Geometric Properties of Starlike and Spiral-like Functions Defined by a Complex Fractional Operator

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-215
Published: Jul 6, 2026
Cite as:
Hameed Ur Rehman, Jamal Salah, Geometric Properties of Starlike and Spiral-like Functions Defined by a Complex Fractional Operator, Int. J. Anal. Appl., 24 (2026), 215.

Main Article Content

Hameed Ur Rehman, Jamal Salah

Abstract

This paper considers subclasses of the analytic function known as starlike and spiral-like functions of real order in the open unit disk. We derive necessary and sufficient coefficient conditions, distortion bounds, and radii of starlikeness and spiral-likeness, and show how the parameters influence the geometric domains in the unit disk. The class is generated via a complex fractional operator, which extends earlier results by allowing a rotation parameter to take complex values. This operator unifies several classical differential and integral operators, including the Sălăgean and Libera operators, with the former generalized from integer to real order. The operator provides a useful approach for generating subclasses of starlike and spiral-like functions, describing both rotational and radial behaviour. We also study geometric properties of these functions, including conjugate symmetry, and examine the effect of the operator on their mapping behaviour.

Article Details

References

  1. W. 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.
  2. P.J. Eenigenburg, S.S. Miller, P.T. Mocanu, M.O. Reade, On a Subclass of Bazilevič Functions, Proc. Am. Math. Soc. 45 (1974), 88-92. https://doi.org/10.1090/S0002-9939-1974-0344441-4.
  3. M.I.S. Robertson, On the Theory of Univalent Functions, Ann. Math. 37 (1936), 374-408. https://doi.org/10.2307/1968451.
  4. R.J. Libera, Some Classes of Regular Univalent Functions, Proc. Am. Math. Soc. 16 (1965), 755-758. https://doi.org/10.1090/S0002-9939-1965-0178131-2.
  5. S. Mahmood, J. Sokół, New Subclass of Analytic Functions in Conical Domain Associated with Ruscheweyh Q-Differential Operator, Results Math. 71 (2016), 1345-1357. https://doi.org/10.1007/s00025-016-0592-1.
  6. N. Khan, K. Khan, F. Tawfiq, J.S. Ro, I. Al-shbeil, Applications of Fractional Differential Operator to Subclasses of Uniformly q-Starlike Functions, Fractal Fract. 7 (2023), 715. https://doi.org/10.3390/fractalfract7100715.
  7. H.Ö. Güney, B. Yıldızhan, Certain Subclasses of Spirallike Univalent Functions Related with Generalized Distribution Series, Afr. Mat. 33 (2022), 16. https://doi.org/10.1007/s13370-021-00956-9.
  8. G. Murugusundaramoorthy, D. Raducanu, Certain Properties for Spiral-Like Functions Associated with Ruscheweyh-Type q-Difference Operator, Konuralp J. Math. 6 (2018), 218-225.
  9. M. Caputo, Linear Models of Dissipation Whose Q Is Almost Frequency Independent-II, Geophys. J. Int. 13 (1967), 529-539. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x.
  10. J. Salah, M. Darus, A Subclass of Uniformly Convex Functions Associated With a Fractional Calculus Operator Involving Caputo's Fractional Differentiation, Acta Univ. Apulensis 24 (2010), 295-304.
  11. J. Salah, M. Darus, A Note on Starlike Functions of Order α Associated with a Fractional Calculus Operator Involving Caputo's Fractional, J. Appl. Comput. Sci. Math. 5 (2011), 97-101.
  12. T. Sekine, S. Owa, Coefficient Inequalities for Certain Univalent Functions, Math. Inequal. Appl. 2 (1999), 535-544. https://doi.org/10.7153/mia-02-44.
