On classes of analytic functions with respect to symmetric points defined using the Mittag-Leffler function of Le Roy type

Kadhavoor R. Karthikeyan; Sakkarai Lakshmi; Dharmaraj Mohankumar; Kadhavoor R. Karthikeyan; Sakkarai Lakshmi; Dharmaraj Mohankumar

doi:10.3934/math.2026947

AIMS Mathematics

2026, Volume 11, Issue 7: 23490-23509. doi: 10.3934/math.2026947

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On classes of analytic functions with respect to symmetric points defined using the Mittag-Leffler function of Le Roy type

1.
Department of Applied Mathematics and Science, National University of Science & Technology, Muscat, P.O. Box 620, Oman
2.
Mathematics Section, College of Computing and Information Sciences, University of Technology and Applied Sciences - Al Musanah, P.O. Box: 191, Postal Code: 314, Al Mulladha, Oman
3.
Department of Mathematics for Innovation, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences (SIMATS), Thandalam, Chennai, 602 105, India

Received: 04 March 2026 Revised: 13 July 2026 Accepted: 22 July 2026 Published: 31 July 2026
MSC : 30C45, 30C80

In this article, we will define an operator using the Hadamard product of an analytic function involving the Mittag-Leffler function of the Le Roy type. The defined operator extends and unifies the well-known operators that were defined to study various subclasses of univalent functions. Using the defined operator, we introduce a new family of analytic functions with respect to symmetric points. The analytic characterization of the defined function class was mainly motivated by the so-called $ \alpha $-exponentially convex functions. Further, the defined function class involves $ (r, \, \ell) $-symmetrical function which is a generalization of the notions such as even, odd, and $ \ell $-symmetrical functions. Estimates of the initial bounds, Fekete-Szegő inequality, and the integral representation of the defined function class are the main results of the paper. Applications of the main results are presented as inclusion relations and corollaries.
- starlike function,
- convex function,
- exponential convex function,
- Mittag-Leffler function of the Le Roy type,
- Fekete-Szegő inequality, subordination
Citation: Kadhavoor R. Karthikeyan, Sakkarai Lakshmi, Dharmaraj Mohankumar. On classes of analytic functions with respect to symmetric points defined using the Mittag-Leffler function of Le Roy type[J]. AIMS Mathematics, 2026, 11(7): 23490-23509. doi: 10.3934/math.2026947

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Abstract

In this article, we will define an operator using the Hadamard product of an analytic function involving the Mittag-Leffler function of the Le Roy type. The defined operator extends and unifies the well-known operators that were defined to study various subclasses of univalent functions. Using the defined operator, we introduce a new family of analytic functions with respect to symmetric points. The analytic characterization of the defined function class was mainly motivated by the so-called $ \alpha $-exponentially convex functions. Further, the defined function class involves $ (r, \, \ell) $-symmetrical function which is a generalization of the notions such as even, odd, and $ \ell $-symmetrical functions. Estimates of the initial bounds, Fekete-Szegő inequality, and the integral representation of the defined function class are the main results of the paper. Applications of the main results are presented as inclusion relations and corollaries.

