Close-to-convexity and partial sums for normalized Le Roy-type $ q $-Mittag-Leffler functions

Khaled Matarneh; Suha B. Al-Shaikh; Mohammad Faisal Khan; Ahmad A. Abubaker; Javed Ali; Khaled Matarneh; Suha B. Al-Shaikh; Mohammad Faisal Khan; Ahmad A. Abubaker; Javed Ali

doi:10.3934/math.2025644

AIMS Mathematics

2025, Volume 10, Issue 6: 14288-14313. doi: 10.3934/math.2025644

Previous Article Next Article

Research article

Close-to-convexity and partial sums for normalized Le Roy-type $ q $-Mittag-Leffler functions

1.
Faculty of Computer Studies, Arab Open University, Riyadh 11681, Saudi Arabia
2.
Department of Basic Sciences, College of Science, and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi Arabia
3.
College of Computing and Informatics, Saudi Electronic University, Riyadh 11673, Saudi Arabia

Received: 09 April 2025 Revised: 25 May 2025 Accepted: 28 May 2025 Published: 23 June 2025
MSC : 05A30, 11B65, 30C45, 47B38

In recent years, researchers have explored the properties of close-to-convexity and partial sums for various Mittag-Leffler functions, including $ q $-Mittag-Leffler, Bernas Mittag-Leffler, and Le Roy-type Mittag-Leffler functions. Building on previous research, this paper explores the Le Roy-type $ q $-Mittag-Leffler function in the open unit disk, with a specific focus on its normalization. We also use the concept of $ q $-close-to-convex functions, and examine whether the Le Roy-type $ q $-Mittag-Leffler function possesses close-to-convexity. Moreover, we determine lower bounds for the normalized Le Roy-type $ q $-Mittag-Leffler and its sequence of partial sums. We also present some lemmas, propositions, examples, and meaningful corollaries that highlight the importance of our findings. The results presented in this paper are new and demonstrate improvements over some existing findings in the literature.
- analytic functions,
- Mittag-Leffler functions,
- $ q $-Mittag-Leffler function,
- $ q $-difference operator,
- close-to-convexity,
- partial sums,
- Le Roy-type $ q $-Mittag-Leffler function
Citation: Khaled Matarneh, Suha B. Al-Shaikh, Mohammad Faisal Khan, Ahmad A. Abubaker, Javed Ali. Close-to-convexity and partial sums for normalized Le Roy-type $ q $-Mittag-Leffler functions[J]. AIMS Mathematics, 2025, 10(6): 14288-14313. doi: 10.3934/math.2025644

Related Papers:

Abstract

In recent years, researchers have explored the properties of close-to-convexity and partial sums for various Mittag-Leffler functions, including $ q $-Mittag-Leffler, Bernas Mittag-Leffler, and Le Roy-type Mittag-Leffler functions. Building on previous research, this paper explores the Le Roy-type $ q $-Mittag-Leffler function in the open unit disk, with a specific focus on its normalization. We also use the concept of $ q $-close-to-convex functions, and examine whether the Le Roy-type $ q $-Mittag-Leffler function possesses close-to-convexity. Moreover, we determine lower bounds for the normalized Le Roy-type $ q $-Mittag-Leffler and its sequence of partial sums. We also present some lemmas, propositions, examples, and meaningful corollaries that highlight the importance of our findings. The results presented in this paper are new and demonstrate improvements over some existing findings in the literature.

