Convolution Properties of q-Janowski-Type Functions Associated with (x,y)-Symmetrical Functions

Abstract

The purpose of this paper is to define new classes of analytic functions by amalgamating the concepts of q-calculus, Janowski type functions and $(x,y)$-symmetrical functions. We use the technique of convolution and quantum calculus to investigate the convolution conditions which will be used as a supporting result for further investigation in our work, we deduce the sufficient conditions, P$\acute{o}$lya-Schoenberg theorem and the application. Finally motivated by definition of the neighborhood, we give analogous definition of neighborhood for the classes ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ and ${\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$, and then investigate the related neighborhood results, which are also pointed out.

1. Introduction

Let $\mathcal{F}\left(k\right)$ denote the family of all functions that are analytic in the open unit disc $k=\left\{w\in \mathbb{C}:\left|w\right|<1\right\}$ and let $\mathcal{F}$ represents a subfamily of class $h\in \mathcal{F}\left(k\right)$ which has the form

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

(1)

and suppose $\tilde{\mathcal{S}}$ containing all the functions in $\mathcal{F}$ that are univalent k. The convolution or Hadamard product of two analytic functions $h,g\in \mathcal{F}$ where h is defined by (1) and $g\left(w\right)=w+{\sum }_{v=2}^{\infty }{b}_v{w}^v$, is

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

In order to define new classes of q-Janowski symmetrical functions defined in k, we first recall the necessary notions and notations concerning, Janowski type functions, the theory of $(x,y)$-symmetrical functions and quantum calculus (or q-calculus).

Janowski in [1] introduced the class $\mathcal{P}[\alpha ,\beta ]$, a given $h\in \mathcal{F}\mathrm{and}h\left(0\right)=1$ is said to be in $\mathcal{P}[\alpha ,\beta ]$ if and only if $p\left(w\right)={\displaystyle \frac{1+\alpha s\left(w\right)}{1+\beta s\left(w\right)}}$, for $-1\le \beta <\alpha \le 1\mathrm{and}s\left(w\right)\in \Delta$ where $\Delta$ denote for the family of Schwarz functions, that is

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

(2)

Let y be an arbitrarily fixed integer and for $\epsilon =e^{\frac{2\pi i}{y}}$, a domain $\mathbf{G}\subset \mathbb{C}$ is said to be y-fold symmetric domain if $\epsilon \mathbf{G}=\mathbf{G}$. A function h is called y-symmetrical function for each $w\in \mathbf{G}$ if $h\left(\epsilon w\right)=\epsilon h\left(w\right).$

In 1995, Liczberski and Polubinski [2] constructed the concept of $(x,y)$-symmetrical functions for $(x=0,1,2,\dots ,y-1),$ and $(y=2,3,\dots )$. If $\mathbf{G}$ is y-fold symmetric domain and x any integer, then a function $h:\mathbf{G}\to \mathbb{C}$ is called $(x,y)$-symmetrical if for each $w\in \mathbf{G}$, $h\left(\epsilon w\right)={\epsilon }^{x}h\left(w\right).$ The family of all $(x,y)$-symmetrical functions will be denoted by ${\mathcal{F}}_y^x$, we note that ${\mathcal{F}}_2^0$, ${\mathcal{F}}_2^1$ and ${\mathcal{F}}_y^1$ are families of even, odd and of y-symmetrical functions, respectively.

Theorem 1

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

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

(3)

Remark 1.

Equivalently, (3) may be written as

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

(4)

where

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

(5)

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

Recently the authors of [3,4] obtained many interesting results for various classes using the concept of $(x,y)$-symmetrical functions and q-derivative.

In [5], Jackson introduced and studied the concept of the q-derivative operator ${\partial }_{q}h\left(w\right)$ as follows:

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

(6)

Equivalently (6), may be written as

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

where

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

(7)

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

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

Then

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

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

The q-integral of a function h presented by Jackson [6] As a right inverse as

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

provided that ${\sum }_{v=0}^{\infty }{q}^{v}h\left(w{q}^v\right)$ is converges.

Proposition 1.

If n and m any real (or complex) constants and $w\in k,$ then we have

1. 

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

2. 

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

3. 

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

In recent years, using quantum (or q-calculus) for studying diverse families of analytic functions. Srivastava et al. [7] found distortion and radius of univalent and starlikenss for several subclasses of q-starlike functions. Naeem et al. [8] investigated subfamilies of q-convex functions with respect to the Janowski functions connected with q-conic domain. Govindaraj and Sivasubramanian in [9] found subclasses connected with q-conic domain. In [10], we use the symmetric q-derivative operator to define a new subclass of analytic and bi-univalent function. Srivastava [11] published survey-cum-expository review paper which is useful for researchers and scholars.

