(\frac{(\zeta -q{\zeta }^3)(1+\beta e^{i\varphi })}{(1-\zeta )(1-q\zeta )(1-q^2\zeta )}-\frac{(1+\gamma e^{i\varphi })\zeta }{(1-u_x\zeta )(1-u_{x}q\zeta )}\right)\right]\ne 0,\left|\zeta \right|<1,$$

where $0<q<1,$ $-1\le \beta <\alpha \le 1,0\le \varphi <2\pi$ and $u_x,\gamma$ are defined by (15) and (13), respectively.

Proof. 

We have, $h\in {\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ if and only if

$$\frac{e^{i\lambda }\frac{{\partial }_q\left(\zeta {\partial }_{q}h\left(\zeta \right)\right)}{{\partial }_q{h}_{x,y}\left(\zeta \right)}-isin\lambda }{cos\lambda }\ne \frac{1+\alpha e^{i\varphi }}{1+\beta e^{i\varphi }},\left(\right|\zeta |<R,)$$

which implies

$${\partial }_q\left(\zeta {\partial }_{q}h\left(\zeta \right)\right)(1+\beta e^{i\varphi })-{\partial }_q{h}_{x,y}\left(\zeta \right)\{1+\gamma e^{i\varphi }\}\ne 0.$$

(12)

$$\gamma =e^{-i\lambda }\{\alpha cos\left(\lambda \right)+i\beta sin\left(\lambda \right)\}.$$

(13)

Putting $h\left(\zeta \right)=\zeta +{\sum }_{v=2}^{\infty }{a}_v{\zeta }^v$, we get

$${\partial }_{q}h=1+\sum _{v=2}^{\infty }{\left[v\right]}_q{a}_v{\zeta }^{v-1},{\partial }_q\left(\zeta {\partial }_{q}h\right)=1+\sum _{v=2}^{\infty }{\left[v\right]}_q^2{a}_v{\zeta }^{v-1}={\partial }_{q}h\ast \frac{1}{(1-\zeta )(1-q\zeta )}.$$

$${\partial }_q{h}_{x,y}\left(\zeta \right)={\partial }_{q}h\ast \frac{1}{(1-u_x\zeta )}=\sum _{v=1}^{\infty }{\left[v\right]}_q{u}_x^v{a}_v{\zeta }^{v-1},$$

(14)

where

$$u_x^v={\delta }_{v,x},\mathrm{and}{\delta }_{v,x}\mathrm{is}\mathrm{given}\mathrm{by}\left(3\right),$$

(15)

the equivalent on the left side of (12) is

$${\partial }_{q}h\ast \left(\frac{1+\beta e^{i\varphi }}{(1-\zeta )(1-q\zeta )}-\frac{1+\gamma e^{i\varphi }}{1-u_x\zeta }\right),$$

(16)

simplifying (16) we get

$$\frac{1}{\zeta }\left[\zeta {\partial }_{q}h\ast \left(\frac{(1+\beta e^{i\varphi })\zeta }{(1-\zeta )(1-q\zeta )}-\frac{(1+\gamma e^{i\varphi })\zeta }{1-u_x\zeta }\right)\right]\ne 0,$$

(17)

since $\zeta {\partial }_{q}h\ast g=h\ast \zeta {\partial }_{q}g$, Equation (17) can be expressed as

$$\frac{1}{\zeta }\left[h\ast \left(\frac{(\zeta -q{\zeta }^3)(1+\beta e^{i\varphi })}{(1-\zeta )(1-q\zeta )(1-q^2\zeta )}-\frac{(1+\gamma e^{i\varphi })\zeta }{(1-u_x\zeta )(1-u_{x}q\zeta )}\right)\right]\ne 0.$$

□

Remark 3. 

Ganesan et al. in [24] and Silverman et al. in [25] demonstrated the following result for $q\to 1^{-}$ and various values of $x,y,\lambda ,\alpha$ and β.

Theorem 3. 

A function $f\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ if and only if

$$\frac{1}{\zeta }\left[h\ast \left(\frac{(1+\beta e^{i\varphi })\zeta }{(1-\zeta )(1-q\zeta )}-\frac{(1+\gamma e^{i\varphi })\zeta }{1-u_x\zeta }\right)\right]\ne 0,\left|\zeta \right|<1,$$

where $0\le \varphi <2\pi ,-1\le \beta <\alpha \le 1$, $0<q<1$ and γ,$u_x$ are defined by (13) and (15), respectively.

Proof. 

