Analytic Estimates for Bi-Univalent Functions Associated with a New Operator Involving the q–Rabotnov Function

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-1
Published: Jan 6, 2026
Cite as:
Ahmad Almalkawi, Ala Amourah, Abdullah Alsoboh, Jamal Salah, khaled Al Mashrafih, Abed Al-Rahman Malkawi, Tala Sasa, Analytic Estimates for Bi-Univalent Functions Associated with a New Operator Involving the q–Rabotnov Function, Int. J. Anal. Appl., 24 (2026), 1.

Main Article Content

Ahmad Almalkawi, Ala Amourah, Abdullah Alsoboh, Jamal Salah, khaled Al Mashrafih, Abed Al-Rahman Malkawi, Tala Sasa

Abstract

In this paper, we introduce and analyze a new subclass of bi-univalent functions associated with a differential operator constructed from the q–Rabotnov function. Motivated by the framework of q–calculus and its interplay with geometric function theory, the proposed operator is defined through convolution with q–Rabotnov kernels, thereby generating novel analytic structures. By applying the subordination principle, we establish sharp coefficient estimates for the initial Taylor–Maclaurin coefficients |α2| and |α3|, and derive Fekete–Szegö type inequalities for the class under consideration. The results presented here extend and generalize several recent contributions in the theory of biunivalent functions, highlighting the central role of q–special functions in the development of new operator-based subclasses. These findings provide deeper insights into the analytic behavior of bi-univalent mappings and suggest further applications of q–calculus in operator theory, convolution structures, and complex analysis.

