Geometric and analytic properties of a $ q $-Lune-type subclass of univalent functions defined by $ q $-calculus

Muqrin A. Almuqrin; Mohammed AbaOud; Mohammad Faisal Khan; Muqrin A. Almuqrin; Mohammed AbaOud; Mohammad Faisal Khan

doi:10.3934/math.20261157

AIMS Mathematics

2026, Volume 11, Issue 9: 29106-29134. doi: 10.3934/math.20261157

Previous Article Next Article

Research article Special Issues

Geometric and analytic properties of a $ q $-Lune-type subclass of univalent functions defined by $ q $-calculus

1.
Department of Mathematics, College of Science in Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia; m.almuqrin@mu.edu.sa
2.
Department of Mathematics and Statistics, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudia Arabia; maabaoud@imamu.edu.sa
3.
Department of Basics Sciences, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi Arabia; f.khan@seu.edu.sa

Received: 23 April 2026 Revised: 22 August 2026 Accepted: 31 August 2026 Published: 10 September 2026
MSC : Primary 05A30, 30C45; Secondary 11B65, 47B38

In this paper, we introduced and explored a new subclass $ \mathcal{R}_L^{\, q} $ of univalent functions in the open unit disk $ \mathbb{E} $ related to the $ q $-difference operator and newly define $ q $-Lune-type domain. The class is defined by the subordination condition
$ D_q f(z)\prec z+\sqrt{1+\beta(q)z^2}, \qquad 0<q<1, $
where $ \beta(q) = \frac{\log(1+q)}{\log 2} $. First, we showed that this class is non-empty for both identity and non-identity functions. Using techniques from subordination theory and Carathéodory functions, we derived sharp coefficient estimates and obtained bounds of the second and third Hankel determinant. Furthermore, we investigated logarithmic coefficients $ \gamma_1 $, $ \gamma_2 $, and $ \gamma_3 $. In addition, graphical illustrations were provided to demonstrate the geometric behavior of the class and to verify the subordination condition. The results presented here extend several results when $ q\to1^{-} $ and contribute to the growing theory of analytic and univalent function classes.
- $ q $-calculus,
- $ q $-difference operator,
- subordination,
- Hankel determinant,
- logarithmic coefficients,
- analytic and univalent functions,
- $ q $-Lune-type domain
Citation: Muqrin A. Almuqrin, Mohammed AbaOud, Mohammad Faisal Khan. Geometric and analytic properties of a $ q $-Lune-type subclass of univalent functions defined by $ q $-calculus[J]. AIMS Mathematics, 2026, 11(9): 29106-29134. doi: 10.3934/math.20261157

Related Papers:

Abstract

In this paper, we introduced and explored a new subclass $ \mathcal{R}_L^{\, q} $ of univalent functions in the open unit disk $ \mathbb{E} $ related to the $ q $-difference operator and newly define $ q $-Lune-type domain. The class is defined by the subordination condition

$ D_q f(z)\prec z+\sqrt{1+\beta(q)z^2}, \qquad 0<q<1, $

where $ \beta(q) = \frac{\log(1+q)}{\log 2} $. First, we showed that this class is non-empty for both identity and non-identity functions. Using techniques from subordination theory and Carathéodory functions, we derived sharp coefficient estimates and obtained bounds of the second and third Hankel determinant. Furthermore, we investigated logarithmic coefficients $ \gamma_1 $, $ \gamma_2 $, and $ \gamma_3 $. In addition, graphical illustrations were provided to demonstrate the geometric behavior of the class and to verify the subordination condition. The results presented here extend several results when $ q\to1^{-} $ and contribute to the growing theory of analytic and univalent function classes.

