Let $0<q<1$. The $q-$bracket ${\left[\kappa \right]}_q$ is formally defined as such

The $q-$derivative, also known as the $q-$difference operator, of a function Φ is defined by and ${⅁}^{\left(\kappa \right)}\langle \mathsf{\Phi }(\Im );q\rangle =⅁\left({⅁}^{(\kappa -1)}\langle \mathsf{\Phi }(\Im );q\rangle \right)$ for $\kappa \ge 1.$

[8] By letting $-1\le \beta <\lambda \le 1$. A function $f\in \mathcal{A}$ is in the class ${\mathcal{S}}^{*}(\lambda ,\beta ;q)$ if it satisfies the following subordination: where the values of ${\lambda }_q$ and ${\beta }_q$ are given as

By letting $-1\le B<\lambda \le 1$. A function $f\in \mathcal{A}$ is in the class $\mathcal{C}(\lambda ,\beta ;q)$ if it satisfies the following subordination: where the values of ${\lambda }_q$ and ${\beta }_q$ are given as

By imposing $-1\le \beta <\lambda \le 1$ and $q\in \left(0,1\right)$. A function f belonging to the family Σ, as defined in equation (1), is considered to be a member of the bi-starlike class, denoted by ${\mathcal{S}}_{\Sigma }^{*}(\lambda ,\beta ;q)$ if it satisfies certain subordination conditions. These conditions can be expressed as follows: and where the function $\hslash \left(\varsigma \right)={\mathsf{\Phi }}^{-1}\left(\varsigma \right)$ is defined by the equation (3), and the values of ${\lambda }_q$ and ${\beta }_q$ are given as

By imposing $-1\le \beta <\lambda \le 1$ and $q\in \left(0,1\right)$. A function f belonging to the family Σ, as defined in equation (1), is considered to be a member of the bi-convex class, denoted by ${\mathcal{C}}_{\Sigma }(\lambda ,\beta ;q)$ if it satisfies certain subordination conditions. These conditions can be expressed as follows: and where the function $\hslash \left(\varsigma \right)={\mathsf{\Phi }}^{-1}\left(\varsigma \right)$ is defined by the equation (3), and the values of ${\lambda }_q$ and ${\beta }_q$ are given as

Let $-1\le \beta <\lambda \le 1$. A function f belonging to the family Σ, as defined in equation (1), is considered to be a member of the bi-starlike class $\underset{q\to 1^{-}}{lim}{\mathcal{S}}_{\Sigma }^{*}(\lambda ,\beta ;q)={\mathcal{S}}_{\Sigma }^{*}(\lambda ,\beta )$ if it satisfies the following subordination conditions: and where the function $\hslash \left(\varsigma \right)={\mathsf{\Phi }}^{-1}\left(\varsigma \right)$ is defined by the equation (3).

Let $-1\le \beta <\lambda \le 1$. A function f belonging to the family Σ, as defined in equation (1), is considered to be a member of the bi-convex class, denoted by $\underset{q\to 1^{-}}{lim}{\mathcal{C}}_{\Sigma }(\lambda ,\beta ;q)={\mathcal{C}}_{\Sigma }(\lambda ,\beta )$ if it satisfies the following subordination conditions: and where the function $\hslash \left(\varsigma \right)={\mathsf{\Phi }}^{-1}\left(\varsigma \right)$ is defined by the equation (3).

If Φ is an element of Σ defined by (1), it can be said that Φ is a member of the class ${\mathcal{S}}_{\Sigma }^{*}(\lambda ,\beta ;q)$, as per the following statement:

If $\mathsf{\Phi }$ belongs to the class ${\mathcal{S}}_{\Sigma }^{*}(\lambda ,\beta ;q)$, according to Definition 5, under the given conditions, there exist analytic functions $\mathsf{{\rm Y}}$ and $\Omega$ such that $\mathsf{{\rm Y}}\left(0\right)=\Omega \left(0\right)=0$, and $\left|\mathsf{{\rm Y}}\right(\Im \left)\right|<1$ and $|\Omega (\varsigma \left)\right|<1$ hold for all ℑ and $\varsigma$ in the unit disk ℧. In light of these conditions, the function $\mathsf{\Phi }$ can be expressed as follows: and For $\mathsf{{\rm Y}}(\Im )=1+{\displaystyle \sum _{\mathbb{k}=1}^{\infty }}{c}_{\mathbb{k}}{\Im }^{\mathbb{k}}=1+c_1\Im +c_2{\Im }^2+c_3{\Im }^3+\cdots$, we have Next we calculate the values of $c_1$ and $c_2$. Taking then, we get Comparing Equations (16) and (17), we have By utilizing equations (14) and (15) along (18), we can derive the following expression. and There is a commonly accepted understanding that if. and then for all $j\in \mathbb{N}$, we have In view of (1), (3), from (19) and (20), we obtain and By comparing the pertinent coefficients in (19) and (20), we arrive at the following. and It follows from (22) and (24) that and Adding (24) and (25), we get Substituting the value of $(c_1^2+d_1^2)$ from (27), we obtain Applying for the coefficients $c_2$ and $d_2$ along the equation (21), we obtain The subtraction of equation (25) from equation (24) yields the following result: Then, in view of (26) and (27), Eq. (30) becomes Thus applying (21), we conclude that This completes the proof of Theorem. □

Given that Φ is an element of Σ defined by (1) and belongs to the class ${\mathcal{S}}_{\Sigma }^{*}(\lambda ,\beta ;q)$, and μ is a real number, we can state the following

If $\mathsf{\Phi }\in {\mathcal{S}}_{\Sigma }^{*}(\lambda ,\beta ;q)$ is given by (1), from (29) and (30), we have where Then, we conclude that Which completes the proof of Theorem 2. □

If Φ is an element of Σ defined by (1), it can be said that Φ is a member of the class ${\mathcal{C}}_{\Sigma }(\lambda ,\beta ;q)$, as per the following statement: and

If $\mathsf{\Phi }$ belongs to the class ${\mathcal{C}}_{\Sigma }(\lambda ,\beta ;q)$, according to Definition 6, under the given conditions, the function $\mathsf{\Phi }$ can be expressed as follows: and By utilizing equations (31) and (32) along (16-18), we can derive the following expression. and In view of (1), (3), from (33) and (34), we obtain and By comparing the pertinent coefficients in (19) and (20), we arrive at the following. and It follows from (35) and (37) that and Adding (37) and (38), we get Substituting the value of $(c_1^2+d_1^2)$ from (40), we obtain Applying for the coefficients $c_2$ and $d_2$ along the equation (21), we obtain The subtraction of equation (38) from equation (37) yields the following result: Then, in view of (39) and (40), Eq. (43) becomes Thus applying (21), we conclude that This completes the proof of Theorem. □

Given that Φ is an element of Σ defined by (1) and belongs to the class ${\mathcal{C}}_{\Sigma }(\lambda ,\beta ;q)$, and ϵ is a real number, we can state the following where

If $\mathsf{\Phi }\in {\mathcal{C}}_{\Sigma }(\lambda ,\beta ;q)$ is given by (1), from (42) and (43), we have where Then, we conclude that Which completes the proof of Theorem 4. □

If Φ is an element of Σ defined by (1), it can be said that Φ is a member of the class ${\mathcal{S}}_{\Sigma }^{*}(\lambda ,\beta )$, as per the following statement: and