**1. Introduction**

The mathematical study of *q*-calculus, particularly *q*-fractional calculus and *q*-integral calculus, *q*-transform analysis has been a topic of grea<sup>t</sup> interest for researchers due to its wide applications in different fields (see [1,2]). Some of the earlier work on the applications of the *q*-calculus was introduced by Jackson [3,4]. Later, *q*-analysis with geometrical interpretation was turned into identified through quantum groups. Due to the applications of *q*-analysis in mathematics and other fields, numerous researchers [3,5–14] did some significant work on *q*-calculus and studied its several other applications. Recently, Srivastava [15] in his survey-cum-expository article, explored the mathematical application of *q*-calculus, fractional *q*- calculus and fractional *q*-differential operators in geometric function theory. Keeping in view the significance of *q*-operators instead of ordinary operators and due to the wide range of applications of *q*-calculus, many researchers comprehensively studied *q*-calculus such as Srivastava et al. [16], Muhammad and Darus [17], Kanas and Reducanu [18] and Muhammad and Sokol [19]. Motivated by [15–21], we consider subfamilies of *q*-convex functions and *q*-close to convex functions with respect to Janowski functions connected with *q*-conic domain.

Let A be the class of functions of the form

$$f(z) = z + \sum\_{n=2}^{\infty} a\_n z^n, \quad a\_n \in \mathbb{C}, z \in \mathcal{U}, \tag{1}$$

which are analytic in the open unit disk *U* = {*z* : *z* ∈ C, |*z*| < <sup>1</sup>}. Let A⊇S, where S represents the set of all univalent functions in *U*. The classes of starlike (*S*∗) and convex (*C*) functions in *U* are the well known subclasses of *S*. Moreover, the class *K* of close to convex functions in *U* consists of normalized functions *f* ∈ A that satisfy the following conditions:

$$f \in \mathcal{A} \quad \text{and} \quad \text{Re}\left(\frac{zf'(z)}{\mathcal{g}(z)}\right) > 0, \quad \text{where} \quad \mathcal{g}(z) \in \mathcal{S}^\*.$$

Now, for *κ* ≥ 0, the classes *κ*-uniformly convex mappings (*κ* − *UCV*) and *κ*-starlike mappings (*κ* − *UST*), explored by Kanas and Wi´sniowska, see [22–28]. Kanas and Wi´sniowska [22,23] also initiated the study of analytic functions on conic domain Ω*<sup>κ</sup>*, *κ* ≥ 0 as:

$$\Omega\_{\mathbb{K}} = \left\{ u + iv : u > \kappa \sqrt{\left(u - 1\right)^2 + v^2} \right\}$$

See [22,23] for geometric interpretation of Ω*<sup>κ</sup>*. These conic regions are images of the unit disk under the extremal functions *hκ* (*z*) given by:

$$h\_{\mathbf{k}}(z) = \begin{cases} \frac{1+z}{1-z} & \mathbf{x} = \mathbf{0}, \\ 1 + \left(\log\frac{\sqrt{\pi}+1}{1-\sqrt{z}}\right)^2 \frac{2}{\pi^2} & \mathbf{x} = 1, \\ 1 + \sinh^2\left\{\arctan h\sqrt{z}\left(\frac{2}{\pi}\arccos\kappa\right)\right\} \frac{2}{1-\kappa^2} & 0 < \mathbf{x} < 1, \\ 1 + \frac{1}{\mathbf{x}^2 - 1}\sin\left(\frac{\pi}{2\mathcal{R}(y)}\int\_0^{\frac{\mu(z)}{\sqrt{y}}} \frac{\mathbf{dx}}{\sqrt{1-\mathbf{x}^2}\sqrt{1-y^2\mathbf{x}^2}}\right) + \frac{1}{\kappa^2 - 1} & \kappa > 1, \end{cases} \tag{2}$$

.

where

$$
\mu(z) = \frac{z - \sqrt{y}}{1 - \sqrt{y}z}, \ z \in \mathcal{U}.
$$

Here, *κ* = cosh (*πR*(*y*)/(4*R*(*y*))) ∈ (0, <sup>1</sup>), where *<sup>R</sup>*(*y*) is Legendre's complete elliptic integral of first kind and *<sup>R</sup>*(*y*) = *R*(1 − *y*2) is its complementary integral, see [22,23,29–34]. If *hκ* (*z*) = 1 + *δ* (*κ*) *z* + *δ*1 (*κ*) *z*2 + ··· is taken from [23] for (2), then

$$\delta\left(\kappa\right) = \begin{cases} \frac{8\left(\arccos\kappa\right)^2}{\pi^2 \left(1 - \kappa^2\right)} & 0 \le \kappa < 1, \\\frac{8}{\pi^2} & \kappa = 1, \\\frac{\pi^2}{4\sqrt{y}(\kappa^2 - 1)R^2(y)(1+y)} & \kappa > 1, \end{cases} \tag{3}$$

$$\delta\_1\left(\kappa\right) = \delta\_2\left(\kappa\right)\delta\left(\kappa\right),$$

where

$$\delta\_2\left(\kappa\right) = \begin{cases} \frac{\mathcal{T}\_1^2 + 2}{3} & 0 \le \kappa < 1, \\\frac{2}{3} & \kappa = 1, \\\frac{4\mathcal{R}^2(y)\left(y^2 + 6y + 1\right) - \pi^2}{24\mathcal{R}^2(y)\left(1 + y\right)\sqrt{\mathcal{Y}}} & \kappa > 1, \end{cases} \tag{4}$$

where T1 = 2*π* arccos *κ*, and *y* ∈ (0, <sup>1</sup>).

