Certain novel generalized subclasses of uniformly starlike and convex functions: coefficient characterizations and inclusion properties through Bessel and Gaussian hypergeometric functions

Muhammad Imran Faisal; Maslina Darus; Georgia Irina Oros; Muhammad Imran Faisal; Maslina Darus; Georgia Irina Oros

doi:10.3934/math.2026127

AIMS Mathematics

2026, Volume 11, Issue 2: 3171-3192. doi: 10.3934/math.2026127

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Research article

Certain novel generalized subclasses of uniformly starlike and convex functions: coefficient characterizations and inclusion properties through Bessel and Gaussian hypergeometric functions

1.
Mathematics Department, Taibah University, Universities Road, P. O. Box 344, Medina, Kingdom of Saudi Arabia
2.
Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia
3.
Department of Mathematics and Computer Science, University of Oradea, Universitatii 1, Oradea 410087, Romania

Received: 11 November 2025 Revised: 13 January 2026 Accepted: 16 January 2026 Published: 02 February 2026
MSC : 30C45, 30C80

In this scholarly article, we present novel generalized subclasses of $ \nu $ uniformly starlike functions of order $ \rho $, designated as $ M(\tau, \rho, \nu) $, alongside $ \nu $ uniformly convex functions of order $ \rho $, referred to as $ N(\tau, \rho, \nu) $. We provide comprehensive coefficient characterizations that delineate the conditions under which analytic functions are classified within the newly established subclasses of uniformly starlike and uniformly convex families, respectively. Furthermore, we conduct an analysis of the implications of the Bessel function and explore the consequences of the Gaussian hypergeometric function on these mathematical classes to substantiate an inclusion property for analytic functions that reside within these subclasses.
- analytic functions,
- starlike functions,
- convex functions,
- Bessel function of the first kind,
- Gaussian hypergeometric function
Citation: Muhammad Imran Faisal, Maslina Darus, Georgia Irina Oros. Certain novel generalized subclasses of uniformly starlike and convex functions: coefficient characterizations and inclusion properties through Bessel and Gaussian hypergeometric functions[J]. AIMS Mathematics, 2026, 11(2): 3171-3192. doi: 10.3934/math.2026127

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Abstract

In this scholarly article, we present novel generalized subclasses of $ \nu $ uniformly starlike functions of order $ \rho $, designated as $ M(\tau, \rho, \nu) $, alongside $ \nu $ uniformly convex functions of order $ \rho $, referred to as $ N(\tau, \rho, \nu) $. We provide comprehensive coefficient characterizations that delineate the conditions under which analytic functions are classified within the newly established subclasses of uniformly starlike and uniformly convex families, respectively. Furthermore, we conduct an analysis of the implications of the Bessel function and explore the consequences of the Gaussian hypergeometric function on these mathematical classes to substantiate an inclusion property for analytic functions that reside within these subclasses.