Utilizing the ideas of q-derivative operator and the concept of $(x,y)$-symmetrical functions we introduce a new subclass ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$. This class is introduced by using the q-derivative operator with the concept to $(x,y)$-symmetric points.

Definition 1.

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

$$\Re \left\{\frac{w{\partial }_{q}h\left(w\right)}{h_{x,y}\left(w\right)}\right\}\in \mathcal{P}[\alpha ,\beta ],w\in k,$$

(8)

where $h_{x,y}$ is defined in (3).

For special cases of the parameters $q,\alpha ,\beta ,x$ and y the class ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ yield several known subclasses of $\mathcal{F}$, namely: ${\tilde{\mathcal{S}}}_1^{x,y}(\alpha ,\beta ):={\tilde{\mathcal{S}}}^{x,y}(\alpha ,\beta )$ introduced by the authors of [12]; ${\tilde{\mathcal{S}}}_1^{1,y}(\alpha ,\beta ):={\tilde{\mathcal{S}}}_y(\alpha ,\beta )$, introduced by the authors Latha and Darus [13]; ${\tilde{\mathcal{S}}}_1^{1,y}(1,-1):={\tilde{\mathcal{S}}}_y$ as defined by Sakaguchi [14]; ${\tilde{\mathcal{S}}}_1^{1,1}(\alpha ,\beta ):=\tilde{\mathcal{S}}[\alpha ,\beta ]$ which reduce to a well-known class defined by Janowski [1]; ${\tilde{\mathcal{S}}}_q^{1,1}(1-2\kappa ,-1)$ = ${\tilde{\mathcal{S}}}_q\left(\kappa \right)$ which was introduced and studied by Agrawal and Sahoo in [15]; ${\tilde{\mathcal{S}}}_q^{1,1}(1,-1)$ = ${\tilde{\mathcal{S}}}_q$ which was first introduced by Ismail et al. [16]; ${\tilde{\mathcal{S}}}_1^{1,1}(1-2\kappa ,-1)$ = $\tilde{\mathcal{S}}\left(\kappa \right)$ the well-known class of starlike function of order $\kappa$ by Robertson [17]; and ${\mathcal{S}}_1^{1,1}(1,-1,0)={\mathcal{S}}^{*}$ the class introduced by Nevanlinna [18], etc.

We denote by ${\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$ the subclass of $\mathcal{F}$ consisting of all functions h such that

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

(9)

We need to recall the following neighborhood concept introduced by Goodman [19] and generalized by Ruscheweyh [20].

Definition 2.

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

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

(10)

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

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

(11)

Ruscheweyh [20] proved, among other results, that for all $\eta \in \mathbb{C}$, with $\left|\eta \right|<\rho$,

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

Lemma 1

([21]). Let ϕ be a convex and g a starlike, for F analytic in $\mathcal{U}$ with $F\left(0\right)=1,$ then

$$\frac{\varphi \ast Fg}{\varphi \ast g}\left(\mathcal{U}\right)\subset \overline{CO}\left(F\left(\mathcal{U}\right)\right),$$

where $\overline{CO}\left(F\left(\mathcal{U}\right)\right)$ denotes the closed convex hull of $F\left(\mathcal{U}\right)$.

The goal of this research to give a convolution conditions for a function h to be in the classes ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ and ${\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$ which will be used to drive a sufficient conditions, P$\acute{o}$lya–Schoenberg theorem and application. In the next section be the motivation of the Definition 2, we give analogous definition of neighborhood for the class ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ and ${\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$, then investigate related neighborhood results.

2. Results

Theorem 2.

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

$$\frac{1}{w}\left[h\ast \left(\frac{(w-q{w}^3)(1+\beta e^{i\varphi })}{(1-w)(1-qw)(1-q^{2}w)}-\frac{(1+\alpha e^{i\varphi })w}{(1-u_{x}w)(1-u_{x}qw)}\right)\right]\ne 0,\left|w\right|<R\le 1,$$

where $0<q<1,$ $-1\le \beta <\alpha \le 1,0\le \varphi <2\pi$ and $u_x$ is defined by (14).

Proof. 

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

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

which implies

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

(12)

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

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

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

(13)

where

$$u_x^v={\delta }_{v,x},\mathrm{and}{\delta }_{v,x}\mathrm{is}\mathrm{given}\mathrm{by}(5).$$

(14)

The left hand side of (12) is equivalent to

$${\partial }_{q}h\ast \left(\frac{1+\beta e^{i\varphi }}{(1-w)(1-qw)}-\frac{1+\alpha e^{i\varphi }}{1-u_{x}w}\right),$$

(15)

simplifying (15) we obtain

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

(16)

since $w{\partial }_{q}h\ast g=h\ast w{\partial }_{q}g$, we can write the Equation (16) as

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

 □

Remark 2.

