Class of 𝓗𝓤𝓕 by fractional derivative operator
*Santosh Dnyandev JadhavCorresponding authorsdjmaths23@gmail.comDepartment of General EngineeringN. K. Orchid College of Engineering and TechnologySolapur, Maharashtra, 413002, IndiaView full profile → , Sopan Raosaheb Shindescholarswapnil@gmail.comDepartment of Computer EngineeringDr. D. Y. Patil College of Engineering and InnovationPune, Maharashtra, 410507, IndiaView full profile → , Kalyani Vishal Dokejivansagharsh@gmail.comDepartment of Engineering Science and HumanitiesSandip Institute of Technology and Research Centre (SITRC)Mahiravani Trimbak Road, Nashik, Maharashtra, 422213, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Feb 2026
- Published Online:
- 25 Aug 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2632
- Pages:
- 2473–2480
Abstract
This research study, we explore a family of CV map both univalent and sensepreserve within Z, and which are expressed in ℏ( ̃) = l( ̃)+(κ( ̃)) whereas l( ̃) with κ( ̃) are analytic throughout Z.
A central aspect newly introduced class defined through a generalized fractional differential operator. By setting in this operator, we define the function class as SHPα (ρ̃1,...,℘) belongs to family of 𝓗𝓤𝓕. Alongwith we derive sufficient conditions on the coefficients for functions in this class, and further these conditions become necessary when the coefficients are negative. Additionally, we examine key function-theoretic features of this class, including estimates for distortion and the characterization of its extreme points.
Keywords
Subject Classifications
References
[1] H. Silverman, “Harmonic univalent functions with negative coefficients,” J. Math. Anal. Appl., vol. 220, no. 1, pp. 283–289 (1998).
[2] H. Lewy, “On the non-vanishing of the Jacobian in certain one-to-one mappings,” Bull. Amer. Math. Soc., vol. 42, pp. 689–692 (1936).
[3] J. S. Liu, “The Noor integral and strongly starlike functions,” J. Math. Anal. Appl., vol. 261, pp. 441–447 (2001).
[4] J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Ann. Acad. Sci. Fenn. Math., vol. 9, pp. 3–25 (1984).
[5] S. R. Shinde, Study of Complex Valued Topological Space and Soft Topological Space in Context with Fixed Point Theory, Ph.D. dissertation, ProQuest Dissertations & Theses Global, ID: 3328763966 (2024).
[6] J. L. Liu and K. I. Noor, “Some properties of Noor integral operator,” J. Nat. Geom., vol. 21, pp. 81–90 (2002).
[7] K. I. Noor and M. A. Noor, “On certain classes of analytic functions defined by Noor integral operator,” J. Math. Anal. Appl., vol. 281, pp. 244–252 (2003).
[8] K. I. Noor, B. Malik, and S. Z. H. Bukhari, “Harmonic functions defined by a generalized fractional differential operator,” J. Adv. Math. Stud., vol. 2, no. 1, pp. 41-52 (2009).
[9] L. Li and S. Ponnusamy, “Injectivity of sections of univalent harmonic mappings,” Nonlinear Anal., vol. 89, pp. 276-283 (2013).
[10] N. E. Cho, “The Noor integral operator and strongly close-to-convex functions,” J. Math. Anal. Appl., vol. 283, pp. 202-212 (2003).
[11] S. Mishra, H. Nagar, N. Mani, and R. Shukla, “NE transform of pathway fractional integrals involving S-function,” AIMS Math., vol. 10, no. 2, pp. 3013-3024 (2025).
[12] A. N. Abdulhadi and H. O. Altaie, “Solutions of fractional partial differential equations by combined new technique NTADM,” J. Interdiscip. Math., vol. 29, no. 5-B, pp. 1683-1688 (2026).
[13] O. P. Ahuja, “Planar harmonic univalent and related mappings,” J. Inequal. Pure Appl. Math., vol. 6, no. 4, Art. no. 122 (2005).
[14] S. R. Shinde, “Convergence: Partially ordered soft topological space,” J. Interdiscip. Math., vol. 27, no. 2, pp. 191-200 (2024).
[15] S. Owa and H. M. Srivastava, “Univalent and starlike generalized hypergeometric functions,” Canad. J. Math., vol. 39, pp. 1057-1077 (1987).
[16] S. R. Shinde, “Complex valued approach to the system of nonlinear second order boundary value problem and multivalued mapping via fixed point method,” Chebyshevskii Sbornik, vol. 24, no. 3, pp. 212-227 (2023).