**Definition 1.** *([35]) Let p* ∈ A *and p*(0) = 1 *be in the class* P (*<sup>λ</sup>*, *α*) *if and only if*

$$p\left(z\right) \prec \frac{1+\lambda z}{1+\alpha z}, \quad (-1 \le \alpha < \lambda \le 1),$$

*where* ≺ *stands for subordination.* *Mathematics* **2020**, *8*, 440

Janowski [35] initiated the class P (*<sup>λ</sup>*, *α*) by showing that *p* ∈ P (*<sup>λ</sup>*, *α*) if and only if there exists a mapping *p* ∈ P such that

$$\frac{p\left(z\right)\left(\lambda+1\right)-\left(\lambda-1\right)}{p\left(z\right)\left(\kappa+1\right)-\left(\alpha-1\right)} \prec \frac{1+\lambda z}{1+\alpha z}.$$

where P is class the of mappings with non-negative real parts.

**Definition 2.** *([36]) Let function f* ∈ A *be in the class* S∗ (*<sup>λ</sup>*, *α*) *if and only if*

$$\frac{z f'(z)}{f(z)} = \frac{p\left(z\right)\left(\lambda + 1\right) - \left(\lambda - 1\right)}{p\left(z\right)\left(\alpha + 1\right) - \left(\alpha - 1\right)}, \quad \left(-1 \le \alpha < \lambda \le 1\right).$$

**Definition 3.** *([36]) Let function f* ∈ A *is in the class* C (*<sup>λ</sup>*, *α*) *if and only if*

$$\frac{\left(zf'(z)\right)'}{\left(f(z)\right)'} = \frac{p\left(z\right)\left(\lambda + 1\right) - \left(\lambda - 1\right)}{p\left(z\right)\left(\alpha + 1\right) - \left(\alpha - 1\right)'} \quad \left(-1 \le \alpha < \lambda \le 1\right).$$

**Definition 4.** *([7]) Let function f* ∈ A*, n* ∈ N0 *and q* ∈ (0, <sup>1</sup>)*, the q-difference* (*or q* − *derivative*) *operator Dq is defined as:*

$$D\_q f(z) = -\frac{f(z) - f(qz)}{(q-1)z}.$$

*Note that*

$$D\_q z^n = [n]\_q z^{n-1}, \; D\_q \left\{ \sum\_{n=1}^{\infty} a\_n z^n \right\} = \sum\_{n=1}^{\infty} [n]\_q a\_n z^{n-1},$$

*where*

$$[n]\_q = \frac{1 - q^n}{1 - q}.$$

**Definition 5.** *([37]) Let function f* ∈ A *is in the class* S∗*q* (*<sup>λ</sup>*, *α*) *if and only if*

$$\frac{zD\_q f(z)}{f(z)} = \frac{(\lambda + 1)\widetilde{p}\left(z\right) - (\lambda - 1)}{(a + 1)\,\widetilde{p}\left(z\right) - (a - 1)}, \quad \left(-1 \le a < \lambda \le 1\right), \; q \in \left(0, 1\right).$$

*By principle of subordination we can be written as follows:*

$$\frac{zD\_{\emptyset}f(z)}{f(z)} \prec \frac{(\lambda+1)z+2+(\lambda-1)\,qz}{\left(\alpha+1\right)z+2+\left(\alpha-1\right)\,qz},$$

*where*

$$
\widetilde{p}\left(z\right) = \frac{1+z}{1-qz}.
$$

**Definition 6.** *([37]) Let function f* ∈ A *is in class* C*q* (*<sup>λ</sup>*, *α*) *if and only if*

$$\frac{D\_q\left(zD\_qf(z)\right)}{D\_qf(z)} = \frac{\widetilde{p}\left(z\right)\left(\lambda + 1\right) - \left(\lambda - 1\right)}{\widetilde{p}\left(z\right)\left(a + 1\right) - \left(a - 1\right)}, \quad \left(-1 \le a < \lambda \le 1\right), \quad q \in \left(0, 1\right)\dots$$

*Similarly, by principle of subordination, we can be written as follows:*

$$\frac{D\_q\left(zD\_qf(z)\right)}{D\_qf(z)} \prec \frac{z(\lambda+1) + (\lambda-1)\,qz+2}{z\left(\alpha+1\right) + \left(\alpha-1\right)\,qz+2}.$$

Mahmood et al. [38] introduced the class *k* − P*q* (*<sup>λ</sup>*, *α*) as: **Definition 7.** *([38]) A function h* ∈ *k* − P*q* (*<sup>λ</sup>*, *<sup>α</sup>*)*, if and only if*

$$h\left(z\right) \prec \frac{\left(\lambda O\_1 + O\_3\right) h\_k\left(z\right) - \left(\lambda O\_1 - O\_3\right)}{\left(\alpha O\_1 + O\_3\right) h\_k\left(z\right) - \left(\alpha O\_1 - O\_3\right)}, \quad k \ge 0, \ q \in \left(0, 1\right),$$