References

[1]	A. Naz, S. Nagpal, V. Ravichandran, Geometric properties of generalized Bessel function associated with the exponential function, Math. Slovaca, 73 (2023), 1459–1478. https://doi.org/10.1515/ms-2023-0106 doi: 10.1515/ms-2023-0106
[2]	A. K. Pandey, L. Parida, A. K. Sahoo, S. Barik, S. K. Paikray, On certain subclass of multivalent functions associated with Bessel functions, Integral Transforms Spec. Funct., 36 (2025), 807–823. https://doi.org/10.1080/10652469.2025.2460011 doi: 10.1080/10652469.2025.2460011
[3]	D. A. Bradac-Vlada, Univalence studies concerning a new integral operator involving Bessel function of the first kind, J. Inequal. Spec. Funct., 16 (2025), 60–71. https://doi.org/10.54379/jiasf-2025-2-5 doi: 10.54379/jiasf-2025-2-5
[4]	L. I. Cotîrlă, R. Szász, On the monotony of Bessel functions of the first kind, Comput. Methods Funct. Theory, 24 (2024), 747–752. https://doi.org/10.1007/s40315-023-00498-0 doi: 10.1007/s40315-023-00498-0
[5]	M. Sharma, N. K. Jain, S. Kumar, Monotonicity, starlikeness and subordination related results for some special functions, Iran. J. Sci., 2025. https://doi.org/10.1007/s40995-025-01847-5
[6]	G. I. Oros, G. Oros, D. A. Bardac-Vlada, Study on the criteria for starlikeness in integral operators involving Bessel functions, Symmetry, 15 (2023), 1976. https://doi.org/10.3390/sym15111976 doi: 10.3390/sym15111976
[7]	D. A. Bradac-Vlada, G. I. Oros, Applications of a new linear operator involving Dziok-Srivastava operator and Bessel function of the first kind, Turk. J. Math., 49 (2025), 649–667. https://doi.org/10.55730/1300-0098.3615 doi: 10.55730/1300-0098.3615
[8]	S. Mandal, M. M. Soren, Fourth-order differential subordination results for analytic functions involving the generalized Bessel functions, TWMS J. Appl. Eng. Math., 15 (2025), 1245–1258.
[9]	A. Bunyan, F. Ghanim, H. F. Al-Janaby, A new investigation into a linear operator connected to Gaussian hypergeometric functions, Iraqi J. Sci., 65 (2024), 3270–3278. https://doi.org/10.24996/ijs.2024.65.6.26 doi: 10.24996/ijs.2024.65.6.26
[10]	P. Yadav, S. Joshi, H. Pawar, Mapping properties of holomorphic function associated with Gaussian hypergeometric function, TWMS J. App. Eng. Math., 14 (2024), 1759–1771.
[11]	N. E. Cho, S. Y. Woo, S. Owa, Uniform convexity properties for hypergeometric functions, Fract. Calculus Appl. Anal., 3 (2002), 303–313.
[12]	G. I. Oros, A. M. Rus, A note on convexity properties for Gaussian hypergeometric function, Contemp. Math., 5 (2024), 2801–2813. https://doi.org/10.37256/cm.5320242739 doi: 10.37256/cm.5320242739
[13]	E. Merkes, B. T. Scott, Starlike hypergeometric functions, Proc. Amer. Math. Soc., 12 (1961), 885–888.
[14]	B. C. Carlson, D. B. Shaffer, Starlike and prestarlike hypergeometric functions, J. Math. Anal. Appl., 15 (1984), 737–745. https://doi.org/10.1137/0515057 doi: 10.1137/0515057
[15]	S. T. Ruscheweyh, V. Singh, On the order of starlikeness of hypergeometric functions, J. Math. Anal. Appl., 113 (1986), 1–11. https://doi.org/10.1016/0022-247X(86)90329-X doi: 10.1016/0022-247X(86)90329-X
[16]	L. F. Preluca, Third-order strong differential subordinations involving fractional integral of confluent (Kummer) and Gaussian hypergeometric functions, Gen. Math., 32 (2024), 28–45. https://doi.org/10.2478/gm-2024-0003 doi: 10.2478/gm-2024-0003
[17]	A. K. Sharma, P. Saurabh, K. K. Dixit, Class mappings properties of convolutions involving certain univalent functions associated with hypergeometric functions, Electron. J. Math. Anal. Appl., 1 (2013), 326–333.
[18]	G. Murugusundaramoorthy, Subclasses of starlike and convex functions involving Poisson distribution series, Afr. Mat., 28 (2017), 1357–1366. https://doi.org/10.1007/s13370-017-0520-x doi: 10.1007/s13370-017-0520-x
[19]	S. Porwal, K. K. Dixit, An application of generalized Bessel functions on certain analytic functions, Acta Univ. Matthiae Belii Ser. Math., 21 (2013), 51–57.