Since $h\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ if and only if $g\left(\zeta \right)={\int }_0^{\zeta }\frac{h\left(w\right)}{w}{d}_q\zeta \in {\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta ,\lambda )$, we have

$$\frac{1}{\zeta }\left[g\ast \left(\frac{(\zeta -q{\zeta }^3)(1+\beta e^{i\varphi })}{(1-\zeta )(1-q\zeta )(1-q^2\zeta )}-\frac{(1+\gamma e^{i\varphi })\zeta }{(1-u_x\zeta )(1-u_{x}q\zeta )}\right)\right]$$

$$=\frac{1}{\zeta }\left[h\ast \left(\frac{(1+\beta e^{i\varphi })\zeta }{(1-\zeta )(1-q\zeta )}-\frac{(1+\gamma e^{i\varphi })\zeta }{1-u_x\zeta }\right)\right],$$

where $\gamma$ is defined by (13). As a result, Theorem 3 leads to the conclusion. □

Remark 4. 

Take note that Theorem 3 makes it simple for us to derive the corresponding condition for a function $h\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ if and only if

$$\frac{(h\ast g)\left(\zeta \right)}{\zeta }\ne 0,g\in \mathcal{F},\zeta \in k,$$

(18)

where $g\left(\zeta \right)$ has the form

$$\begin{array}{cc}\hfill g\left(\zeta \right)=\zeta +\sum _{v=2}^{\infty }{t}_v{\zeta }^v,& \\ \hfill t_v=\frac{{\left[v\right]}_q-{\delta }_{v,x}+({\left[v\right]}_q\beta -{\delta }_{v,x}\gamma )e^{i\varphi }}{(\beta -\gamma )e^{i\varphi }}.\end{array}$$

(19)

Corollary 1. 

Let $h\left(\zeta \right)=\zeta +w{\zeta }^v,v\ge 2$, then $h\left(\zeta \right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ if and only if

$$\left|w\right|\le \frac{|\beta -\gamma |}{{\left[v\right]}_q-{\delta }_{v,x}+|{\left[v\right]}_q\beta -{\delta }_{v,x}\gamma |}.$$

(20)

Proof. 

Let $h\left(\zeta \right)=\zeta +w{\zeta }^v,$ with satisfying inequality (20).

Let g given in Remark 4 and

$$\left|\frac{(h\ast g)\left(\zeta \right)}{\zeta }\right|\ge 1-\sum _{v=2}^{\infty }\left|w\right||t_v{\left|\right|\zeta |}^{v-1}>1-\left|\zeta \right|>0,\zeta \in k.$$

Applying Remark 4, we get $h\left(\zeta \right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$. Conversely, let $h\left(\zeta \right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ and let

$$g\left(w\zeta \right)=\zeta +\sum _{v=2}^{\infty }\frac{{\left[v\right]}_q-{\delta }_{v,x}+|{\left[v\right]}_q\beta -{\delta }_{v,x}\gamma |}{|\beta -\gamma |}{\left|\zeta \right|}^v.$$

Then, using Theorem 3, we obtain

$$\frac{(h\ast g)\left(\zeta \right)}{\zeta }=1+w\frac{{\left[v\right]}_q-{\delta }_{v,x}+|{\left[v\right]}_q\beta -{\delta }_{v,x}\gamma |}{|\beta -\gamma |}{\zeta }^{v-1}\ne 0.$$

Let $\left|w\right|>\frac{|\beta -\gamma |}{{\left[v\right]}_q-{\delta }_{v,x}+|{\left[v\right]}_q\beta -{\delta }_{v,x}\gamma |}$, then there exists $\vartheta \in k$, such that

$$\frac{(h\ast g)\left(\vartheta \right)}{\vartheta }=0,\vartheta \in k,$$

which is a contradiction. Hence, $\left|w\right|\le \frac{|\beta -\gamma |}{{\left[v\right]}_q-{\delta }_{v,x}+|{\left[v\right]}_q\beta -{\delta }_{v,x}\gamma |}.$ □

The sufficient condition theorem can be driven by using Remark 4.

Theorem 4. 

Let $h\left(\zeta \right)=\zeta +{\sum }_{v=2}^{\infty }{a}_v{\zeta }^v,$ be analytic in k, for $\left|\lambda \right|<\frac{\pi }{2}$, $0<q<1$ and $-1\le \beta <\alpha \le 1$, if

$$\sum _{v=2}^{\infty }\left\{\frac{({\left[v\right]}_q-{\delta }_{v,x})+\left|\gamma {\delta }_{v,x}-\beta {\left[v\right]}_q\right|}{|\gamma -\beta |}\right\}\left|a_v\right|\le 1,$$

(21)

where γ is given by (13), then $h\left(\zeta \right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$.