*where*

$$O\_1 = 1 + q \text{ and } O3 = 3 - q.$$

In addition, *hk*(*z*) is defined in Label (2). Geometrically, the mapping *h* ∈ *k* − P*q* (*<sup>λ</sup>*, *α*) takes all domain values <sup>Ω</sup>*k*,*<sup>q</sup>* (*<sup>λ</sup>*, *<sup>α</sup>*), 1 ≤ *α* < *λ* ≤ 1, *k* ≥ 0, which is definable as:

$$\Omega\_{k,q}(\lambda,\mathfrak{a}) = \{r = u + iv : \Re\left(\Psi\right) > k\left|\Psi - 1\right|\},$$

where

$$\Psi = \frac{\left(\varkappa O\_1 - O\_3\right)r\left(z\right) - \left(\lambda O\_1 - O\_3\right)}{\left(\varkappa O\_1 + O\_3\right)r\left(z\right) - \left(\lambda O\_1 + O\_3\right)}.$$

This domain describes the conic type domain; for details, see [38]. Note that


**Definition 8.** *([38]) Let f* ∈ A *be in the class k* −ST *<sup>q</sup>*(*β*, *<sup>γ</sup>*)*, if and only if*

$$\begin{aligned} &\Re\left(\frac{\left(\gamma O\_1 - O\_3\right)\frac{zD\_qf(z)}{f(z)} - \left(\beta O\_1 - O\_3\right)}{\left(\gamma O\_1 + O\_3\right)\frac{zD\_qf(z)}{f(z)} - \left(\beta O\_1 + O\_3\right)}\right) \\ &> k \left|\frac{\left(\gamma O\_1 - O\_3\right)\frac{zD\_qf(z)}{f(z)} - \left(\beta O\_1 - O\_3\right)}{\left(\gamma O\_1 + O\_3\right)\frac{zD\_qf(z)}{f(z)} - \left(\beta O\_1 + O\_3\right)} - 1\right|.\end{aligned}$$

*or, equivalently,*

$$\frac{zD\_q f(z)}{f(z)} \in k - \mathcal{P}\_q(\beta, \gamma)\_\*$$

*where k* ≥ 0, −1 ≤ *γ* < *β* ≤ 1.

We can see that, when *q* → 1, then *κ* − ST *<sup>q</sup>*(*β*, *γ*) diminishes to the renowned class which is stated in [39].

Motivated by the definition above, we introduced new classes *κ* −UCV*q*(*β*, *<sup>γ</sup>*), *κ* −UK*q*(*<sup>λ</sup>*, *α*, *β*, *γ*) and *κ* −UQ*q*(*<sup>λ</sup>*, *α*, *β*, *γ*) of analytic functions.

**Definition 9.** *Let f* ∈ A*, be in the class k* − UCV*q*(*β*, *γ*) *if and only if*

$$\begin{split} & \left| \Re \left( \frac{\left( \gamma O\_{1} - O\_{3} \right) \frac{D\_{4} \left( z D\_{4} f(z) \right)}{D\_{4} f(z)} - \left( \beta O\_{1} - \left( O\_{3} \right) \right)}{\left( \gamma O\_{1} + O\_{3} \right) \frac{D\_{4} \left( z D\_{4} f(z) \right)}{D\_{4} f(z)} - \left( \beta O\_{1} + O\_{3} \right)} \right) \right| \\ & \left| \frac{\left( \gamma O\_{1} - O\_{3} \right) \frac{D\_{4} \left( z D\_{4} f(z) \right)}{D\_{4} f(z)} - \left( \beta O\_{1} - O\_{3} \right)}{\left( \gamma O\_{1} + O\_{3} \right) \frac{D\_{4} \left( z D\_{4} f(z) \right)}{D\_{4} f(z)} - \left( \beta O\_{1} + O\_{3} \right)} - 1 \right|, \end{split}$$

*Mathematics* **2020**, *8*, 440

*or, equivalently,*

$$\frac{D\_q\left(zD\_q f(z)\right)}{D\_\emptyset f(z)} \in k - \mathcal{P}\_q(\beta, \gamma)\_\prime$$

*where k* ≥ 0, −1 ≤ *γ* < *β* ≤ 1.

> One can clearly see that

$$f \in \mathbb{x} - \mathcal{U}\mathcal{C}\mathcal{V}\_q(\boldsymbol{\beta}, \boldsymbol{\gamma}) \Leftrightarrow \boldsymbol{z}D\_q(\boldsymbol{z}) \in \mathbb{x} - \mathcal{S}\mathcal{T}\_q(\boldsymbol{\beta}, \boldsymbol{\gamma}).\tag{5}$$

Note that, when *q* → 1, then the class *κ* − UCV*q*(*β*, *γ*) reduces to a well-known class defined in [39].

