# Certain Coefficient Problems for q-Starlike Functions Associated with q-Analogue of Sine Function

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1.**

**Lemma**

**1**

**Lemma**

**2.**

**Lemma**

**3**

**Lemma**

**4**

**Lemma**

**5**

- First, four coefficient bounds $\left(\right)open="|"\; close="|">{a}_{n}$
- The Zalcman inequality $\left(\right)open="|"\; close="|">{a}_{n}^{2}-{a}_{2n-1}$ for $n=2.$
- The generalized Zalcman inequality $\left(\right)open="|"\; close="|">{a}_{n}{a}_{m}-{a}_{n+m-1}\left(\right)open="("\; close=")">m-1$ for certain values of m and n.
- The upper bounds of the second Hankel $\left(\right)$ and the third Hankel determinant $\left(\right)open="|"\; close="|">{H}_{3,1}\left(f\right)$

## 2. Main Results

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

## 3. Zalcman and Generalized Zalcman Conjecture

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 4. Hankel Determinants

**Theorem**

**6.**

**Proof.**

**Corollary**

**3.**

**Theorem**

**7.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Fekete, M.; Szegö, G. Eine bemerkung uber ungerade schlichten funktionene. J. Lond. Math. Soc.
**1993**, 8, 85–89. [Google Scholar] - Pommerenke, C. On the Hankel determinants of univalent functions. Mathematika
**1967**, 14, 108–112. [Google Scholar] [CrossRef] - Pommerenke, C. On starlike and close-to-convex functions. Proc. Lond. Math. Soc.
**1963**, 3, 290–304. [Google Scholar] [CrossRef] - Noonan, J.W.; Thomas, D.K. On the Hankel determinants of a really mean p-valent functions. Proc. Lond. Math.
**1972**, 3, 503–524. [Google Scholar] [CrossRef] - Noor, K.I. On subclasses of close-to-convex functionsof higher order. Int. J. Math. Math. Sci.
**1983**, 6, 327–334. [Google Scholar] [CrossRef] - Noor, K.I. On quasi-convex univalent functions and related topics. Internat. J. Math. Sci.
**1987**, 2, 241–258. [Google Scholar] [CrossRef] - Noor, K.I. Higher order close-to-convex functions. Math Jpn.
**1992**, 37, 1–8. [Google Scholar] - Ehrenborg, R. The Hankel determinant of exponential polynomials. Am. Math. Mon.
**2000**, 107, 557–560. [Google Scholar] [CrossRef] - Layman, J.W. The Hankel transform and some of its properties. J. Integer Seq
**2001**, 4, 1–11. [Google Scholar] - Janteng, A.; Suzeini, A.H.; Darus, M. Hankel determinant for starlike and convex functions. Int. J. Math. Anal.
**2007**, 1, 619–625. [Google Scholar] - Mahmood, S.; Srivastava, H.M.; Khan, N.; Ahmad, Q.Z.; Khan, B.; Ali, I. Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions. Symmetry
**2019**, 11, 347. [Google Scholar] [CrossRef] [Green Version] - Srivastava, H.M.; Ahmad, Q.Z.; Darus, M.; Khan, N.; Khan, B.; Zaman, N.; Shah, H.H. Upper Bound of the Third Hankel Determinant for a Subclass of Close-to-Convex Functions Associated with the Lemniscate of Bernoulli. Mathematics
**2019**, 7, 848. [Google Scholar] [CrossRef] [Green Version] - Arif, M.; Raza, M.; Tang, H.; Hussain, S.; Khan, H. Hankel determinant of order three for familiar subsets of analytic functions related with sine function. Open Math.
**2019**, 17, 1615–1630. [Google Scholar] [CrossRef] - Srivastava, H.M.; Ahmad, Q.Z.; Khan, N.; Khan, N.; Khan, B. Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a General Conic Domain. Mathematics
**2019**, 7, 181. [Google Scholar] [CrossRef] - Shafiq, M.; Srivastava, H.M.; Khan, N.; Ahmad, Q.Z.; Darus, M.; Kiran, S. An Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions Associated with k-Fibonacci Numbers. Symmetry
**2020**, 12, 1043. [Google Scholar] [CrossRef] - Murugusundaramoorthy, G.; Bulboacă, T. Hankel Determinants for New Subclasses of Analytic Functions Related to a Shell Shaped Region. Mathematics
**2020**, 8, 1041. [Google Scholar] [CrossRef] - Guney, H.O.; Korfeci, B. Fourth Hankel Determinant for a subclass of analytic functions related to modified sigmoid functions. Int. J. Open Probl. Comput. Sci. Math.
**2021**, 14, 41–49. [Google Scholar] - Zhang, H.-Y.; Tang, H. A study of fourth-order Hankel determinants for starlike functions connected with the sine function. J. Funct. Spaces
**2021**, 2021, 9991460. [Google Scholar] [CrossRef] - Srivastava, H.M.; Khan, B.; Khan, N.; Tahir, M.; Ahmad, S.; Khan, N. Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the q-exponential function. Bull. Sci. Math.
**2021**, 167, 102942. [Google Scholar] [CrossRef] - Saliu, A.; Noor, K.I. On Coefficients Problems for Certain Classes of Analytic Functions. J. Math. Anal.
**2021**, 12, 13–22. [Google Scholar] - Raza, M.; Riaz, A.; Xin, Q.; Malik, S.N. Hankel Determinants and Coefficient Estimates for Starlike Functions Related to Symmetric Booth Lemniscate. Symmetry
**2022**, 14, 1366. [Google Scholar] [CrossRef] - Khan, B.; Aldawish, I.; Araci, S.; Khan, M.G. Third Hankel Determinant for the Logarithmic Coefficients of Starlike Functions Associated with Sine Function. Fractal Fract.
**2022**, 6, 261. [Google Scholar] [CrossRef] - Riaz, A.; Raza, M.; Thomas, D.K. The Third Hankel determinant for starlike functions associated with sigmoid functions. Forum Math.
**2022**, 34, 137–156. [Google Scholar] [CrossRef] - Riaz, A.; Raza, M. Hankel determinants for starlike and convex functions associated with lune. submitted.
- Riaz, A.; Raza, M.; Thomas, D.K. Hankel determinants for starlike and convex functions associated with a cardioid domain. submitted.
- Afis, S.; Khalida, I.N. On Quantum Differential Subordination Related with Certain Family of Analytic Functions. J. Math.
**2020**, 2020, 6675732. [Google Scholar] [CrossRef] - Saliu, A.; Jabeen, K.; Al-shbeil, I.; Oladejo, S.O.; Cătaş, A. Radius and Differential Subordination Results for Starlikeness Associated with Limaçon Class. J. Funct. Spaces
**2022**, 2022, 8264693. [Google Scholar] [CrossRef] - Al-Shbeil, I.; Saliu, A.; Cătaş, A.; Malik, S.N.; Oladejo, S.O. Some Geometrical Results Associated with Secant Hyperbolic Functions. Mathematics
**2022**, 10, 2697. [Google Scholar] [CrossRef] - Jackson, F.H. On q-definite integrals. Q. J. Pure Appl. Math.
**1910**, 41, 193–203. [Google Scholar] - Jackson, F.H. q-difference equations. Am. J. Math.
**1910**, 32, 305–314. [Google Scholar] [CrossRef] - Ismail, M.E.H.; Merkes, E.; Styer, D. A generalization of starlike functions. Complex Var.
**1990**, 14, 77–84. [Google Scholar] [CrossRef] - Srivastava, H.M. Univalent Functions, Fractional Calculus, and Associated Generalized Hypergeometric Functions. In Univalent Functions, Fractional Calculus, and Their Applications; Srivastava, H.M., Owa, S., Eds.; Halsted Press (Ellis Horwood Limited): Chichester, UK; John Wiley and Sons: New York, NY, USA, 1989; pp. 329–354. [Google Scholar]
- Mahmood, M.; Jabeen, M.; Malik, S.N.; Srivastava, H.M.; Manzoor, R.; Riaz, S.M.J. Some coefficient inequalities of q-starlike functions associated with conic domain defined by q-derivative. J. Funct. Spaces