Proof. 

To prove Theorem 4, it is sufficient to demonstrate that $\frac{(h\ast g)\left(\zeta \right)}{\zeta }\ne 0$ where g is given by (19). Let $h\left(\zeta \right)=\zeta +{\sum }_{v=2}^{\infty }{a}_v{\zeta }^v$ and $g\left(\zeta \right)=\zeta +{\sum }_{v=2}^{\infty }{t}_v{\zeta }^v.$ The convolution

$$\frac{(h\ast g)\left(\zeta \right)}{\zeta }=1+\sum _{v=2}^{\infty }{t}_v{a}_v{\zeta }^{v-1},\zeta \in k.$$

From Theorem 3, $h\left(\zeta \right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ if and only if $\frac{(h\ast g)\left(\zeta \right)}{\zeta }\ne 0$, for g given by (19). Using (19) and (21), we get

$$\left|\frac{(f\ast g)\left(\zeta \right)}{\zeta }\right|\ge 1-\sum _{v=2}^{\infty }\frac{{\left[v\right]}_q-{\delta }_{v,x}+|{\left[v\right]}_q\beta -{\delta }_{v,x}\gamma |}{|\beta -\gamma |}|a_v{\left|\right|\zeta |}^{v-1}>0,\zeta \in k.$$

Thus, $h\left(\zeta \right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$. □

Theorem 5. 

Let f be a convex function and let $h\left(\zeta \right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ and satisfies inequality

$$\sum _{v=2}^{\infty }\left\{\frac{\left|\gamma {\delta }_{v,x}-\beta {\left[n\right]}_q\right|+({\left[v\right]}_q-{\delta }_{v,x})}{|\gamma -\beta |}\right\}\left|a_v\right|<1,$$

(22)

then, $(h\ast f)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$.

Proof. 

Let $f\left(\zeta \right)=\zeta +{\sum }_{v=2}^{\infty }{b}_v{\zeta }^v$ be convex and $h\left(\zeta \right)=\zeta +{\sum }_{v=2}^{\infty }{a}_v{\zeta }^v\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ and satisfies inequality (22). Therefore,

$$1-\sum _{v=2}^{\infty }\frac{{\left[v\right]}_q-{\delta }_{v,x}+|{\left[v\right]}_q\beta -{\delta }_{v,x}\gamma |}{|\beta -\gamma |}\left|a_v\right|>0.$$

(23)

In order to demonstrate $(h\ast f)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$, it is sufficient to demonstrate that $\frac{(h\ast f\ast g)\left(\zeta \right)}{\zeta }\ne 0$ where g is given by (19). Consider

$$\left|\frac{(h\ast f\ast g)\left(\zeta \right)}{\zeta }\right|\ge 1-\sum _{v=2}^{\infty }|a_v\left|\right|b_v\left|\right|t_v{\left|\right|\zeta |}^{v-1}.$$

As $\zeta \in k$ and g is convex, we get $|b_v|\le 1$. Using (23), we get

$$\left|\frac{(h\ast g\ast f)\left(\zeta \right)}{\zeta }\right|\ge 1-\sum _{v=2}^{\infty }\frac{{\left[v\right]}_q-{\delta }_{v,x}+|{\left[v\right]}_q\beta -{\delta }_{v,x}\gamma |}{|\beta -\gamma |}\left|a_v\right|>0,\zeta \in k.$$

Thus, $h\ast f\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda ).$ □

Corollary 2. 

Let $h\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$, and satisfies inequality (22), then

$$F_i\left(\zeta \right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda ),(i=1,2,3,4),$$

where

$$F_1\left(\zeta \right)={\int }_0^{\zeta }\frac{h\left(t\right)}{t}dt,F_2\left(\zeta \right)={\int }_0^{\zeta }\frac{h\left(t\right)-h\left(zt\right)}{t-zt}dt,\left|z\right|\le 1,z\ne 1,$$

$$F_3\left(\zeta \right)=\frac{2}{\zeta }{\int }_0^{\zeta }h\left(t\right)dt,F_4\left(\zeta \right)=\frac{m+1}{m}{\int }_0^{\zeta }{t}^{m-1}h\left(t\right)dt,\Re m>0.$$

Proof. 