**Definition 10.** *Let f* ∈ A*, be in the class k* −UK*q*(*<sup>λ</sup>*, *α*, *β*, *γ*) *if and only if there exists g* ∈ *k* −ST *<sup>q</sup>*(*β*, *<sup>γ</sup>*), *such that*

$$\begin{aligned} \Re \left( \frac{(aO\_1 - O\_3) \frac{zD\_4 f(z)}{\mathcal{S}(z)} - (\lambda O\_1 - O\_3)}{(aO\_1 + O\_3) \frac{zD\_4 f(z)}{\mathcal{S}(z)} - (\lambda O\_1 + O\_3)} \right) \\\\ > k \left| \frac{(aO\_1 - O\_3) \frac{zD\_4 f(z)}{\mathcal{S}(z)} - (\lambda O\_1 - O\_3)}{(aO\_1 + O\_3) \frac{zD\_4 f(z)}{\mathcal{S}(z)} - (\lambda O\_1 + O\_3)} - 1 \right|. \end{aligned}$$

*We can write equivalently*

$$\frac{zD\_q f(z)}{g(z)} \in k - \mathcal{P}\_q(\lambda, a),$$

*where k* ≥ 0, −1 ≤ *γ* < *β* ≤ 1, −1 ≤ *α* < *λ* ≤ 1.

Note that, when *q* → 1, then, the class *k* −UK*q*(*<sup>λ</sup>*, *α*, *β*, *γ*) reduces into the well-known class that is defined in (see [40]).

**Definition 11.** *Let f* ∈ A*, belong to the class k* −UQ*q*(*<sup>λ</sup>*, *α*, *β*, *γ*) *if and only if there exist g* ∈ *k* −CV*q*(*β*, *<sup>γ</sup>*), *such that*

$$\begin{aligned} \Re \left( \frac{\left( aO\_1 - O\_3 \right) \frac{D\_q \left( zD\_q f(z) \right)}{D\_q \mathfrak{g}(z)} - \left( \lambda O\_1 - O\_3 \right)}{\left( aO\_1 + O\_3 \right) \frac{D\_q \left( zD\_q f(z) \right)}{D\_q \mathfrak{g}(z)} - \left( \lambda O\_1 + O\_3 \right)} \right) \\\\ > k \left| \frac{\left( aO\_1 - O\_3 \right) \frac{D\_q \left( zD\_q f(z) \right)}{D\_q \mathfrak{g}(z)} - \left( \lambda O\_1 - O\_3 \right)}{\left( aO\_1 + O\_3 \right) \frac{D\_q \left( zD\_q f(z) \right)}{D\_q \mathfrak{g}(z)} - \left( \lambda O\_1 + O\_3 \right)} - 1 \right|, \end{aligned}$$

*or, equivalently,*

$$\frac{D\_{\mathfrak{q}}\left(zD\_{\mathfrak{q}}f(z)\right)}{D\_{\mathfrak{q}}\mathfrak{g}(z)} \in k - \mathcal{P}\_{\mathfrak{q}}(\lambda, \mathfrak{a})\_{\mathfrak{q}}$$

*where, for k* ≥ 0, −1 ≤ *γ* < *β* ≤ 1, −1 ≤ *α* < *λ* ≤ 1.

> It is simple to verify this

$$f \in \kappa - \mathcal{U}\mathcal{Q}\_q(\lambda, \mathfrak{a}, \mathfrak{b}, \gamma) \Leftrightarrow \mathfrak{z}D\_q f \in \kappa - \mathcal{U}\mathcal{K}\_q(\lambda, \mathfrak{a}, \mathfrak{b}, \gamma). \tag{6}$$

*Mathematics* **2020**, *8*, 440

A special case arises when *q* → 1, then the class *κ* −UQ*q*(*<sup>λ</sup>*, *α*, *β*, *γ*) reduces to a well known class defined in [40].

#### **2. Set of Lemmas**

**Lemma 1.** *([41]) Suppose* 1 + ∑∞*<sup>n</sup>*=<sup>1</sup> *cnz<sup>n</sup>* = *d*(*z*) ≺ *<sup>H</sup>*(*z*) = 1 + ∑∞*<sup>n</sup>*=<sup>1</sup> *Cnzn*. *If H*(*U*) *is convex and <sup>H</sup>*(*z*) ∈ A*, then*

$$|c\_n| \le |\mathbb{C}\_1|\_\prime, \quad n \ge 1.$$

**Lemma 2.** *([38]) Suppose d*(*z*) = 1 + ∑∞*<sup>n</sup>*=<sup>1</sup> *cnz<sup>n</sup>* ∈ *k* − <sup>P</sup>*q*(*<sup>λ</sup>*, *<sup>α</sup>*), *then*

$$|c\_n| \le |\delta\left(k, \lambda, \mathfrak{a}\right)| = \frac{O\_1(\lambda - \mathfrak{a})}{4} \delta(k),$$

*where δ* (*k*) *is given by (3).*

**Lemma 3.** *([38]) Suppose d* ∈ *k* −ST *<sup>q</sup>*(*β*, *<sup>γ</sup>*), *k* ≥ 0 *is given by*

$$d(z) = z + \sum\_{n=2}^{\infty} b\_n z^n, \quad z \in \mathcal{U}\_\prime$$

*then*

$$|b\_{\mathfrak{n}}| \le \prod\_{m=0}^{n-2} \left( \frac{\left| \delta(k) O\_1(\beta - \gamma) - 4q \left[ m \right]\_q \gamma \right|}{4q \left[ m + 1 \right]\_q} \right) \lambda$$