References

[1]	D. Breaz, K. R. Karthikeyan, A. Senguttuvan, D. Mohankumar, Properties of a subclass of starlike functions involving the quantum derivative operator, MethodsX, 15 (2025), 103740. https://doi.org/10.1016/j.mex.2025.103740 doi: 10.1016/j.mex.2025.103740
[2]	D. Breaz, K. R. Karthikeyan, D. Mohankumar, On classes of non-univalent functions influenced by the multiplicative derivative, Front. Appl. Math. Stat., 12 (2026), 1777576. https://doi.org/10.3389/fams.2026.1777576 doi: 10.3389/fams.2026.1777576
[3]	S. S. Varma, T. Rosy, U. Vadivelan, Radius of exponential convexity of certain subclass of analytic functions, Creat. Math. Inform., 29 (2020), 109–112. https://doi.org/10.37193/CMI.2020.01.13 doi: 10.37193/CMI.2020.01.13
[4]	H. Tang, M. Abbas, R. K. Alhefthi, M. Arif, Improvement on Hankel determinant bounds for specific holomorphic functions, Acta Math. Sci., 46 (2026), 39–61. https://doi.org/10.1007/s10473-026-0103-8 doi: 10.1007/s10473-026-0103-8
[5]	H. Tang, M. Abbas, R. K. Alhefthi, M. Arif, The multi-index Mittag-Leffler-Le Roy functions as I- and $\overline{H}$-functions and new fractional calculus operators, IFAC-PapersOnLine, 58 (2024), 67–72. https://doi.org/10.1016/j.ifacol.2024.08.168 doi: 10.1016/j.ifacol.2024.08.168
[6]	S. Gerhold, Asymptotics for a variant of the Mittag-Leffler function, Integr. Transf. Spec. F., 23 (2012), 397–403. https://doi.org/10.1080/10652469.2011.596151 doi: 10.1080/10652469.2011.596151
[7]	R. Garra, F. Polito, On some operators involving Hadamard derivatives, Integr. Transf. Spec. F., 24 (2013), 773–782. https://doi.org/10.1080/10652469.2012.756875 doi: 10.1080/10652469.2012.756875
[8]	T. R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7–15.
[9]	H. M. Srivastava, Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput., 211 (2009), 198–210. https://doi.org/10.1016/j.amc.2009.01.055 doi: 10.1016/j.amc.2009.01.055
[10]	Ž. Tomovski, K.Mehrez, Some families of generalized Mathieu-type power series, associated probability distributions and related inequalities involving complete monotonicity and log-convexity, Math. Inequal. Appl., 20 (2017), 973–986. https://doi.org/10.7153/mia-2017-20-61 doi: 10.7153/mia-2017-20-61
[11]	K. R. Karthikeyan, K. A. Reddy, G. Murugusundaramoorthy, On classes of Janowski functions associated with a conic domain, Ital. J. Pure Appl. Mat., 47 (2019), 684–698.
[12]	K. A. Reddy, K. R. Karthikeyan, G. Murugusundaramoorthy, Inequalities for the Taylor coefficients of spiralike functions involving $q$-differential operator, Eur. J. Pure Appl. Math., 12 (2019), 846–856. https://doi.org/10.29020/nybg.ejpam.v12i3.3429 doi: 10.29020/nybg.ejpam.v12i3.3429
[13]	İ. Aktaş, M. Görgülü, Further geometric properties of the Mittag-Leffler-Prabhakar functions of Le Roy type, J. Inequal. Appl., 2025 (2025), 155. https://doi.org/10.1186/s13660-025-03403-6 doi: 10.1186/s13660-025-03403-6
[14]	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 doi: 10.1155/S0161171204108090
[15]	G. Ş. Sălăgean, Subclasses of univalent functions, In: Complex analysis—fifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981), Berlin: Springer, 1013 (1981), 362–372. http://dx.doi.org/10.1007/BFb0066543
[16]	A. W. Goodman, Univalent functions, Mariner Publishing Co., Inc., Tampa, FL, 1983.
[17]	I. Graham, G. Kohr, Geometric function theory in one and higher dimensions, Boca Raton: CRC Press, 2003. http://dx.doi.org/10.1201/9780203911624
[18]	D. K. Thomas, N. Tuneski, A. Vasudevarao, Univalent functions: A primer, Berlin, Boston: De Gruyter, 2018. http://dx.doi.org/10.1515/9783110560961
[19]	W. Ma, D. Minda, A unified treatment of some special classes of univalent functions, In: Proceedings of the Conference on Complex Analysis, International Press Inc., 1992,157–169.
[20]	K. R. Karthikeyan, S. Lakshmi, S. Varadharajan, D.Mohankumar, E. Umadevi, Starlike functions of complex order with respect to symmetric points defined using higher order derivatives, Fractal Fract., 6 (2022), 116. https://doi.org/10.3390/fractalfract6020116 doi: 10.3390/fractalfract6020116