For $q\to 1^{-}$ and spacial values of $x,y,\alpha$ and β, we have following result proved by Ganesan and et al in [22] Silverman and et al in [23].

Theorem 3.

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

$$\frac{1}{w}\left[h\ast \left(\frac{(1+\beta e^{i\varphi })w}{(1-w)(1-qw)}-\frac{(1+\alpha e^{i\varphi })w}{1-u_{x}w}\right)\right]\ne 0,\left|w\right|<1,$$

where $0<q<1,$ $-1\le \beta <\alpha \le 1,0\le \varphi <2\pi$ and $u_x$ is defined by (14).

Proof. 

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

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

$$=\frac{1}{w}\left[h\ast \left(\frac{(1+\beta e^{i\varphi })w}{(1-w)(1-qw)}-\frac{(1+\alpha e^{i\varphi })w}{1-u_{x}w}\right)\right].$$

Thus the result follows from Theorem 3. □

Note that we can easily from Theorem 3 obtain that the equivalent condition for a function $h\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ in the following Corollary.

Corollary 1.

For $q\in (0,1),-1\le \beta <\alpha \le 1$ and $\varphi \in [0,2\pi ),$ then

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

(17)

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

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

(18)

By using Corollary 1 we drive the sufficient condition theorem.

Theorem 4.

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

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

(19)

then $h\left(w\right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$.

Proof. 

For the proof of Theorem 4, it suffices to show that $\frac{(h\ast g)\left(w\right)}{w}\ne 0$ where g is given by (18). Let $h\left(w\right)=w+{\sum }_{v=2}^{\infty }{a}_v{w}^v$ and $g\left(w\right)=w+{\sum }_{v=2}^{\infty }{t}_v{w}^v.$ The convolution

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

From Corollary 1 that $h\left(w\right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ if and only if $\frac{(h\ast g)\left(w\right)}{w}\ne 0$, for g given by (18). Using (18) and (19), we obtain

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

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

Theorem 5.

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

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

(20)

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

Proof. 

Let $f\left(w\right)=w+{\sum }_{v=2}^{\infty }{b}_v{w}^v$ is a convex and $h\left(w\right)=w+{\sum }_{v=2}^{\infty }{a}_v{w}^v\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ and satisfies inequality (20), therefore

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

(21)

To prove that $(h\ast f)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ it is enough to show that $\frac{(h\ast f\ast g)\left(w\right)}{w}\ne 0$ where g is given by (18). Consider

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

Since $w\in k$ and g is convex, we obtain $|b_v|\le 1$. Using (21), we obtain

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

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

3. Applications

Corollary 2.

Let $h\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$, and satisfies the inequality (20). Then

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

where

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

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

Proof. 

Since

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

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

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

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

We note that ${\varphi }_i,i=1,2,3,4$. can easily be verified to be convex. Now, using Theorem 5 to obtain $F_i\left(w\right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ),(i=1,2,3,4)$. □

4. $(\mathit{\rho },\mathit{q})$-Neighborhoods for Functions in the Classes ${\tilde{\mathcal{S}}}_{\mathit{q}}^{\mathit{x},\mathit{y}}(\mathsf{\alpha },\mathsf{\beta })$ and ${\tilde{\mathcal{K}}}_{\mathit{q}}^{\mathit{x},\mathit{y}}(\mathsf{\alpha },\mathsf{\beta })$

By taking motivation from Definition 2 and to find some neighborhood results for our classes, we introduce the following concepts of neighborhood that analogous to those obtained by Ruscheweyh [20].

Definition 3.

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

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

(22)

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

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

(23)

Remark 3.

1. 

For ${\gamma }_v=v$ of Definition 3 we obtain Definition 2 of the neighborhood concept introduced by Goodman [19] and generalized by Ruscheweyh [20].

2. 

For ${\gamma }_v={\left[v\right]}_q$ of Definition 3 we obtain the definition of neighborhood with q-derivative

${\mathcal{N}}_{q,\rho }^{\lambda }\left(h\right),{\mathcal{N}}_{q,\rho }^{\lambda }\left(e\right)$, where ${\left[v\right]}_q$ is given by Equation (7).

3. 

For ${\gamma }_v=\frac{({\left[v\right]}_q-{\delta }_{v,x})+\left|\alpha {\delta }_{v,x}-\beta {\left[v\right]}_q\right|}{|\alpha -\beta |}$ of Definition 3 we obtain the definition of the neighborhood for the classes ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ and ${\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$ which is ${\mathcal{N}}_{q,\rho }^{x,y}(\alpha ,\beta ;h)$.

Theorem 6.

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

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

(24)

Then

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

Proof. 