*where δ*(*k*) *is given by (3).*

**Lemma 4.** *([42]) Suppose d* ∈ S∗*, f* ∈ C *and G* ∈ S*, then we have*

$$\frac{f(z) \* d(z)G(z)}{f(z) \* d(z)} \in \overline{c\rho}(G(\mathcal{U})), \quad z \in \mathcal{U}.$$

*Here, "\*" means convolution and co*(*G*(*U*) *means the closed convex hull <sup>G</sup>*(*U*)*.*

**Lemma 5.** *([38]) The function f* ∈ A *will belong to the class k* −ST *<sup>q</sup>*(*β*, *<sup>γ</sup>*), *if the following inequality holds:*

$$\begin{aligned} &\sum\_{n=2}^{\infty} \left\{ 2O\_3(1+k)q\left[n-1\right]\_q + \left| \left(\gamma O\_1 + O\_3\right)\left[n\right]\_q - \left(\beta O\_1 + \left(O\_3\right)\right) \right| \right\} |a\_n| \\ &\le O\_1\left|\gamma - \beta\right|. \end{aligned}$$

*Throughout this paper, we assume that k* ≥ 0, −1 ≤ *γ* < *β* ≤ 1, −1 ≤ *α* < *λ* ≤ 1, *and q* ∈ (0, <sup>1</sup>), *unless otherwise specified.*

#### **3. Main Results**

**Theorem 1.** *Let f* ∈ A*; then, f is in the class k* − UCV*q*(*β*, *<sup>γ</sup>*), *if the following inequality holds:*

$$\begin{aligned} &\sum\_{n=2}^{\infty} \left[ n \right]\_q \left\{ 2O\_3(k+1)q \left[ n - 1 \right]\_q + \left\lfloor \left( \gamma O\_1 + O\_3 \right) \left[ n \right]\_q - \left( \beta O\_1 + O\_3 \right) \right\rfloor \right\} |a\_n| \\ &\le O\_1 \left| \gamma - \beta \right|. \end{aligned}$$

**Proof.** By Lemma 5 and relation (5), the proof is straightforward.

For *q* → 1<sup>−</sup>, in Theorem 1, then we obtained following corollary, proved by Malik and Noor [39]. **Corollary 1.** *Let f* ∈ A*; then, f belongs to k* − UCV(*β*, *<sup>γ</sup>*), *if the following inequality holds*

$$\sum\_{n=2}^{\infty} n \left\{ 2(k+1)\left(n-1\right) + \left| n\left(\gamma+1\right) - \left(\beta+1\right) \right| \right\} \left| a\_n \right| \le \left| \gamma - \beta \right| $$

**Theorem 2.** *Let f* ∈ A*, then f is in the class k* −UK*q*(*<sup>λ</sup>*, *α*, *β*, *<sup>γ</sup>*), *if the condition (7) holds*

$$\sum\_{n=2}^{\infty} \left\{ 2O\_3(k+1) \left| b\_n - \left[ n \right]\_q a\_n \right| + \left| \left( aO\_1 + O\_3 \right) \left[ n \right]\_q a\_n - \left( \lambda O\_1 + O\_3 \right) b\_n \right| \right\} $$

$$\leq O\_1 \left| a - \lambda \right|. \tag{7}$$

.

**Proof.** Presuming that (7) holds, then it is enough to show that

$$\begin{aligned} k \left\lvert \frac{\left(\alpha O\_1 - O\_3\right) \frac{zD\_4 f(z)}{\mathfrak{E}(z)} - \left(\lambda O\_1 - O\_3\right)}{\left(\alpha O\_1 + O\_3\right) \frac{zD\_4 f(z)}{\mathfrak{E}(z)} - \left(\lambda O\_1 + O\_3\right)} - 1 \right\rvert \\\\ - \operatorname{Re} \left\{ \frac{\left(\alpha O\_1 - O\_3\right) \frac{zD\_4 f(z)}{\mathfrak{E}(z)} - \left(\lambda O\_1 - O\_3\right)}{\left(\alpha O\_1 + O\_3\right) \frac{zD\_4 f(z)}{\mathfrak{E}(z)} - \left(\lambda O\_1 + O\_3\right)} - 1 \right\} \\\\ < 1. \end{aligned}$$

We have

$$\begin{aligned} &k \left| \frac{\left(\varkappa O\_1 - O\_3\right) \frac{zD\_df(z)}{\mathcal{S}(z)} - \left(\lambda O\_1 - O\_3\right)}{\left(\varkappa O\_1 + O\_3\right) \frac{zD\_df(z)}{\mathcal{S}(z)} - \left(\lambda O\_1 + O\_3\right)} - 1\right| \\\\ &- \text{Re}\left\{ \frac{\left(\varkappa O\_1 - O\_3\right) \frac{zD\_df(z)}{\mathcal{S}(z)} - \left(\lambda O\_1 - O\_3\right)}{\left(\varkappa O\_1 + O\_3\right) \frac{zD\_df(z)}{\mathcal{S}(z)} - \left(\lambda O\_1 + O\_3\right)} - 1 \right\}, \end{aligned}$$

