A NEW MULTIVALENT ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS IN DISTORTION BOUNDS AND ITS EXTREMAL PROPERTIES

Stalin Thangamani; Dumitru Baleanu; Majeed Ahmad Yousif; Pshtiwan Othman Mohammed; Meraa Arab

doi:10.11948/20250180

2026 Volume 16 Issue 6

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Stalin Thangamani, Dumitru Baleanu, Majeed Ahmad Yousif, Pshtiwan Othman Mohammed, Meraa Arab. A NEW MULTIVALENT ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS IN DISTORTION BOUNDS AND ITS EXTREMAL PROPERTIES[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 3182-3200. doi: 10.11948/20250180

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Stalin Thangamani, Dumitru Baleanu, Majeed Ahmad Yousif, Pshtiwan Othman Mohammed, Meraa Arab. A NEW MULTIVALENT ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS IN DISTORTION BOUNDS AND ITS EXTREMAL PROPERTIES[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 3182-3200. doi: 10.11948/20250180

A NEW MULTIVALENT ANALYTIC FUNCTIONS WITH NEGATIVE COEFFICIENTS IN DISTORTION BOUNDS AND ITS EXTREMAL PROPERTIES

1.
Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi, Chennai 600062, India
2.
Department of Computer Science and Mathematics, Lebanese American University, Beirut 11022801, Lebanon
3.
Department of Mathematics, College of Education, University of Zakho, Duhok 42001, Iraq
4.
Research and Development Center, University of Sulaimani, Sulaymaniyah 46001, Iraq
5.
Department of Computer Engineering, Biruni University, 34010 Istanbul, Turkey
6.
Section of Mathematics, Università Telematica Internazionale Uninettuno, Rome 00186, Italy
7.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa 31982, Saudi Arabia

Author Bio: Email: drstalint@veltech.edu.in(S. Thangamani); Email: dumitru.baleanu@lau.edu.lb(D. Baleanu); Email: majeed.yousif@uoz.edu.krd(M. A. Yousif)
Corresponding authors: Email: pshtiwan.muhammad@univsul.edu.iq(P. O. Mohammed); Email: marab@kfu.edu.sa(M. Arab)

Abstract

Abstract

This work investigates a new subclass $ \mathcal{SM}^n_{b, \beta, \tau, p}(\mu) $ of multivalent analytic functions with negative coefficients, characterized by a generalized multiplier transformation operator $ \mathcal{M}_{\beta, \tau, p}^{n} $. Various properties of this subclass are investigated, including coefficient bounds, extreme points of extremal functions, distortion bounds, radii of convexity and starlikeness, and sharp lower bounds for partial sums. Additionally, graphical representations of extremal functions are provided to visually illustrate the theoretical results. These findings contribute to the ongoing development of geometric function theory.
- Multivalent functions /
- negative coefficient multivalent functions /
- convolution /
- derivative operators
MSC: 30C45, 30C50, 30C55