References

[1]	P. L. Duren, Univalent functions, New York: Springer, 1983.
[2]	J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. Math., 17 (1915), 12–22. http://doi.org/10.2307/2007212 doi: 10.2307/2007212
[3]	M. E. H. Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Variables, Theory and Application, 14 (1990), 77–84. https://doi.org/10.1080/17476939008814407 doi: 10.1080/17476939008814407
[4]	H. M. Srivastava, S. Owa, Univalent functions, fractional calculus, and their applications, New York: John Wiley and Sons, 1989.
[5]	G. Gasper, M. Rahman, Basic hypergeometric series, 2 Eds., Cambridge: Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511526251
[6]	H. M. Srivastava, P. W. Karlsson, Multiple Gaussian hypergeometric series, Chichester: Ellis Horwood, 1984.
[7]	F. H. Jackson, $q$-Difference equations, Am. J. Math., 32 (1910), 305–314. https://doi.org/10.2307/2370183 doi: 10.2307/2370183
[8]	V. Kac, P. Cheung, Quantum calculus, New York: Springer, 2002. https://doi.org/10.1007/978-1-4613-0071-7
[9]	F. H. Jackson, On $q$-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193–203.
[10]	K. Raghavendar, A. Swaminathan, Close-to-convexity properties of basic hypergeometric functions using their Taylor coefficients, J. Math. Appl., 35 (2012), 53–67. https://doi.org/10.7862/rf.2012.5 doi: 10.7862/rf.2012.5
[11]	S. A. Shah, K. I. Noor, Study on the $q$-analogue of a certain family of linear operators, Turk. J. Math., 43 (2019), 2707–2714. https://doi.org/10.3906/mat-1907-41 doi: 10.3906/mat-1907-41
[12]	D. Bansal, J. K. Prajapa, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic, 61 (2016), 338–350. https://doi.org/10.1080/17476933.2015.1079628 doi: 10.1080/17476933.2015.1079628
[13]	B. Bira, T. R. Sekhar, D. Zeidan, Exact solutions for some time-fractional evolution equations using Lie group theory, Math. Method. Appl. Sci., 41 (2018), 6717–6725. https://doi.org/10.1002/mma.5186 doi: 10.1002/mma.5186
[14]	B. Bira, H. Mandal, Chennai, D. Zeidan, Exact solution of the time fractional variant Boussinesq-Burgers equations, Appl. Math., 66 (2021), 437–449. https://doi.org/10.21136/AM.2021.0269-19 doi: 10.21136/AM.2021.0269-19
[15]	K. Mehrez, S. Das, A. Kumar, Geometric properties of the products of modified Bessel functions of the first kind, Bull. Malays. Math. Sci. Soc., 44 (2021), 2715–2733. https://doi.org/10.1007/s40840-021-01082-2 doi: 10.1007/s40840-021-01082-2
[16]	K. Mehrez, Some geometric properties of a class of functions related to the Fox-Wright functions, Banach J. Math. Anal., 14 (2020), 1222–1240. https://doi.org/10.1007/s43037-020-00059-w doi: 10.1007/s43037-020-00059-w
[17]	K. Mehrez, M. Raza, The Mittag-Leffler-Prabhakar functions of Le Roy-type and its geometric properties, Iran. J. Sci., 49 (2025), 745–756. https://doi.org/10.1007/s40995-024-01738-1 doi: 10.1007/s40995-024-01738-1
[18]	I. Aktas, On some geometric properties and Hardy class of $q$-Bessel functions, AIMS Mathematics, 5 (2020), 3156–3168. https://doi.org/10.3934/math.2020203 doi: 10.3934/math.2020203
[19]	I. Aktas, H. Orhan, Bounds for radii of convexity of some $q$-Bessel functions, B. Korean Math. Soc., 57 (2020), 355–369. https://doi.org/10.4134/BKMS.b190242 doi: 10.4134/BKMS.b190242
[20]	I. Aktas, A. Baricz, Bounds for radii of starlikeness of some $q$-Bessel functions, Results Math., 72 (2017), 947–963. https://doi.org/10.1007/s00025-017-0668-6 doi: 10.1007/s00025-017-0668-6
[21]	P. Agarwal, S. Jain, J. Choi, Certain q-series identities, RACSAM Rev. R. Acad. A, 111 (2017), 139–146. https://doi.org/10.1007/s13398-016-0281-7 doi: 10.1007/s13398-016-0281-7
[22]	S. Kumar, D. Zeidan, An efficient Mittag-Leffler kernel approach for time-fractional advection-reaction-diffusion equation, Appl. Numer. Math., 170 (2021), 190–207. https://doi.org/10.1016/j.apnum.2021.07.025 doi: 10.1016/j.apnum.2021.07.025
[23]	M. Izadi, D. Zeidan, A convergent hybrid numerical scheme for a class of nonlinear diffusion equations, Comp. Appl. Math., 41 (2022), 318. https://doi.org/10.1007/s40314-022-02033-8 doi: 10.1007/s40314-022-02033-8
[24]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204 (2006), ⅶ–ⅹ. https://doi.org/10.1016/S0304-0208(06)80001-0 doi: 10.1016/S0304-0208(06)80001-0