$$\leq (k+1) \left| \frac{(aO\_1 - O\_3) \frac{zD\_tf(z)}{\frac{g(z)}{g(z)}} - (\lambda O\_1 - O\_3)}{(aO\_1 + O\_3) \frac{zD\_tf(z)}{g(z)} - (\lambda O\_1 + O\_3)} - 1 \right|, \tag{8}$$

$$\begin{aligned} \mathcal{I} &= 2O\_3(k+1) \left| \frac{g(z) - zD\_q f(z)}{\left(aO\_1 + O\_3\right) zD\_q f(z) - \left(\lambda O\_1 + O\_3\right) g(z)} \right|, \\\\ &= 2O\_3(k+1) \left| \frac{\sum\_{n=2}^{\infty} \left\{ b\_n - \left[n\right]\_q a\_{\text{fl}} \right\} z^n}{O\_1\left(\kappa - \lambda\right) z + \sum\_{n=2}^{\infty} \left\{ \left(aO\_1 + O\_3\right) \left[n\right]\_q a\_{\text{fl}} - \left(\lambda O\_1 + O\_3\right) b\_{\text{fl}} \right\} z^n} \right|, \end{aligned}$$

$$\leq \frac{2O\_3(k+1)\sum\_{n=2}^{\infty} \left\{ \left| b\_{\mathcal{U}} - [n]\_q a\_{\mathcal{U}} \right| \right\}}{O\_1\left| a - \lambda \right| - \sum\_{n=2}^{\infty} \left| (aO\_1 + O\_3) \left[ n \right]\_q a\_{\mathcal{U}} - (\lambda O\_1 + O\_3) b\_{\mathcal{U}} \right|}.$$

The expression (8) is bounded above by 1 if

$$\begin{aligned} &\sum\_{n=2}^{\infty} \left[ 2O\_3(k+1) \left| b\_{\text{ll}} - \left[ n \right]\_q a\_{\text{ll}} \right| + \left| (aO\_1 + O\_3) \left[ n \right]\_q a\_{\text{ll}} - (\lambda O\_1 + O\_3) b\_{\text{ll}} \right| \right] \\ &\le \left( O\_1 \right) \left| \alpha - \lambda \right|. \end{aligned}$$

**Corollary 2.** *([40]) Let f* ∈ A*. Then, f is in the class k* − UK*q*→<sup>1</sup>(*<sup>λ</sup>*, *α*, *β*, *γ*) = *k* − UK(*<sup>λ</sup>*, *α*, *β*, *<sup>γ</sup>*)*, if the following condition holds:*

$$\sum\_{n=2}^{\infty} \left\{ 2(k+1) \left| b\_{\text{ll}} - n a\_{\text{ll}} \right| + \left| (n+1) n a\_{\text{ll}} - (\lambda + 1) b\_{\text{ll}} \right| \right\} \le \left| \alpha - \lambda \right|.$$

*Here, q* → 1 *represents the limiting value of q as it approaches* 1*.*

**Theorem 3.** *Let f* ∈ A*. Then, f is in the class k* −UQ*q*(*<sup>λ</sup>*, *α*, *β*, *<sup>γ</sup>*), *if the following condition holds:*

$$\begin{aligned} &\sum\_{n=2}^{\infty} \left[ n \right]\_q \left[ 2O\_3(k+1) \left| b\_n - \left[ n \right]\_q a\_n \right| + \left| \left( aO\_1 + O\_3 \right) \left[ n \right]\_q a\_n - \left( \lambda O\_1 + O\_3 \right) b\_n \right| \right] \\ &\le O\_1 \left| \alpha - \lambda \right| . \end{aligned}$$

**Proof.** By Theorem 2 and relation (6), the proof is straightforward.

**Corollary 3.** *([40]) Let f* ∈ A*. Then, f is in the class k* −UK*q*→<sup>1</sup>(*<sup>λ</sup>*, *α*, *β*, *γ*) = *k* −UQ(*<sup>λ</sup>*, *α*, *β*, *<sup>γ</sup>*), *if*

$$\sum\_{n=2}^{\infty} n \left\{ 2(k+1) \left| b\_{\text{ll}} - n a\_{\text{ll}} \right| + \left| (\alpha + 1) n a\_{\text{ll}} - (\lambda + 1) b\_{\text{ll}} \right| \right\} \le \left| \alpha - \lambda \right|.$$

.

**Corollary 4.** *([43]) Let f* ∈ A*. Then, f is in the class* 1 −UK*q*→<sup>1</sup>(<sup>1</sup> − 2*τ*, −1, 1, −<sup>1</sup>) = UK(*τ*) *if*

$$\sum\_{n=2}^{\infty} n^2 \left| a\_n \right| \le \frac{1-\tau}{2}.$$

.

**Theorem 4.** *Let f* ∈ *k* − UCV*q*(*β*, *<sup>γ</sup>*), *is of the form* (1)*. Then,*

$$|a\_n| \le \frac{1}{[n]\_q} \Pi\_{m=0}^{n-2} \left( \frac{\left| \delta(k) O\_1(\beta - \gamma) - 4q \left[ m \right]\_q \gamma \right|}{4q \left[ m + 1 \right]\_q} \right) \dots$$

*where δ*(*k*) *is given by (3).*

**Proof.** By Lemma 3 and relation (5), the proof is straightforward.

For *q* → 1<sup>−</sup>, Theorem 4 brings to the following corollary, proved by Noor [39].