Article Details

References

  1. S.K. Sharma, R. Jain, On Some Properties of Generalized $q$-Mittag-Leffler Function, Math. Aeterna, 4 (2014), 613–619.
  2. Y.N. Rabotnov, Equilibrium of an Elastic Medium with After-Effect, Fract. Calc. Appl. Anal. 17 (2014), 684–696. https://doi.org/10.2478/s13540-014-0193-1.
  3. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, World Scientific, (2010).
  4. A. Kilbas, H.M. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, (2006).
  5. M.H. Annaby, Z.S. Mansour, $q$-Fractional Calculus and Equations, Springer, (2012).
  6. V.G. Kac, P. Cheung, Quantum Calculus, Springer, 2002.
  7. R. Askey, M.E. Ismail, A Generalization of Ultraspherical Polynomials, Birkhäuser Basel, 1983. https://doi.org/10.1007/978-3-0348-5438-2_6.
  8. R. Askey, The $q$-Gamma and $q$-Beta Functions, Appl. Anal. 8 (1978), 125–141. https://doi.org/10.1080/00036817808839221.
  9. R. Chakrabarti, R. Jagannathan, S.S.N. Mohammed, New Connection Formulae for the $q$-Orthogonal Polynomials via a Series Expansion of the $q$-Exponential, J. Phys.: Math. Gen. 39 (2006), 12371–12380. https://doi.org/10.1088/0305-4470/39/40/006.
  10. G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
  11. P.L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften Series, Springer, New York, 1983.
  12. W.C. Ma, D. Minda, a Unified Treatment of Some Special Classes of Univalent Functions, in: Proceedings of the International Conference on Complex Analysis at the Nankai Institute of Mathematics, pp. 157–169, 1992.
  13. A. Alsoboh, A. Amourah, O. Alnajar, M. Ahmed, T.M. Seoudy, Exploring $q$-Fibonacci Numbers in Geometric Function Theory: Univalence and Shell-Like Starlike Curves, Mathematics 13 (2025), 1294. https://doi.org/10.3390/math13081294.
  14. A. Alsoboh, A. Amourah, W.G. Atshan, J. Salah, T. Sasa, A Study of Bi-Univalent Class in Leaf-Like Domains Using Quantum Calculus Through Subordination, Int. J. Anal. Appl. 23 (2025), 293. https://doi.org/10.28924/2291-8639-23-2025-293.
  15. A. Alsoboh, A. Amourah, M. Darus, C.A. Rudder, Investigating New Subclasses of Bi-Univalent Functions Associated with $q$-Pascal Distribution Series Using the Subordination Principle, Symmetry 15 (2023), 1109. https://doi.org/10.3390/sym15051109.
  16. 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), 2366–2380. https://doi.org/10.37256/cm.5220243737.
  17. A. Amourah, A. Alsoboh, J. Salah, K. Al Kalbani, Bounds on Initial Coefficients for Bi-Univalent Functions Linked to $q$-Analog of le Roy-Type Mittag-Leffler Function, WSEAS Trans. Math. 23 (2024), 714–722. https://doi.org/10.37394/23206.2024.23.73.
  18. A. Alsoboh, A. Amourah, F.M. Sakar, O. Ogilat, G.M. Gharib, et al., Coefficient Estimation Utilizing the Faber Polynomial for a Subfamily of Bi-Univalent Functions, Axioms 12 (2023), 512. https://doi.org/10.3390/axioms12060512.
  19. A. Alsoboh, G.I. Oros, A Class of Bi-Univalent Functions in a Leaf-Like Domain Defined Through Subordination via $q$-Calculus, Mathematics 12 (2024), 1594. https://doi.org/10.3390/math12101594.
  20. A. Amourah, A. Alsoboh, D. Breaz, S.M. El-Deeb, A Bi-Starlike Class in a Leaf-Like Domain Defined Through Subordination via $q$-Calculus, Mathematics 12 (2024), 1735. https://doi.org/10.3390/math12111735.
  21. A. Alsoboh, A. Amourah, W.G. Atshan, J. Salah, T. Sasa, A Study of Bi-Univalent Class in Leaf-Like Domains Using Quantum Calculus Through Subordination, Int. J. Anal. Appl. 23 (2025), 293. https://doi.org/10.28924/2291-8639-23-2025-293.
  22. A. Alsoboh, A.S. Tayyah, A. Amourah, A.A. Al-Maqbali, K. Al Mashraf, et al., Hankel Determinant Estimates for Bi-Bazilevič-Type Functions Involving Q-Fibonacci Numbers, Eur. J. Pure Appl. Math. 18 (2025), 6698. https://doi.org/10.29020/nybg.ejpam.v18i3.6698.
  23. T. Al-Hawary, A. Amourah, A. Alsoboh, O. Ogilat, I. Harny, et al., Applications of $q$-Ultraspherical Polynomials to Bi-Univalent Functions Defined by $q$-Saigo's Fractional Integral Operators, AIMS Math. 9 (2024), 17063–17075. https://doi.org/10.3934/math.2024828.
  24. T. Al-Hawary, A. Amourah, A. Alsoboh, A.M. Freihat, O. Ogilat, I. Harny, M. Darus, Subclasses of Yamakawa-type bi-starlike functions subordinate to Gegenbauer polynomials associated with quantum calculus, Results Nonlinear Anal. 7 (2024), 75–83.
  25. W. Janowski, Extremal Problems for a Family of Functions with Positive Real Part and for Some Related Families, Ann. Pol. Math. 23 (1970), 159–177. https://doi.org/10.4064/ap-23-2-159-177.
  26. W. Janowski, Some Extremal Problems for Certain Families of Analytic Functions I, Ann. Pol. Math. 28 (1973), 297–326. https://doi.org/10.4064/ap-28-3-297-326.
  27. J. Sokół, On Starlike Functions Connected With Fibonacci Numbers, Zesz. Nauk. Politech. Rzeszow. Mat., 23(157), 1999, 111–116.
  28. J. Sokół, A Certain Class of Starlike Functions, Comput. Math. Appl. 62 (2011), 611–619. https://doi.org/10.1016/j.camwa.2011.05.041.
  29. V.S. Masih, A. Ebadian, S. Yalçin, Some Properties Associated to a Certain Class of Starlike Functions, Math. Slovaca 69 (2019), 1329–1340. https://doi.org/10.1515/ms-2017-0311.
  30. M.S. Robertson, Certain Classes of Starlike Functions, Mich. Math. J. 32 (1985), 135–140. https://doi.org/10.1307/mmj/1029003181.