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References

[1]	R. Aghalary, R. M. Ali, S. B. Joshi and V. Ravichandran, Inequalities for analytic functions defined by certain linear operators, Int. J. Math. Sci., 2005, 4(2), 267–274. Google Scholar
[2]	J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. Math., 1915, 17, 12–22. doi: 10.2307/2007212 CrossRef Google Scholar
[3]	E. E. Ali, R. M. El-Ashwah and R. Sidaoui, Application of subordination and superordination for multivalent analytic functions associated with differintegral operator, AIMS Math., 2023, 8(5), 11440–11459. doi: 10.3934/math.2023579 CrossRef Google Scholar
[4]	A. A. Amourah and F. Yousef, Some properties of a class of analytic functions involving a new generalized differential operator, Bol. Soc. Paran. Mat., 2020, 38(6), 33–42. Google Scholar
[5]	F. Azhar, M. I. Asjad, M. A. Yousif, I. A. Baloch, D. Baleanu and P. O. Mohammed, Extensions and improvements of Jensen's and majorization type inequalities for p-harmonic convex functions, Contemp. Math., 2026, 7(2), 1915–1944. Google Scholar
[6]	F. Azhar, M. I. Asjad, M. A. Yousif, A. A. Lupas, I. A. Baloch and P. O. Mohammed, A novel family of discrete variant inequality for Jensen and Hermite–Hadamard types in the p-harmonic convex function setting, Contemp. Math., 2026, 7(1), 940–964. Google Scholar
[7]	S. D. Bernardi, A survey of the development of the theory of schlicht functions, 1952, 263–287. Google Scholar
[8]	S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc., 1969, 135, 429–446. doi: 10.1090/S0002-9947-1969-0232920-2 CrossRef Google Scholar
[9]	L. Bieberbach, Uber einige extremal problem in gebiete der konformen abbildung, Math. Ann., 1916, 77, 153–172. doi: 10.1007/BF01456900 CrossRef Google Scholar
[10]	M. Biernacki, Les fonctions multivalentes, Bull. Amer. Math. Soc., 1942, 48, 640–641. doi: 10.1090/S0002-9904-1942-07722-6 CrossRef Google Scholar
[11]	L. de Branges, A proof of the Bieberbach conjecture, Acta Math., 1984, 154, 137–152. Google Scholar
[12]	S. Bulut, Coefficient bounds for p-valently close-to-convex functions associated with vertical strip domain, Korean J. Math., 2021, 29(2), 395–407. Google Scholar
[13]	H. Byeon, M. Balamurugan, T. Stalin, V. Govindan, J. Ahmad and W. Emam, Some properties of subclass of multivalent functions associated with a generalized differential operator, Sci. Rep., 2024, 14(1), 8760. doi: 10.1038/s41598-024-58781-6 CrossRef Google Scholar
[14]	N. E. Cho, J. Patel and G. P. Mohapatra, Argument estimates of certain multivalent functions involving a linear operator, Int. J. Math. Math. Sci., 2002, 31, 659–673. doi: 10.1155/S0161171202112300 CrossRef Google Scholar
[15]	N. E. Cho and H. M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Model., 2003, 37(1–2), 39–49. doi: 10.1016/S0895-7177(03)80004-3 CrossRef Google Scholar
[16]	T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl., 1972, 38(3), 746–765. doi: 10.1016/0022-247X(72)90081-9 CrossRef Google Scholar
[17]	G. M. Goluzin, Geometric theory of functions of a complex variable, American Mathematical Society, 1969, 26. Google Scholar
[18]	S. H. Hadi, M. Darus and J. R. Lee, Some geometric properties of multivalent functions associated with a new generalized q-Mittag-Leffler function, AIMS Math., 2022, 7(7), 11772–11783. doi: 10.3934/math.2022656 CrossRef Google Scholar
[19]	W. K. Hayman, Multivalent Functions, Cambridge University Press, Cambridge, 1958. Google Scholar
[20]	W. K. Hayman, Survey article–coefficient problems for univalent functions and related function classes, J. London Math. Soc., 1966, 1(1), 550–550. Google Scholar
[21]	J. A. Jenkins, Univalent Functions and Conformal Mapping, Springer-Verlag, Berlin, 1958; Russian Translation, IL, Moscow, 1962. Google Scholar
[22]	N. Khan, B. Khan, Q. Z. Ahmad and S. Ahmad, Some convolution properties of multivalent analytic functions, AIMS Math., 2017, 2(2), 260–268. doi: 10.3934/Math.2017.2.260 CrossRef Google Scholar