**Corollary 5.** *Let f* ∈ *k* − UCV(*β*, *<sup>γ</sup>*)*. Then,*

$$|a\_{\mathcal{U}}| \le \frac{1}{n} \Gamma \Gamma\_{m=0}^{n-2} \left( \frac{|\delta(k)(\beta - \gamma) - 2m\gamma|}{2\left(m + 1\right)} \right),$$

*where δ*(*k*) *is given by (3).* **Theorem 5.** *If f* ∈ *k* −UK*q*(*<sup>λ</sup>*, *α*, *β*, *γ*) *and g* ∈ *k* −ST *<sup>q</sup>*(*β*, *<sup>γ</sup>*)*, then,*

$$|a\_{\mathrm{il}}| \leq \begin{cases} \frac{1}{[n]\_q} \prod\_{m=0}^{n-2} \frac{\left| \delta(k) O\_1(\beta - \gamma) - 4q[m]\_q \gamma \right|}{4q[m+1]\_q} \\\\ + \frac{\delta(k) O\_1(\lambda - a)}{4[n]\_q} \sum\_{j=1}^{n-1} \prod\_{m=0}^{j-2} \frac{\left| \delta(k) O\_1(\beta - \gamma) - 4q[j]\_q \gamma \right|}{4q[j+1]\_q}, & n \geq 2, \end{cases}$$

*where δ*(*k*) *is given in (3).*

**Proof.** Let us take

$$\frac{zD\_{\emptyset}f(z)}{g(z)} = h(z),\tag{9}$$

where

$$h \in k - \mathcal{P}\_q(\lambda, \alpha) \text{ and } \mathfrak{g} \in k - \mathcal{ST}\_q(\beta, \gamma).$$

Now, from (9), we have

$$zD\_q f(z) = \mathfrak{g}(z)h(z),$$

which implies that

$$z + \sum\_{n=2}^{\infty} [n]\_q a\_n z^n = \left(1 + \sum\_{n=1}^{\infty} c\_n z^n\right) \left(z + \sum\_{n=2}^{\infty} b\_n z^n\right) \dots$$

By equating *z<sup>n</sup>* coefficients

$$\begin{aligned} \left[n\right]\_q a\_n = b\_n + \sum\_{j=1}^{n-1} b\_j c\_{n-j}, \ a = 1, \ b\_1 = 1. \end{aligned} $$

This implies that

$$\left| \left[ n \right]\_q \left| a\_n \right| \le \left| b\_n \right| + \sum\_{j=1}^{n-1} \left| b\_j \right| \left| c\_{n-j} \right| \,. \tag{10}$$

Since *h* ∈ *k* − <sup>P</sup>*q*(*<sup>λ</sup>*, *<sup>α</sup>*), therefore, by using Lemma 2 on (10), we have

$$\left| [n]\_q \left| a\_{\boldsymbol{n}} \right| \leq \left| b\_{\boldsymbol{n}} \right| + \frac{\delta(k) O\_1(\lambda - a)}{4} \sum\_{j=1}^{n-1} \left| b\_j \right|. \tag{11}$$

Again *g* ∈ *k* −ST *<sup>q</sup>*(*β*, *<sup>γ</sup>*), therefore, by using Lemma 3 on (11), we have

$$|a\_{ll}| \leq \begin{cases} \frac{1}{\left[n\right]\_q} \prod\_{m=0}^{n-2} \left( \frac{\left[\delta(k)O\_1(\beta-\gamma)-4q[m]\_q\gamma\right]}{4q[m+1]\_q} \right) \\\\ + \frac{\delta(k)O\_1(\lambda-n)}{4\left[n\right]\_q} \sum\_{j=1}^{n-1} \prod\_{m=0}^{j-2} \left( \frac{\left[\delta(k)O\_1(\beta-\gamma)-4q[m]\_q\gamma\right]}{4q[m+1]\_q} \right) \end{cases}$$

**Corollary 6.** *([40]) If f* ∈ *k* −UK*q*→<sup>1</sup>(*<sup>λ</sup>*, *α*, *β*, *γ*) = *k* −UK(*<sup>λ</sup>*, *α*, *β*, *<sup>γ</sup>*)*, then*

$$|a\_{n}| \leq \begin{cases} \frac{1}{n} \prod\_{m=0}^{n-2} \left( \frac{|\delta(k)(\beta-\gamma)-2m\gamma|}{2(m+1)} \right) \\\\ + \frac{\delta(k)(\lambda-a)}{2n} \sum\_{j=1}^{n-1} \prod\_{m=0}^{j-2} \left( \frac{|\delta(k)(\beta-\gamma)-2m\gamma|}{2(m+1)} \right), n \geq 2, \end{cases}$$

*Mathematics* **2020**, *8*, 440

where *δ*(*k*) is defined by (3).