[20]	V. Srinivas, P. T. Reddy, H. Niranjan, Some properties of certain class of uniformly convex functions defined by Bessel functions, In: V. Madhu, A. Manimaran, D. Easwaramoorthy, D. Kalpanapriya, M. M. Unnissa, Advances in algebra and analysis, Springer, 2019. https://doi.org/10.1007/978-3-030-01120-8_16
[21]	H. Silverman, Starlike and convexity properties for hypergeometric functions, J. Math. Anal. Appl., 172 (1993), 574–581. https://doi.org/10.1006/jmaa.1993.1044 doi: 10.1006/jmaa.1993.1044
[22]	M. I. Faisal, M. Darus, A study of a special family of analytic functions at infinity, Appl. Math. Lett. 25 (2012), 654–657. https://doi.org/10.1016/j.aml.2011.10.007 doi: 10.1016/j.aml.2011.10.007
[23]	M. I. Faisal, Study of analytic functions related with simply connected convex set, Iran. J. Sci. Technol. Trans. A Sci., 43 (2019), 2595–2600. https://doi.org/10.1007/s40995-019-00751-z doi: 10.1007/s40995-019-00751-z
[24]	M. I. Faisal, Study of simply connected convex domain and its geometric properties (Ⅱ), J. Taibah Univ. Sci., 13 (2019), 597–603. https://doi.org/10.1080/16583655.2019.1611368 doi: 10.1080/16583655.2019.1611368
[25]	M. S. Robertson, On the theory of univalent functions, Ann. Math., 37 (1936), 374–408. https://doi.org/10.2307/1968451 doi: 10.2307/1968451
[26]	H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), 109–116.
[27]	K. K. Dixit, S. K. Pal, On a class of univalent functions related to complex order, Indian J. Pure Appl. Math., 26 (1995), 889–896.
[28]	A. Baricz, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen, 73 (2008), 155–178. https://doi.org/10.5486/pmd.2008.4126 doi: 10.5486/pmd.2008.4126
[29]	L. M. Fatunsin, T. O. Opoola, On new subclasses of analytic functions associated with generalized Bessel functions, Int. J. Nonlinear Anal. Appl., 2024. https://doi.org/10.22075/ijnaa.2024.34524.5155
[30]	B. A. Frasin, F. Yousef, T. Al-Hawary, Application of generalized Bessel functions to classes of analytic functions, Afr. Mat., 32 (2021), 431–439. https://doi.org/10.1007/s13370-020-00835-9 doi: 10.1007/s13370-020-00835-9
[31]	M. U. Nawaz, D. Breaz, M. Raza, L. Cotîrlă, Starlikeness and convexity of generalized Bessel-maitland function, Axioms, 13 (2024), 69. https://doi.org/10.3390/axioms13100691 doi: 10.3390/axioms13100691
[32]	G. Murugusundaramoorthy, T. Janani, An application of generalized Bessel functions on certain subclasses of analytic functions, Turk. J. Anal. Number Theory, 3 (2015), 1–6. https://doi.org/10.12691/tjant-3-1-1 doi: 10.12691/tjant-3-1-1
[33]	A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, McGraw-Hill, 1953.
[34]	N. M. Temme, Special functions, New York, Wiley, 1996.
[35]	P. Hästö, S. Ponnusamy, M. Vuorinen, Starlikeness of the Gaussian hypergeometric functions, Complex. Var. Elliptic Equations, 55 (2010), 173–184. https://doi.org/10.1080/17476930903276134 doi: 10.1080/17476930903276134
[36]	S. Shams, S. R. Kulkarni, J. M. Jahangiri, Classes of uniformly starlike and convex functions, Int. J. Math. Math. Sci., 55 (2004), 2959–2961. https://doi.org/10.1155/S0161171204402014 doi: 10.1155/S0161171204402014
[37]	S. Owa, Y. Polatoglu, E. Yavuz, Coefficient inequalities for classes of uniformly starlike and convex functions, J. Inequal. Pure Appl. Math., 7 (2006), 160.
[38]	R. Bharati, R. Parvatham, A. Swaminathan, On subclasses of uniformly convex functions and corresponding class of starlike functions, Tamkang J. Math., 6 (1997), 17–32.
[39]	K. G. Subramanian, T. V. Sudharsan, P. Balasubrahmanyam, H. Silverman, Classes of uniformly starlike functions, Publ. Math. Debrecen, 53 (1998), 309–315.
[40]	K. G. Subramanian, G. Murugusundaramoorthy, P. Balasubrahmanyam, H. Silverman, Subclasses of uniformly convex and uniformly starlike functions, Math. Jpn., 42 (1995), 517–522.
[41]	O. Altintas, S. Owa, On subclasses of univalent functions with negative coefficients, Pusan Kyongnam Math. J., 4 (1988), 41–56.
[42]	A. Baricz, Generalized Bessel functions of the first kind, Springer-Verlag, 2010. https://doi.org/10.1007/978-3-642-12230-9

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