References

[1]	P. L. Duren, Univalent functions, New York: Springer, 1983.
[2]	C. Pommerenke, Univalent functions, Göttingen: Vandenhoeck & Ruprecht, 1975.
[3]	M. Obradović, N. Tuneski, Hankel determinants of second and third order for the class $\mathcal{S}$ of univalent functions, Math. Slovaca, 71 (2021), 649–654. https://doi.org/10.1515/ms-2021-0010 doi: 10.1515/ms-2021-0010
[4]	K. Jabeen, A. Saliu, J. Gong, S. Hussain, Majorization problem for $q$-general family of functions with bounded radius rotations, Mathematics, 12 (2024), 2605. https://doi.org/10.3390/math12172605 doi: 10.3390/math12172605
[5]	V. Allu, N. L. Sharma, On logarithmic coefficients for classes of analytic functions associated with convex functions, B. Sci. Math., 191 (2024), 103384. https://doi.org/10.1016/j.bulsci.2024.103384 doi: 10.1016/j.bulsci.2024.103384
[6]	H. Hankel. Zur Theorie der komplexen Zahlensysteme, Math. Ann., 11, (1877), 384–436.
[7]	P. L. Duren, Y. J. Leung, Logarithmic coefficients of univalent functions, J. Anal. Math., 36 (1979), 36–43. https://doi.org/10.1007/BF02798766 doi: 10.1007/BF02798766
[8]	N. E. Cho, B. Kowalczyk, O. S. Kwon, A. Lecko, Y. J. Sim, On the third logarithmic coefficient in some subclasses of close-to-convex functions, RACSAM Rev. R. Acad. A, 114 (2020), 52. https://doi.org/10.1007/s13398-020-00786-7 doi: 10.1007/s13398-020-00786-7
[9]	S. Ponnusamy, N. L. Sharma, K. J. Wirths, Logarithmic coefficients problems in families related to starlike and convex functions, J. Aust. Math. Soc., 109 (2020), 230–249. https://doi.org/10.1017/S1446788719000065 doi: 10.1017/S1446788719000065
[10]	B. Kowalczyk, A. Lecko, Second Hankel determinant of logarithmic coefficients of convex and starlike functions, B. Aust. Math. Soc., 105 (2022), 458–467. https://doi.org/10.1017/S0004972721000836 doi: 10.1017/S0004972721000836
[11]	S. S. Eker, B. Şeker, B. Çekiç, M. Acu, Sharp bounds for the second Hankel determinant of logarithmic coefficients for strongly starlike and strongly convex functions, Axioms, 11 (2022), 369. https://doi.org/10.3390/axioms11080369 doi: 10.3390/axioms11080369
[12]	H. M. Srivastava, S. S. Eker, B. Şeker, B. Çekiç, Second Hankel determinant of logarithmic coefficients for a subclass of univalent functions, Miskolc Math. Notes, 25 (2024), 479–488. https://doi.org/10.18514/MMN.2024.4314 doi: 10.18514/MMN.2024.4314
[13]	F. H. Jackson, XI.–On $q$-functions and a certain difference operator, T. Roy. Soc. Edin., 46 (1909), 253–281. https://doi.org/10.1017/S0080456800002751 doi: 10.1017/S0080456800002751
[14]	V. Kac, P. Cheung, Quantum calculus, New York: Springer, 2002.
[15]	M. E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Cambridge: Cambridge University Press, 2005. https://doi.org/10.1017/CBO9781107325982
[16]	T. Panigrahi, T. Bulboacă, S. P. Dhal, Bounds for the second Hankel determinant and its inverse in specific function classes, Axioms, 15 (2026), 130. https://doi.org/10.3390/axioms15020130 doi: 10.3390/axioms15020130
[17]	M. A. Mamon, S. Alyusof, R. Alyusof, A. H. El-Qadeem, Sharpness estimation of Hankel determinants and logarithmic coefficients for a family of analytic functions related to a lung-shaped domain, Mathematics, 14 (2026), 1240. https://doi.org/10.3390/math14081240 doi: 10.3390/math14081240
[18]	P. O. Sabir, R. P. Agarwal, S. J. Mohammedfaeq, P. O. Mohammed, N. Chorfi, T. Abdeljawad, Hankel determinant for a general subclass of $m$-fold symmetric bi-univalent functions defined by Ruscheweyh operators, J. Inequal. Appl., 2024 (2024), 14. https://doi.org/10.1186/s13660-024-03088-3 doi: 10.1186/s13660-024-03088-3
[19]	R. Nawaz, R. Fayyaz, D. Breaz, L. I. Cotîrlă, Sharp coefficient estimates for analytic functions associated with lemniscate of Bernoulli, Mathematics, 12 (2024), 2309. https://doi.org/10.3390/math12152309 doi: 10.3390/math12152309
[20]	J. Sokół, G. Murugusundaramoorthy, K. Vijaya, On $\lambda$-pseudo starlike functions associated with vertical strip domain, Asian-Eur. J. Math., 16 (2023), 2350135. https://doi.org/10.1142/S1793557123501358 doi: 10.1142/S1793557123501358
[21]	A. Alsoboh, M. Çağlar, M. Buyankara, Fekete–Szegö inequality for a subclass of bi-univalent functions linked to $q$-ultraspherical polynomials, Contemp. Math., 5 (2024), 2531–2545. https://doi.org/10.37256/cm.5220243737 doi: 10.37256/cm.5220243737
