A study on third-order subordination of holomorphic functions using a combined convolutions operator
*Mustafa I. HameedCorresponding authormustafa8095@uoanbar.edu.iqDepartment of MathematicsCollege of Education for Pure SciencesUniversity of AnbarRamadi, 31001, Iraq0000-0003-0726-3180View full profile → , Muthanna Mishlishmuthana.kh.m@uoanbar.edu.iqDepartment of MathematicsCollege of Education for Pure SciencesUniversity of AnbarRamadi, 31001, Iraq0009-0002-6071-5705View full profile → , Kayode Oshinubioshinubik@gmail.comSchool of InformaticsComputing and Cyber SystemNorthern Arizona UniversityFlagstaff AZ, U.S.A.0000-0003-4598-8510View full profile → , Hussaini Joshuajhussaini@unimaid.edu.ngDepartment of MathematicsFaculty of ScienceUniversity of MaiduguriBorno, 600004, Nigeria0000-0001-8236-6900View full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Dec 2025
- Published Online:
- 16 Jul 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2601
- Pages:
- 2363–2373
Abstract
In the present study, our team demonstrated novel findings through the use of the new operator, which was obtained via the convolution product of the four amora-arus. Includes certain differential subordination results of the third degree, including the operator MNω,μβ,δ,α,γ (τ, ϱ)Υ(ρ), and this is obtaining the dependence. We also looked into the group for embraced functions, such as the government relations office’s subordination derivative as well as the Srivastava operator. Additionally, find the operator’s third difference level MNω,μβ,δ,α,γ (τ, ϱ)Υ(ρ) and prove a number of theorems to identify the optimal subordinate. In the final section of the study, some sandwich-type theories were addressed, as well as determining the best dominant and subordinate ones. keeps its geometric functions. Initialization results can be obtained within the unit disk by imposing limitations on functions that belong to subclass properties.
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References
[1] T.H. Gronwall, “On the distortion in conformal mapping when the second coefficient in the mapping function has an assignedvalue”, Proceedings of the National Academy of Sciences, vol. 6, pp. 300-302 (1920).
[2] M. Finkelstein, “Growth estimates of convex functions”, Proceedings of the American Mathematical Society, vol. 18, no. 3, pp. 412-418 (1967).
[3] D.E. Tepper, “On the radius of convexity and boundary distortion of schlicht functions”, Transactions of the American Mathematical Society, vol. 150, no. 2, pp. 519-528 (1970).
[4] K.S. Padmanabhan, “Estimates of growth for certain convex and close-to-convex functions in the unit disc”, Journal of the Indian Mathematical Society, vol. 33, pp. 37-47 (1969).
[5] A.A. Kilbas, M. Saigo and R.K. Saxena, “Generalized Mittag-Leffler function and generalized fractional calculus operators”, Integral Transform. Spec. Funct, vol. 15, pp. 31–49 (2004).
[6] H. Irmak, and M. Şan, “Some relations between certain inequalities concerning analytic and univalent functions”, Applied Mathematics Letters, vol. 23, no. 8, pp. 897-901 (2010).
[7] S.S. Miller and P.T. Mocanu, “Differential Subordination”, Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Marcel-Dekker Inc, New York and Bassel, vol. 225 (2000).
[8] S.S. Miller and P.T. Mocanu, “Subordinants of differential superordinations”, Complex Variables, Theory and Application: An International Journal, vol. 48, no. 10, pp. 815-826 (2003).
[9] A.F. Talak, M.I. Hameed and A.K. Yaqoob, “Study certain subclasses of holomorphic functions for quasi-subordination”, Journal of Interdisciplinary Mathematics, vol. 28, no. 3, pp. 865-871 (2025).
[10] S. Ruscheweyh, “New criteria for univalent functions”, Proceedings of the AMS., vol. 10, no. 9, pp. 10-15 (1975).
[11] M.I. Hameed, B.N. Shihab and K.A. Jassim, “Some properties of subclass of P-valent function with new generalized operator”, In AIP Conference Proceedings, vol. 2394, no. 1, pp. 070006 (2022).
[12] F.N. Hammad and M.I. Hameed, “Certain meromorphic functions with the convolution operator”, Journal of Interdisciplinary Mathematics, vol. 28, no. 3, pp. 945–951 (2025).
[13] H.F. Al-Janaby and F. Ghanim, “Sandwich-type outcome based on a dual linear operator”, Int J pure Appl. Math., vol. 118, no. 3, pp. 819-835 (2018).
[14] A. Amourah and M. Darus, “Some Properties of a New class of univalent functions involving a new generalized differential operator with negative coefficient”, Indian J. Sci. Tech., vol. 9, no. 36, pp. 1-7 (2016).
[15] H.M. Srivastava, A. Prajapat and P. Gachhayat, “Third-order differential subordination and differential subordinations results for analytic functions involving the Srivastava-Attiya operator”, Appl. Math. Inf. Sci., vol. 12, no. 3, pp. 469-481 (2018).
[16] S.J. Al-Dulaimi, M.I. Hameed and I.I. Hameed, “Results of third order differential subordination for holomorphic functions”, Journal of Interdisciplinary Mathematics, vol. 27, no. 4, pp. 881-886 (2024).