Since

$$F_1\left(\zeta \right)={\varphi }_1\left(\zeta \right)\ast h\left(\zeta \right),{\varphi }_1\left(\zeta \right)=\sum _1^{\infty }\frac{1}{v}{\zeta }^v=log{(1-\zeta )}^{-1},$$

$$F_2\left(\zeta \right)={\varphi }_2\left(\zeta \right)\ast h\left(\zeta \right),{\varphi }_2\left(\zeta \right)=\sum _1^{\infty }\frac{1-z^v}{v(1-z)}{\zeta }^v=\frac{1}{1-z}log\left(\frac{1-z\zeta }{1-\zeta }\right),\left|z\right|\le 1,z\ne 1,$$

$$F_3\left(\zeta \right)={\varphi }_3\left(\zeta \right)\ast h\left(\zeta \right),{\varphi }_3\left(\zeta \right)=\sum _0^{\infty }\frac{2}{v+1}{\zeta }^v=\frac{-2[\zeta +log(1-\zeta \left)\right]}{\zeta },$$

$$F_4\left(\zeta \right)={\varphi }_4\left(\zeta \right)\ast h\left(\zeta \right),{\varphi }_4\left(\zeta \right)=\sum _0^{\infty }\frac{1+m}{v+m}{\zeta }^v,\Re \left\{m\right\}>0.$$

We note that ${\varphi }_i,i=1,2,3,4$. It is simple to confirm that it is convex. Now, use Theorem 5 to get $F_i\left(\zeta \right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda ),(i=1,2,3,4)$. □

Remark 5. 

The results in [17] are obtained for $q\to 1^{-}$ and $\lambda =0$ in Theorem 4.

Theorem 6. 

Let $h\in \mathcal{F}$, and for any number η that is complex with $\left|\mu \right|<\rho$, if

$$\frac{h\left(\zeta \right)+\eta \zeta }{1+\eta }\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda ).$$

(24)

Then

$${\mathcal{N}}_{q,{\rho }_1}^{x,y}(\alpha ,\beta ,\lambda ;h)\subset {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda ),$$

where ${\rho }_1=\frac{\rho |\beta -\gamma |}{1+\left|\beta \right|}$.

Proof. 

We suppose that a function f defined by $f\left(\zeta \right)=\zeta +{\sum }_{v=2}^{\infty }{b}_v{\zeta }^v$ is in the class ${\mathcal{N}}_{q,{\rho }_1}^{x,y}(\alpha ,\beta ,\lambda ;h)$. We only need to prove that $f\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ to prove Theorem 6. This claim will be proved in the next three steps.

Theorem 3 provides us with this

$$h\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )\iff \frac{1}{\zeta }\left[(h\ast g\left(\zeta \right))\right]\ne 0,\zeta \in k,$$

(25)

where

$$g\left(\zeta \right)=\zeta +\sum _{v=2}^{\infty }\frac{({\left[v\right]}_q\beta -{\delta }_{v,x}\gamma )e^{i\varphi }+{\left[v\right]}_q-{\delta }_{v,x}}{(\beta -\gamma )e^{i\varphi }}{\zeta }^n,$$

where $\gamma$ is given by (13) and for $0\le \alpha <1,0\le \varphi <2\pi ,-1\le \beta <\alpha \le 1$. We can write $g\left(\zeta \right)=\zeta +{\sum }_{v=2}^{\infty }{t}_v{\zeta }^v$, where $t_v$ is given by (19). So that $|t_v|\le \frac{{\left[v\right]}_q(1+|\beta \left|\right)}{|\beta -\gamma |}.$

Secondly we obtain that (24) is equivalent to

$$\left|\frac{h\left(\zeta \right)\ast g\left(\zeta \right)}{\zeta }\right|\ge \rho ,$$

(26)

because, if $h\left(\zeta \right)=\zeta +{\sum }_{v=2}^{\infty }{a}_v{\zeta }^v\in \mathcal{F}$ and (24) is satisfied, then (25) is equivalent to

$$g\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )\iff \frac{1}{\zeta }\left[\frac{h\left(\zeta \right)\ast g\left(\zeta \right)}{1+\eta }\right]\ne 0,\left|\eta \right|<\rho .$$