[22]	K. Marimuthu, J. Uma, T. Bulboacă, Coefficient estimates for starlike and convex functions associated with cosine function, Hacet. J. Math. Stat., 52 (2023), 596–618.
[23]	K. Marimuthu, U. Jayaraman, T. Bulboacă, Fekete–Szegö and Zalcman functional estimates for subclasses of alpha-convex functions related to trigonometric functions, Mathematics, 12 (2024), 234. https://doi.org/10.3390/math12020234 doi: 10.3390/math12020234
[24]	H. M. Srivastava, Operators of basic (or $q$-) calculus and fractional $q$-calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. A, 44 (2020), 327–344. https://doi.org/10.1007/s40995-019-00815-0 doi: 10.1007/s40995-019-00815-0
[25]	S. Khan, S. Hussain, M. Naeem, M. Darus, A. Rasheed, A subclass of $q$-starlike functions defined by using a symmetric $q$-derivative operator and related with generalized symmetric conic domains, Mathematics, 9 (2021), 917. https://doi.org/10.3390/math9090917 doi: 10.3390/math9090917
[26]	C. Swarup, Sharp coefficient bounds for a new subclass of $q$-starlike functions associated with $q$-analogue of the hyperbolic tangent function, Symmetry, 15 (2023), 763. https://doi.org/10.3390/sym15030763 doi: 10.3390/sym15030763
[27]	L. Zhang, Z. Wang, L. Shi, Sharp bounds on Hankel determinant of $q$-starlike and $q$-convex functions subordinate to secant hyperbolic functions, Fractal Fract., 9 (2025), 346. https://doi.org/10.3390/fractalfract9060346 doi: 10.3390/fractalfract9060346
[28]	M. Li, A. Zhu, Hankel determinant of $q$-starlike functions, J. South China Norm. Univ., 55 (2023), 116–122. https://dx.doi.org/10.6054/j.jscnun.2023085 doi: 10.6054/j.jscnun.2023085
[29]	B. Khan, F. Tchier, M. Oliveira, Regions of variability for generalized Janowski functions, AIMS Math., 11 (2026), 3499–3511. https://doi.org/10.3934/math.2026142 doi: 10.3934/math.2026142
[30]	L. Parida, T. Bulboacă, A. K. Sahoo, Hankel determinants for new subclasses of analytic functions involving a linear operator, Kragujev. J. Math., 46 (2022), 605–616. https://doi.org/10.46793/KgJMat2204.605P doi: 10.46793/KgJMat2204.605P
[31]	B. Khan, G. Murugusundaramoorthy, S. Asif, A subclass of analytic functions involving certain Mathieu-type series, Adv. Anal. Appl. Math., 2 (2025), 73–88. https://doi.org/10.62298/advmath.29 doi: 10.62298/advmath.29
[32]	H. Tang, S. Khan, S. Hussain, N. Khan, Hankel and Toeplitz determinant for a subclass of multivalent $q$-starlike functions of order $\alpha$, AIMS Math., 6 (2021), 5421–5439. https://doi.org/10.3934/math.2021320 doi: 10.3934/math.2021320
[33]	N. S. Almutairi, A. Shahen, A. Cătaş, H. Darwish, Convolution properties of meromorphic $p$-valent functions with coefficients of alternating type defined using $q$-difference operator, Mathematics, 12 (2024), 2104. https://doi.org/10.3390/math12132104 doi: 10.3390/math12132104
[34]	S. Özcan, L. I. Cotîrlă, Generalized $n$-polynomial $p$-convexity and related inequalities, Mathematics, 12 (2024), 1042. https://doi.org/10.3390/math12071042 doi: 10.3390/math12071042
[35]	R. J. Libera, E. J. Złotkiewicz, Early coefficient of the inverse of a regular convex function, P. Am. Math. Soc., 85 (1982), 225–230. https://doi.org/10.1090/S0002-9939-1982-0652447-5 doi: 10.1090/S0002-9939-1982-0652447-5
[36]	V. Ravichandran, S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Math., 353 (2015), 505–510. https://doi.org/10.1016/j.crma.2015.03.003 doi: 10.1016/j.crma.2015.03.003
[37]	J. H. Choi, Y. C. Kim, T. Sugawa, A general approach to the Fekete–Szegö problem, J. Math. Soc. Jpn, 59 (2007), 707–727. https://doi.org/10.2969/jmsj/05930707 doi: 10.2969/jmsj/05930707
[38]	M. Arif, M. Raza, H. Tang, S. Hussain, H. Khan, Hankel determinant of order three for familiar subsets of analytic functions related with sine function, Open Math., 17 (2019), 1615–1630. https://doi.org/10.1515/math-2019-0132 doi: 10.1515/math-2019-0132
[39]	B. Seker, B. Cekic, S. Sumer, O. Akcicek. Hankel determinants of logarithmic coefficients for the class of bounded turning functions associated with lune domain, Math. Sci. Appl. E-Notes, 13 (2025), 72–83.

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)