**Corollary 7.** *([26]) If f* ∈ *k* −UK*q*→<sup>1</sup>(1, −1, 1, −<sup>1</sup>) = *k* −UK*, then*

$$|a\_n| \le \frac{(\delta(k))\_{n-1}}{n!} + \frac{\delta(k)}{n} \Sigma\_{j=0}^{n-1} \frac{(\delta(k))\_{j-1}}{(j-1)!}, \ n \ge 2.1$$

**Corollary 8.** *([44]) If f* ∈ 0 −UK*q*→<sup>1</sup>(1, −1, 1, −<sup>1</sup>) = K*, then*

$$|a\_n| \le n, \quad n \ge 2.$$

**Theorem 6.** *If f* ∈ *k* −UQ*q*(*<sup>λ</sup>*, *α*, *β*, *<sup>γ</sup>*)*, then*

$$|a\_{\mathrm{il}}| \leq \begin{cases} \frac{1}{\left( [n]\_{q} \right)^{2}} \prod\_{m=0}^{n-2} \frac{\left| \delta(k) O\_{1}(\beta-\gamma) - 4q[m]\_{q} \gamma \right|}{4q[m+1]\_{q}} \\\\ + \frac{\delta(k) O\_{1}(\lambda-\alpha)}{4\left( [n]\_{q} \right)^{2}} \sum\_{j=1}^{n-1} \prod\_{m=0}^{j-2} \frac{\left| \delta(k) O\_{1}(\beta-\gamma) - 4q[j]\_{q} \gamma \right|}{4q[j+1]\_{q}}, & n \geq 2, \alpha \end{cases}$$

*where δ*(*k*) *is defined by (3).*

**Proof.** By Theorem 5 and relation (6), the proof is straightforward.

**Corollary 9.** *([40]) If f* ∈ *k* −UQ*q*→<sup>1</sup>(*<sup>λ</sup>*, *α*, *β*, *γ*) = UQ(*<sup>λ</sup>*, *α*, *β*, *γ*) *and is of the form* (1)*, then*

$$|a\_{n}| \leq \begin{cases} \frac{1}{n^{2}} \prod\_{m=0}^{n-2} \left( \frac{|\delta(k)(\beta-\gamma)-2m\gamma|}{2(m+1)} \right) \\\\ + \frac{\delta(k)(\lambda-a)}{2n^{2}} \sum\_{j=1}^{n-1} \prod\_{m=0}^{j-2} \left( \frac{|\delta(k)(\beta-\gamma)-2m\gamma|}{2(m+1)} \right), n \geq 2. \end{cases}$$

**Theorem 7.** *If f* ∈ *k* − <sup>P</sup>*q*(*β*, *γ*) *and χ* ∈ C*, then f* ∗ *χ* ∈ *k* − <sup>P</sup>*q*(*β*, *<sup>γ</sup>*).

**Proof.** Here, we prove that

$$\frac{zD\_q\left(\chi(z)\*f(z)\right)}{\left(\chi(z)\*f(z)\right)} \in k - \mathcal{P}\_q(\beta, \gamma).$$

Consider

$$\begin{aligned} \frac{z D\_q\left(\chi(z) \ast f(z)\right)}{\left(\chi(z) \ast f(z)\right)} &= \frac{\chi(z) \ast f(z) \left(\frac{z D\_q f(z)}{f(z)}\right)}{\chi(z) \ast f(z)},\\ &= \frac{\chi(z) \ast f(z) \Psi(z)}{\chi(z) \ast f(z)}, \end{aligned}$$

where *zDq f*(*z*) *f*(*z*) = <sup>Ψ</sup>(*z*) ∈ <sup>P</sup>*q*(*β*, *<sup>γ</sup>*). By using Lemma 4, we obtain the required result.

**Theorem 8.** *If f* ∈ *k* −UK*q*(*<sup>λ</sup>*, *α*, *β*, *γ*) *and χ* ∈ C*, then f* ∗ *χ* ∈ *k* −UK*q*(*<sup>λ</sup>*, *α*, *β*, *<sup>γ</sup>*).

**Proof.** Since *f* ∈ *k* −UK*q*(*<sup>λ</sup>*, *α*, *β*, *<sup>γ</sup>*), there exist *g* ∈ *k* −ST *<sup>q</sup>*(*β*, *<sup>γ</sup>*), such that *zDq f*(*z*) *g*(*z*) ∈ *k* − <sup>P</sup>*q*(*<sup>λ</sup>*, *<sup>α</sup>*). It follows from Lemma 4 that *χ* ∗ *g* ∈ *k* −ST *<sup>q</sup>*(*β*, *<sup>γ</sup>*).

Consider

$$\begin{aligned} \frac{zD\_q\left(\chi(z)\*f(z)\right)}{\left(\chi(z)\*g(z)\right)} &= \frac{\chi(z)\*\left(zD\_q f(z)\right)}{\left(\chi(z)\*g(z)\right)},\\\\ &= \frac{\chi(z)\*\left(\frac{zD\_q f(z)}{\chi(z)}\right)g(z)}{\chi(z)\*f(z)},\\ &= \frac{\chi(z)\*F(z)g(z)}{\chi(z)\*g(z)}.\end{aligned}$$

where *F* ∈ *k* −ST *<sup>q</sup>*(*<sup>λ</sup>*, *<sup>α</sup>*). By using Lemma 4, we obtain the required result.