  31. H.E. Özkan Uçar, Coefficient Inequality for Q-Starlike Functions, Appl. Math. Comput. 276 (2016), 122–126. https://doi.org/10.1016/j.amc.2015.12.008.
  32. F.H. Jackson, XI.–On $q$-Functions and a Certain Difference Operator, Trans. R. Soc. Edinb. 46 (1909), 253–281. https://doi.org/10.1017/s0080456800002751.
  33. F.H. Jackson, On $q$-Definite Integrals, Q. J. Pure Appl. Math. 41 (1910), 193–203.
  34. A. Aral, V. Gupta, Generalized $q$-Baskakov Operators, Math. Slovaca 61 (2011), 619–634. https://doi.org/10.2478/s12175-011-0032-3.
  35. A. Aral, V. Gupta, On the Durrmeyer Type Modification of the -Baskakov Type Operators, Nonlinear Anal.: Theory Methods Appl. 72 (2010), 1171–1180. https://doi.org/10.1016/j.na.2009.07.052.
  36. A. Aral, V. Gupta, R.P. Agarwal, Applications of $q$-Calculus in Operator Theory, Springer, New York, 2013. https://doi.org/10.1007/978-1-4614-6946-9.
  37. S. Elhaddad, H. Aldweby, M. Darus, Some Properties on a Class of Harmonic Univalent Functions Defined by $q$-Analogue of Ruscheweyh Operator, J. Math. Anal. 9 (2018), 28–35.
  38. R.M. Ali, S.K. Lee, V. Ravichandran, S. Supramaniam, Coefficient Estimates for Bi-Univalent Ma-Minda Starlike and Convex Functions, Appl. Math. Lett. 25 (2012), 344–351. https://doi.org/10.1016/j.aml.2011.09.012.
  39. M. Cağlar, H. Orhan, N. Yağmur, Coefficient Bounds for New Subclasses of Bi-Univalent Functions, Filomat 27 (2013), 1165–1171. https://doi.org/10.2298/fil1307165c.
  40. G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, 1999. https://doi.org/10.1017/CBO9781107325937.
  41. H.Ö. Güney, G. Murugusundaramoorthy, J. Sokół. Subclasses of Bi-Univalent Functions Related to Shell-Like Curves Connected With Fibonacci Numbers, Acta Univ. Sapient. Math. 10 (2018), 70–84.
  42. J. Dziok, R.K. Raina, J. Sokół, Certain Results for a Class of Convex Functions Related to a Shell-Like Curve Connected with Fibonacci Numbers, Comput. Math. Appl. 61 (2011), 2605–2613. https://doi.org/10.1016/j.camwa.2011.03.006.
  43. J. Dziok, R.K. Raina, J. Sokół, On $alpha$-Convex Functions Related to Shell-Like Functions Connected with Fibonacci Numbers, Appl. Math. Comput. 218 (2011), 996–1002. https://doi.org/10.1016/j.amc.2011.01.059.
  44. B. Khan, H.M. Srivastava, N. Khan, M. Darus, M. Tahir, et al., Coefficient Estimates for a Subclass of Analytic Functions Associated with a Certain Leaf-Like Domain, Mathematics 8 (2020), 1334. https://doi.org/10.3390/math8081334.
  45. M. Arif, O. Barkub, H. Srivastava, S. Abdullah, S. Khan, Some Janowski Type Harmonic Q-Starlike Functions Associated with Symmetrical Points, Mathematics 8 (2020), 629. https://doi.org/10.3390/math8040629.
  46. H.M. Srivastava, M.K. Aouf, A.O. Mostafa, Some Properties of Analytic Functions Associated with Fractional q-Calculus Operators, Miskolc Math. Notes 20 (2019), 1245. https://doi.org/10.18514/mmn.2019.3046.
  47. H.M. Srivastava, S.M. El-Deeb, A Certain Class of Analytic Functions of Complex Order Connected with a Q-Analogue of Integral Operators, Miskolc Math. Notes 21 (2020), 417–433. https://doi.org/10.18514/mmn.2020.3102.
  48. M. Shafiq, H.M. Srivastava, N. Khan, Q.Z. Ahmad, M. Darus, et al., An Upper Bound of the Third Hankel Determinant for a Subclass of Q-Starlike Functions Associated with K-Fibonacci Numbers, Symmetry 12 (2020), 1043. https://doi.org/10.3390/sym12061043.
  49. S. Mahmood, Q.Z. Ahmad, H.M. Srivastava, N. Khan, B. Khan, et al., A Certain Subclass of Meromorphically q-Starlike Functions Associated with the Janowski Functions, J. Inequal. Appl. 2019 (2019), 88. https://doi.org/10.1186/s13660-019-2020-z.
  50. S. Mahmood, H. M. Srivastava, N. Khan, Q. Z. Ahmad, B. Khan, and I. Ali, Upper bound of the third Hankel determinant for a subclass of q-starlike functions, Symmetry, 11, 2019, 347.
  51. S. Mahmood, H.M. Srivastava, N. Khan, Q.Z. Ahmad, B. Khan, et al., Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions, Symmetry 11 (2019), 347. https://doi.org/10.3390/sym11030347.
  52. A.S. Tayyah, W.G. Atshan, Starlikeness and Bi-Starlikeness Associated with a New Carathéodory Function, J. Math. Sci. 290 (2025), 232–256. https://doi.org/10.1007/s10958-025-07604-8.
  53. A.S. Tayyah, W.G. Atshan, G.I. Oros, Third-Order Differential Subordination Results for Meromorphic Functions Associated with the Inverse of the Legendre Chi Function via the Mittag-Leffler Identity, Mathematics 13 (2025), 2089. https://doi.org/10.3390/math13132089.
  54. S.A. AL-Ameedee, W.G. Atshan, F.A. AL-Maamori, Second Hankel Determinant for Certain Subclasses of Bi-Univalent Functions, J. Phys.: Conf. Ser. 1664 (2020), 012044. https://doi.org/10.1088/1742-6596/1664/1/012044.
  55. S.A. Al-Ameedee, W. Galib Atshan, F. Ali Al-Maamori, Coefficients Estimates of Bi-Univalent Functions Defined by New Subclass Function, J. Phys.: Conf. Ser. 1530 (2020), 012105. https://doi.org/10.1088/1742-6596/1530/1/012105.
  56. W. Galib Atshan, E. Ibraham Badawi, Results on Coefficient Estimates for Subclasses of Analytic and Bi-Univalent Functions, J. Phys.: Conf. Ser. 1294 (2019), 032025. https://doi.org/10.1088/1742-6596/1294/3/032025.