**2018**, 2018, 8492072. [Google Scholar] [CrossRef] [Green Version] - Mahmood, S.; Raza, N.; AbuJarad, E.S.A.; Srivastava, G.; Srivastava, H.M.; Malik, S.N. Geometric Properties of Certain Classes of Analytic Functions Associated with a q-Integral Operator. Symmetry
**2019**, 11, 719. [Google Scholar] [CrossRef] [Green Version] - Raza, M.; Naz, H.; Malik, S.N.; Islam, S. On q-ANALOGUE of Differential Subordination Associated with Lemniscate of Bernoulli. J. Math.
**2021**, 2021, 5353372. [Google Scholar] [CrossRef] - Zainab, S.; Raza, M.; Xin, Q.; Jabeen, M.; Malik, S.N.; Riaz, S. On q-Starlike Functions Defined by q-Ruscheweyh Differential Operator in Symmetric Conic Domain. Symmetry
**2021**, 13, 1947. [Google Scholar] [CrossRef] - Riaz, S.; Nisar, U.A.; Xin, Q.; Malik, S.N.; Raheem, A. On Starlike Functions of Negative Order Defined by q-Fractional Derivative. Fractal Fract.
**2022**, 6, 30. [Google Scholar] [CrossRef] - Saliu, A.; Al-Shbeil, I.; Gong, J.; Malik, S.N.; Aloraini, N. Properties of q-Symmetric Starlike Functions of Janowski Type. Symmetry
**2022**, 14, 1907. [Google Scholar] [CrossRef] - Makhlouf, A.B.; Naifar, O.; Hammami, M.A.; Wu, B. FTS and FTB of Conformable Fractional Order Linear Systems. Math. Probl. Eng.
**2018**, 2018, 2572986. [Google Scholar] [CrossRef] - Naifar, O.; Jmal, A.; Nagy, A.M.; Makhlouf, A.B. Improved Quasiuniform Stability for Fractional Order Neural Nets with Mixed Delay. Math. Probl. Eng.
**2020**, 2020, 8811226. [Google Scholar] [CrossRef] - Cho, N.E.; Kumar, V.; Kumar, S.S. Radius Problems for Starlike Functions Associated with the Sine Function. Bull. Iran. Math. Soc.
**2019**, 45, 213–232. [Google Scholar] [CrossRef] - Libera, R.J.; Złotkiewicz, E.J. Early coefficient of the inverse of a regular convex function. Proc. Am. Math. Soc.
**1982**, 85, 225–230. [Google Scholar] [CrossRef] - Duren, P.L. Univalent Functions, Grundlehren der Mathematischen Wissenschaften; Springer: New York, NY, USA; Berlin/Heidelberg, Germany; Tokyo, Iapan, 1983; Volume 259. [Google Scholar]
- Ravichandran, V.; Verma, S. Bound for the fifth coefficient of certain starlike functions. Comptes Rendus Math. Acad. Sci.
**2015**, 353, 505–510. [Google Scholar] [CrossRef] - Choi, J.H.; Kim, Y.C.; Sugawa, T. A general approach to the Fekete-Szegö problem. J. Math. Soc.
**2007**, 59, 707–727. [Google Scholar] [CrossRef] - Zhang, H.-Y.; Srivastava, R.; Tang, H. Third-Order Hankel and Toeplitz Determinants for Starlike Functions Connected with the Sine Function. Mathematics
**2019**, 7, 404. [Google Scholar] [CrossRef] [Green Version] - Ma, W. Generalized Zalcman conjecture for starlike and typically real functions. J. Math. Anal. Appl.
**1999**, 234, 328–339. [Google Scholar] [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Taj, Y.; Zainab, S.; Xin, Q.; Tawfiq, F.M.O.; Raza, M.; Malik, S.N.
Certain Coefficient Problems for *q*-Starlike Functions Associated with *q*-Analogue of Sine Function. *Symmetry* **2022**, *14*, 2200.
https://doi.org/10.3390/sym14102200

**AMA Style**

Taj Y, Zainab S, Xin Q, Tawfiq FMO, Raza M, Malik SN.
Certain Coefficient Problems for *q*-Starlike Functions Associated with *q*-Analogue of Sine Function. *Symmetry*. 2022; 14(10):2200.
https://doi.org/10.3390/sym14102200

**Chicago/Turabian Style**

Taj, Yusra, Saira Zainab, Qin Xin, Ferdous M. O. Tawfiq, Mohsan Raza, and Sarfraz Nawaz Malik.
2022. "Certain Coefficient Problems for *q*-Starlike Functions Associated with *q*-Analogue of Sine Function" *Symmetry* 14, no. 10: 2200.
https://doi.org/10.3390/sym14102200