# Applications of Certain Conic Domains to a Subclass of q-Starlike Functions Associated with the Janowski Functions

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## Abstract

**:**

## 1. Introduction, Motivation and Definitions

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

- I.
- Upon setting$$k=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}A=1-2\alpha \phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}B=-1\phantom{\rule{2.em}{0ex}}(0\leqq \alpha <1),$$
- II.
- If, after putting$$A=1\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}B=-1,$$
- III.
- If we first put$$A=1-2\alpha \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\left(0\leqq \alpha <1\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}B=-1,$$
- IV.
- By virtue of (16), in its special case when$$k=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}A=1\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}B=-1,$$
- V.
- If, in Definition 12, we let $q\to 1-$, we are led to the class k-${\mathcal{S}}^{\ast}\left[A,B\right]$, which was introduced and studied by Noor and Sarfaraz [10].
- VI.
- If, in Definition 12, we put $k=0$, we are led to the class ${\mathcal{S}}_{q}^{\ast}\left[A,B\right]$, which was introduced and studied by Srivastava et al. [27].

## 2. Sufficient Conditions

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

**Corollary**

**4.**

## 3. Closure Theorems

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. The Fekete-Szegö Functional

**Lemma**

**1.**

**Theorem**

**4.**

**Proof.**

## 5. Partial Sums for the Function Class ${\mathcal{S}}^{\ast}\left(\mathit{q},\mathit{k},\mathit{A},\mathit{B}\right)$

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 6. Analytic Functions with Negative Coefficients

**Theorem**

**7.**

**Proof.**

**Corollary**

**5.**

**Theorem**

**8.**

## 7. Concluding Remarks and Observations

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Khan, B.; Srivastava, H.M.; Khan, N.; Darus, M.; Ahmad, Q.Z.; Tahir, M.
Applications of Certain Conic Domains to a Subclass of *q*-Starlike Functions Associated with the Janowski Functions. *Symmetry* **2021**, *13*, 574.
https://doi.org/10.3390/sym13040574

**AMA Style**

Khan B, Srivastava HM, Khan N, Darus M, Ahmad QZ, Tahir M.
Applications of Certain Conic Domains to a Subclass of *q*-Starlike Functions Associated with the Janowski Functions. *Symmetry*. 2021; 13(4):574.
https://doi.org/10.3390/sym13040574

**Chicago/Turabian Style**

Khan, Bilal, Hari Mohan Srivastava, Nazar Khan, Maslina Darus, Qazi Zahoor Ahmad, and Muhammad Tahir.
2021. "Applications of Certain Conic Domains to a Subclass of *q*-Starlike Functions Associated with the Janowski Functions" *Symmetry* 13, no. 4: 574.
https://doi.org/10.3390/sym13040574