We assume that a function f defined by $f\left(w\right)=w+{\sum }_{v=2}^{\infty }{b}_v{w}^v$ is in the class ${\mathcal{N}}_{q,\rho }^{x,y}(\alpha ,\beta ;h)$. In order to prove the theorem, we only need to prove that $f\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$. We would prove this claim in next three steps.

From Theorem 3 we have

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

(25)

where

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

where $0\le \varphi <2\pi ,-1\le \beta <\alpha \le 1$. We can write $g\left(w\right)=w+{\sum }_{v=2}^{\infty }{t}_v{w}^v$, where $t_v$ is given by (18).

Secondly, we obtain that (24) is equivalent to

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

(26)

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

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

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

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

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

by using (26),

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

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

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

This proves that

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

In view of our observations (25), it follows that $f\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$. This completes the proof of the theorem. □

When $q\to 1^{-},x=y=\alpha =1$ and $\beta =-1$ in the above theorem we obtain the well-known result proved by Ruscheweyh in [20].

Theorem 7.

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

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

where c is a non-zero real number with $c\le \left|\frac{(h\ast g)\left(w\right)}{w}\right|,w\in k$ and g is defined in Remark 1.

Proof. 

Let $f\left(w\right)=w+{\sum }_{v=2}^{\infty }{b}_v{w}^v\in {\mathcal{N}}_{q,{\rho }_1}^{x,y}(\alpha ,\beta ;h)$. For the proof of Theorem 7, it suffices to show that $\frac{(f\ast g)\left(w\right)}{w}\ne 0$ where g is given by (18). Consider

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

(27)

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

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

(28)

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

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

(29)

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

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

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

Theorem 8.

Let $h\in {\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$, and for all complex number η, with $\left|\mu \right|<\frac{1}{4}$, we have

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

(30)

Proof. 

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

$$H_{\eta }\left(w\right)=\frac{h\left(w\right)+\eta w}{1+\eta }$$

$$=\left(h\left(w\right)\ast \psi \left(w\right)\right),w\in k.$$

where

$$\psi \left(w\right)=\frac{w-\frac{\eta }{1+\eta }{w}^2}{1-w}.$$

Using the principle of convolution we obtain

$$h\left(w\right)\ast \psi \left(w\right)=w{\partial }_{q}h\ast \left(\psi \left(w\right)\ast log\left(\frac{1}{1-w}\right)\right).$$

Since $h\in {\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$, $w{\partial }_{q}h\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ and for $\left|\eta \right|<\frac{1}{4}$, $\psi$ is in the class of starlike functions $\tilde{\mathcal{S}}$, applying the convolution we obtain

$$\psi \left(w\right)\ast log\left(\frac{1}{1-w}\right)={\int }_0^w\frac{\psi \left(\zeta \right)}{\zeta }{d}_q\zeta .$$

(31)

Applying the Alexander relation in (31), we obtain $\psi \left(w\right)\ast log\left(\frac{1}{1-w}\right)$ is in the class of convex functions $\tilde{\mathcal{K}}$. Using Lemma 1 one can prove that $\tilde{\mathcal{K}}\ast {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )\subset {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$. Hence

$$H_{\eta }\left(w\right)=w{\partial }_{q}h\ast \left(\psi \left(w\right)\ast log\left(\frac{1}{1-w}\right)\right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ),\left|\eta \right|<\frac{1}{4}.$$

This completes the proof. □

Theorem 9.

Let $h\in {\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$. Then

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

where $\rho =\frac{|\beta -\alpha |}{4(1+|\beta \left|\right)}$.

Proof. 

Let $h\in {\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$, then by Theorem 8 $H_{\varsigma }\left(w\right)\in {\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta ),\left|\eta \right|<\frac{1}{4}.$ Choosing $\rho =\frac{1}{4}$ and applying Theorem 6, we obtain our required result. □

5. Conclusions

Applications of the q-calculus have been the focal point in the recent times in various mentioned branches of mathematics and physics [11]. In this paper, we have applied the q-calculus for classes of analytic functions with respect to $(x,y)$-symmetric points. The new classes have been defined and studied. In particular, we have investigated some of its geometric properties such as a convolution conditions for the functions h to be in the classes ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ and ${\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$ and a sufficient conditions, application of P$\acute{o}$lya–Schoenberg by spatial examples and the neighborhood results related to the functions in the classes ${\tilde{\mathcal{S}}}_q^{x,y}(\alpha ,\beta )$ and ${\tilde{\mathcal{K}}}_q^{x,y}(\alpha ,\beta )$. The idea used in this article can easily be implemented to define several subclasses of analytic (odd-even-k-symmetrical) functions connected with different image domains. This will open up a lot of new opportunities for research in this and related fields. The generalized Janowski class and symmetric functions or using symmetric q-derivative operator, basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials are applicable particularly in several diverse areas.