[25]	J. Sokół, L. Trojnar-Spelina, On a sufficient condition for strongly starlikeness, J. Inequal. Appl., 2013 (2013), 383. https://doi.org/10.1186/1029-242X-2013-383 doi: 10.1186/1029-242X-2013-383
[26]	S. Owa, H. M. Srivastava, N. Saito, Partial sums of certain classes of analytic functions, Int. J. Comput. Math., 81 (2004), 1239–1256. https://doi.org/10.1080/00207160412331284042 doi: 10.1080/00207160412331284042
[27]	J. J. Foncannon, Classical and quantum orthogonal polynomials in one variable, Math. Intell., 30 (2008), 54–60. https://doi.org/10.1007/BF02985757 doi: 10.1007/BF02985757
[28]	G. M. Mittag-Leffler, Sur la nouvelle fonction $Ea(x)$, C. R. Acad. Sci. Paris, 137 (1903), 554–558.
[29]	A. Wiman, Über den Fundamental satz in der Theorie der Funcktionen $E_{a}(x)$, Acta Math., 29 (1905), 191–201. https://doi.org/10.1007/BF02403202 doi: 10.1007/BF02403202
[30]	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
[31]	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
[32]	É. Le Roy, Valéurs asymptotiques de certaines séries procédant suivant les puissances entères et positives, dune variable réelle, Paris: Bulletin des Sciences Mathématiques, 1900,245–268.
[33]	S. Das, K. Mehrez, On the geometric properties of the Mittag-Leffler and Wright function, J. Korean Math. Soc., 58 (2021), 949–965. https://doi.org/10.4134/JKMS.j200333 doi: 10.4134/JKMS.j200333
[34]	D. Bansal, J. K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic, 61 (2016), 338–350. https://doi.org/10.1080/17476933.2015.1079628 doi: 10.1080/17476933.2015.1079628
[35]	K. Mehrez, S. Das, On some geometric properties of the Le Roy-type Mittag-Leffler function, Hacet. J. Math. Stat., 51 (2022), 1085–1103.
[36]	H. M. Srivastava, J. Choi, Zeta and $q$-Zeta functions and associated series and integrals, Amsterdam: Elsevier Science Publishers, 2012.
[37]	S. K. Sahoo, N. L. Sharma, On a generalization of close-to-convex functions, Ann. Pol. Math., 113 (2015), 93–108. https://doi.org/10.4064/ap113-1-6 doi: 10.4064/ap113-1-6
[38]	H. M. Srivastava, D. Bansal, Close-to-convexity of a certain family of $q$-Mittag-Leffler functions, J. Nonlinear Var. Anal., 1 (2017), 61–69.
[39]	G. Szegö, Zur theorie der schlichten abbildungen, Math. Ann., 100 (1928), 188–211. https://doi.org/10.1007/BF01448843 doi: 10.1007/BF01448843
[40]	M. S. Robertson, The partial sums of multivalently starlike functions, Ann. Math., 42 (1941), 829–838. https://doi.org/10.2307/1968770 doi: 10.2307/1968770
[41]	E. M. Silvia, On partial sums of convex functions of order $\alpha$, Houston J. Math., 11 (1985), 397–404.
[42]	H. Silverman, Partial sums of starlike and convex functions, J. Math. Anal. Appl., 209 (1997), 221–227. https://doi.org/10.1006/jmaa.1997.5361 doi: 10.1006/jmaa.1997.5361
[43]	T. Sheil-Small, A note on the partial sums of convex schlicht functions, B. Lond. Math. Soc., 2 (1970), 165–168. https://doi.org/10.1112/blms/2.2.165 doi: 10.1112/blms/2.2.165
[44]	H. Orhan, E. Gunes, Neighborhoods and partial sums of analytic functions based on Gaussian hypergeometric functions, Indian Journal of Mathematics, 51 (2009), 489–510.
[45]	L. Brickman, D. J. Hallenbeck, T. H. Mac Gregor, D. R. Wilken, Convex hulls and extreme points of families of starlike and convex mappings, Trans. Amer. Math. Soc., 185 (1973), 413–428. https://doi.org/10.2307/1996448 doi: 10.2307/1996448
[46]	N. Yăgmur, H. Orhan, Partial sums of generalized Struve functions, Miskolc Math. Notes, 17 (2016), 657–670. https://doi.org/10.18514/MMN.2016.1419 doi: 10.18514/MMN.2016.1419
[47]	I. Aktas, H. Orhan, Partial sums of normalized Dini functions, Journal of Classical Analysis, 9 (2016), 127–135. https://doi.org/10.7153/jca-09-13 doi: 10.7153/jca-09-13
[48]	M. Din, M. Raza, N. Yagmur, S. N. Malik, On partial sums of Wright functions, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 80 (2018), 79–90.
[49]	B. A. Frasin, L.-I. Cotîrlă, Partial sums of the normalized Le Roy-Type Mittag-Leffler function, Axioms, 12 (2023), 441. https://doi.org/10.3390/axioms12050441 doi: 10.3390/axioms12050441
[50]	H. M. Srivastava, S. Owa, Current topics in analytic function theory, Singapore: World Scientific Publishing Company, 1992. https://doi.org/10.1142/1628
[51]	A. W. Goodman, Univalent functions, Tampa: Mariner Publishing Company, 1983.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)