Thirdly, letting $f\left(\zeta \right)=\zeta +{\sum }_{v=2}^{\infty }{b}_v{\zeta }^v$, we notice that

$$\left|\frac{f\left(\zeta \right)\ast g\left(\zeta \right)}{\zeta }\right|=\left|\frac{\left(f\right(\zeta )-h(\zeta \left)\right)\ast g\left(\zeta \right)}{\zeta }+\frac{h\left(\zeta \right)\ast g\left(\zeta \right)}{\zeta }\right|$$

$$\ge \rho -\left|\frac{\left(f\right(\zeta )-h(\zeta \left)\right)\ast g\left(\zeta \right)}{\zeta }\right|,(\mathrm{by}\mathrm{using}\left(26\right))$$

$$=\rho -\left|\sum _{v=2}^{\infty }(b_v-a_v)t_v{\zeta }^v\right|,$$

$$\ge \rho -\left|\zeta \right|\sum _{v=2}^{\infty }\frac{{\left[v\right]}_q(1+|\beta \left|\right)}{|\beta -\gamma |}|b_v-a_v|$$

$$\ge \rho -\rho \left|\zeta \right|>0.$$

This proves that

$$\frac{(f\ast g)\left(\zeta \right)}{\zeta }\ne 0,\zeta \in k.$$

Our observations (25) indicate that $f\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$. This concludes the proof of the theorem. □

The well-known result proved by Ruscheweyh in [23] can be obtained when $q\to 1^{-},x=y=\alpha =-\beta =1$ and $\lambda =0$, as shown in the above theorem.

Theorem 7. 

Let $h\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$, for ${\rho }_1<c$. Then

$${\mathcal{N}}_{q,{\rho }_1}^{x,y}(\alpha ,\beta ,\lambda ;h)\subset {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda ),$$

where, c is a non-zero real number with $c\le \left|\frac{(h\ast g)\left(\zeta \right)}{\zeta }\right|,\zeta \in k$ and g is defined in Remark 4.

Proof. 

Let $f\left(\zeta \right)=\zeta +{\sum }_{v=2}^{\infty }{b}_v{\zeta }^v\in {\mathcal{N}}_{q,{\rho }_1}^{x,y}(\alpha ,\beta ,\lambda ;h)$. To prove Theorem 7, it is sufficient to demonstrate that $\frac{(f\ast g)\left(\zeta \right)}{\zeta }\ne 0$ when g is given by (19). Consider

$$\left|\frac{f\left(\zeta \right)\ast g\left(\zeta \right)}{\zeta }\right|\ge \left|\frac{h\left(\zeta \right)\ast g\left(\zeta \right)}{\zeta }\right|-\left|\frac{\left(f\right(\zeta )-h(\zeta \left)\right)\ast g\left(\zeta \right)}{\zeta }\right|.$$

(27)

Since $h\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$, therefore applying Theorem 4, we obtain

$$\left|\frac{(h\ast g)\left(\zeta \right)}{\zeta }\right|\ge c,$$

(28)

where c is a non-zero real number and $\zeta \in k$. Now

$$\begin{array}{c}\hfill \left|\frac{\left(f\right(\zeta )-h(\zeta \left)\right)\ast g\left(\zeta \right)}{\zeta }\right|=\left|\sum _{v=2}^{\infty }(b_v-a_v)t_v{\zeta }^v\right|\\ \hfill \le \sum _{v=2}^{\infty }\frac{({\left[v\right]}_q-{\delta }_{v,x})+\left|\gamma {\delta }_{v,x}-\beta {\left[v\right]}_q\right|}{|\gamma -\beta |}|b_v-a_v|\\ \hfill \le \sum _{v=2}^{\infty }\frac{{\left[v\right]}_q(1+|\beta \left|\right)}{|\beta -\gamma |}|b_v-a_v|\\ \hfill \le \frac{\rho |\beta -\gamma |}{{\left[v\right]}_q(1+|\beta \left|\right)}={\rho }_1,\end{array}$$

(29)

using (28) and (29) in (27), we obtain

$$\left|\frac{f\left(\zeta \right)\ast g\left(\zeta \right)}{\zeta }\right|\ge c-{\rho }_1>0,$$

where ${\rho }_1<c$. This concludes the proof. □

3. Conclusions

Motivated by recent applications of the theory of $(x,y)$-symmetrical functions and the concept of q-calculus in geometric function theory, we have employed these two concepts to define and analyze the classes ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ and ${\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta ,\lambda )$ for $\lambda$-spiral-like functions in the open unit disk. Specifically, we have explored a convolution property, which will serve as a foundation for further investigation. We have also derived the sufficient condition and equivalent condition. Additionally, we have examined various neighborhood results for the class ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ,\lambda )$. The methodology presented in this paper can be readily applied to establish multiple classes with distinct image domains. The utilization of symmetric q-calculus, the Janowski class, and the essential q-hypergeometric functions offers ample opportunities for research in various fields.