[21]	K. Sharma, N. K. Jain, V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat., 27 (2016), 923–939. https://doi.org/10.1007/s13370-015-0387-7 doi: 10.1007/s13370-015-0387-7
[22]	K. Arora, S. S. Kumar, Starlike functions associated with a petal shaped domain, B. Korean Math. Soc., 59 (2022), 993–1010.
[23]	G. Murugusundaramoorthy, T. Bulboacă, Hankel determinants for new subclasses of analytic functions related to a shell shaped region, Mathematics, 8 (2020), 1041. https://doi.org/10.3390/math8061041 doi: 10.3390/math8061041
[24]	R. K. Raina, J. Sokół, Some properties related to a certain class of starlike functions, C. R. Math. Acad. Sci. Paris, 353 (2015), 973–978. https://doi.org/10.1016/j.crma.2015.09.011 doi: 10.1016/j.crma.2015.09.011
[25]	B. Khan, H. M. Srivastava, N. Khan, M. Darus, M. Tahir, Q. Z. Ahmad, 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 doi: 10.3390/math8081334
[26]	J. H. Arango, D. Mejía, S. Ruscheweyh, Exponentially convex univalent functions, Complex Var. Theory Appl., 33 (1997), 33–50. https://doi.org/10.1080/17476939708815010 doi: 10.1080/17476939708815010
[27]	K. R. Karthikeyan, E. Umadevi, G. Thirupathi, D. Mohankumar, Properties of a class of analytic functions associated with exponentially convex functions, J. Mech. Con. Math. Sci., 20 (2025), 1–10. https://doi.org/10.26782/jmcms.2025.08.00001 doi: 10.26782/jmcms.2025.08.00001
[28]	K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Jpn., 11 (1959), 72–75. https://doi.org/10.2969/jmsj/01110072 doi: 10.2969/jmsj/01110072
[29]	Z. G. Wang, C. Y. Gao, S. M. Yuan, On certain subclasses of close-to-convex and quasi-convex functions with respect to $k$-symmetric points, J. Math. Anal. Appl., 322 (2006), 97–106. https://doi.org/10.1016/j.jmaa.2005.08.060 doi: 10.1016/j.jmaa.2005.08.060
[30]	S. M. Yuan, Z. M. Liu, Some properties of $\alpha$-convex and $\alpha$-quasiconvex functions with respect to $n$-symmetric points, Appl. Math. Comput., 188 (2007), 1142–1150. https://doi.org/10.1016/j.amc.2006.10.060 doi: 10.1016/j.amc.2006.10.060
[31]	P. Liczberski, J. Połubiński, On $(j, k)$-symmetrical functions, Math. Bohem., 120 (1995), 13–28.
[32]	K. R. Karthikeyan, Some classes of analytic functions with respect to $(j, k)$-symmetric points, ROMAI J., 9 (2013), 51–60.
[33]	P. Gochhayat, A. Prajapati, Geometric properties on $(j, k)$-symmetric functions related to starlike and convex function, Commun. Korean Math. S., 37 (2022), 455–472.
[34]	H. M. Srivastava, S. Z. H. Bukhari, M. Nazir, A subclass of $\alpha$-convex functions with respect to $(2j, k)$-symmetric conjugate points, B. Iran. Math. Soc., 44 (2018), 1227–1242. https://doi.org/10.1007/s41980-018-0086-x doi: 10.1007/s41980-018-0086-x
[35]	L. D. Branges, A proof of the Bieberbach conjecture, Acta Math., 154 (1985), 137–152. https://doi.org/10.1007/BF02392821 doi: 10.1007/BF02392821
[36]	I. Efraimidis, A generalization of Livingston's coefficient inequalities for functions with positive real part, J. Math. Anal. Appl., 435 (2016), 369–379. https://doi.org/10.1016/j.jmaa.2015.10.050 doi: 10.1016/j.jmaa.2015.10.050
[37]	A. W. Pommerenke, Univalent functions, Göttingen: Vandenhoeck & Ruprecht, 1975.
[38]	S. S. Kumar, G. Kamaljeet, A cardioid domain and starlike functions, Anal. Math. Phys., 11 (2021), 54. https://doi.org/10.1007/s13324-021-00483-7 doi: 10.1007/s13324-021-00483-7
[39]	A. Ahmad, J. Gong, I. Al-Shbeil, A. Rasheed, A. Ali, S. Hussain, Analytic functions related to a Balloon-Shaped domain, Fractal Fract., 7 (2023), 865. https://doi.org/10.3390/fractalfract7120865 doi: 10.3390/fractalfract7120865
[40]	K. R. Karthikeyan, Starlike functions associated with $Q$-multiplicative derivative, B. Transilvania Univ. Brasov Series III: Math. Comput. Sci., 6(68) (2026), 99–112. https://doi.org/10.31926/but.mif.2026.6.68.1.8 doi: 10.31926/but.mif.2026.6.68.1.8

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