  13. G.S. Salagean, Subclasses of Univalent Functions, in: C.A. Cazacu, N. Boboc, M. Jurchescu, I. Suciu, (eds) Complex Analysis-Fifth Romanian-Finnish Seminar, Lecture Notes in Mathematics, Vol 1013, Springer, Berlin, Heidelberg, (1983). https://doi.org/10.1007/BFb0066543.
  14. H. Silverman, Univalent Functions with Negative Coefficients, Proc. Amer. Math. Soc. 51 (1975), 109-116. https://doi.org/10.1090/s0002-9939-1975-0369678-0.
  15. H.U. Rehman, M. Darus, J. Salah, Graphing Examples of Starlike and Convex Functions of Order β, Appl. Math. Inf. Sci. 12 (2018), 509-515. https://doi.org/10.18576/amis/120305.
  16. H. Silverman, E.M. Silvia, D. Telage, Convolution Conditions for Convexity, Starlikeness and Spiral-Likeness, Math. Z. 162 (1978), 125-130. https://doi.org/10.1007/BF01215069.
  17. S. Altinkaya, A. Cetinkaya, Some Properties for Spirallike Functions Involving Generalized q-Integral Operator, Honam Math. J. 45 (2023), 689-700. https://doi.org/10.5831/HMJ.2023.45.4.689.
  18. T. Al-Hawary, M.O. Massa'deh, A.O. Fallatah, Some Inclusion Properties for Hohlov Operator to be in Comprehensive Subfamilies of Analytic Functions, Eur. J. Pure Appl. Math. 17 (2024), 3386-3398. https://doi.org/10.29020/nybg.ejpam.v17i4.5407.
  19. M. Darus, A Note on Geometric Function Theory and Recent Studies, AIP Conf. Proc. 1522 (2013), 29-32. https://doi.org/10.1063/1.4801100.
  20. F. Ghanim, H.F. Al-Janaby, O. Bazighifan, Geometric Properties of the Meromorphic Functions Class through Special Functions Associated with a Linear Operator, Adv. Contin. Discrete Models 2022 (2022), 17. https://doi.org/10.1186/s13662-022-03691-y.
  21. J.W. Alexander, Functions Which Map the Interior of the Unit Circle upon Simple Regions, Ann. Math. 17 (1915), 12-22. https://doi.org/10.2307/2007212.
  22. S.D. Bernardi, Convex and Starlike Univalent Functions, Trans. Am. Math. Soc. 135 (1969), 429-446. https://doi.org/10.1090/s0002-9947-1969-0232920-2.
  23. A.E. Livingston, on the Radius of Univalence of Certain Analytic Functions, Proc. Amer. Math. Soc. 17 (1966), 352-357.
  24. L.E. Dunduchenko, S.V. Goncharenko, Mapping of n-Connected Region onto a Plane with Cuts along the Arcs of Logarithmic Spirals, Math. Notes Acad. Sci. USSR 21 (1977), 183-185. https://doi.org/10.1007/BF01106741.
  25. M.G. Khan, B. Ahmad, G. Murugusundaramoorthy, R. Chinram, W.K. Mashwani, Applications of Modified Sigmoid Functions to a Class of Starlike Functions, J. Funct. Spaces 2020 (2020), 8844814. https://doi.org/10.1155/2020/8844814.
  26. J. Salah, H.U. Rehman, I. Al Buwaiqi, A. Al Azab, M. Al Hashmi, Subclasses of Spiral-like Functions Associated with the Modified Caputo's Derivative Operator, AIMS Math. 8 (2023), 18474-18490. https://doi.org/10.3934/math.2023939.
  27. R.M. Hicks, P.A. Henne, Wing Design by Numerical Optimization, J. Aircr. 15 (1978), 407-412. https://doi.org/10.2514/3.58379.
  28. A. Filippone, Airfoil Inverse Design and Optimization by Means of Viscous-Inviscid Techniques, J. Wind Eng. Ind. Aerodyn. 56 (1995), 123-136. https://doi.org/10.1016/0167-6105(94)00095-U.
  29. F.M. Al-Oboudi, On Univalent Functions Defined by a Generalized Sălăgean Operator, Int. J. Math. Math. Sci. 2004 (2004), 1429-1436. https://doi.org/10.1155/S0161171204108090.
  30. NIST Digital Library of Mathematical Functions, Chapter 5 Gamma Function, (2024). https://dlmf.nist.gov/5.
  31. E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge University Press, 1996. https://doi.org/10.1017/CBO9780511608759.
  32. H.M. Fahad, A. Fernandez, M.U. Rehman, M. Siddiqi, Tempered and Hadamard-Type Fractional Calculus with Respect to Functions, Mediterr. J. Math. 18 (2021), 143. https://doi.org/10.1007/s00009-021-01783-9.