[23]	Q. Khan, M. Arif, B. Ahmad and H. Tang, On analytic multivalent functions associated with lemniscate of Bernoulli, AIMS Math., 2020, 5(3), 2261–2271. doi: 10.3934/math.2020149 CrossRef Google Scholar
[24]	P. Koebe, Uber die Uniformisierung beliebiger analytischer Kurven, Nachrichten von der Gesellschaft der Wissenschaften zu Gottingen, Mathematisch-Physikalische Klasse, 1907, 191–210. Google Scholar
[25]	R. J. Libera, Some classes of regular univalent functions, Proc. Math. Soc., 1965, 16, 755–758. doi: 10.1090/S0002-9939-1965-0178131-2 CrossRef Google Scholar
[26]	S. Mehmood, D. Baleanu, M. A. Yousif, P. O. Mohammed, A. Abbas and N. Chorfi, Some new inequalities involving generalized convex functions in the Katugampola fractional setting, Contemp. Math., 2025, 6(4), 4483–4500. Google Scholar
[27]	S. Mehmood, P. O. Mohammed, A. Kashuri, N. Chorfi, S. A. Mahmood and M. A. Yousif, Some new fractional inequalities defined using cr-log-h-convex functions and applications, Symmetry, 2024, 16(4), 407. doi: 10.3390/sym16040407 CrossRef Google Scholar
[28]	P. Montel, Leçons sur les Fonctions Univalentes ou Multivalentes, Gauthier-Villars, Paris, 1933. Google Scholar
[29]	K. I. Noor, N. Khan and Q. Z. Ahmad, Coefficient bounds for a subclass of multivalent functions of reciprocal order, AIMS Math., 2017, 2(2), 322–335. doi: 10.3934/Math.2017.2.322 CrossRef Google Scholar
[30]	F. M. Al-Oboudi, On univalent functions defined by a generalized Salagean operator, Int. J. Math. Math. Sci., 2004, 27–30, 1429–1436. Google Scholar
[31]	A. M. Rashid and A. R. S. Juma, Subordination properties for classes of analytic univalent involving linear operator, Kyungpook Math. J., 2023, 63(2). Google Scholar
[32]	S. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc., 1975, 49, 109–115. doi: 10.1090/S0002-9939-1975-0367176-1 CrossRef Google Scholar
[33]	G. S. Sălăgean, Subclasses of univalent functions, Complex Analysis—Fifth Romanian-Finnish Seminar, Bucharest, June 28–July 3, 1981, Springer, Berlin–Heidelberg, 2006, 362–372. Google Scholar
[34]	A. Saliu, K. I. Noor, S. Hussain and M. Darus, Some results for the family of univalent functions related with limaçon domain, AIMS Math., 2021, 6(4), 3410–3431. doi: 10.3934/math.2021204 CrossRef Google Scholar
[35]	A. C. Schaeffer and D. C. Spencer, Coefficient regions for schlicht functions, American Mathematical Society, 2016, 35. Google Scholar
[36]	T. N. Shanmugam, S. Sivasubramanian and S. Owa, Argument estimates of certain multivalent functions involving Dziok–Srivastava operator, Gen. Math., 2007, 15. Google Scholar
[37]	K. A. Shaqsi and M. Darus, An operator defined by convolution involving the polylogarithms functions, J. Math. Stat., 2008, 4(1), 46–50. doi: 10.3844/jmssp.2008.46.50 CrossRef Google Scholar
[38]	K. Al-Shaqsi and M. Darus, Differential subordination with generalized derivative operator, Int. J. Comput. Math. Sci., 2008, 2(2), 75–78. Google Scholar
[39]	H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 1975, 51(1), 109–116. doi: 10.1090/S0002-9939-1975-0369678-0 CrossRef Google Scholar
[40]	T. Stalin, M. Thirucheran and A. Anand, Obtain subclass of analytic functions connected with convolution of polylogarithm function, Adv. Math. Sci. J., 2020, 9(11), 9639–9645. doi: 10.37418/amsj.9.11.69 CrossRef Google Scholar
[41]	S. Thangamani, D. Baleanu, S. Authimoolam, M. A. Yousif, T. Abdeljawad and P. O. Mohammed, Analytic function classes defined by Mittag–Leffler inspired Poisson-type series, Alexandria Engineering Journal, 2026, 136, 62–72. doi: 10.1016/j.aej.2026.01.001 CrossRef Google Scholar
[42]	S. Thangamani, D. Baleanu, P. Lourdu, M. A. Yousif and P. O. Mohammed, New horizons in analytic function classes induced by the Erdélyi–Kober fractional integral operators, IJOCTA, 2026, 16(1), 176–190. Google Scholar
[43]	S. Thangamani, P. O. Mohammed, J. F. Devadoss and M. A. Yousif, Representation of bi-univalent functions related to Lucas balancing polynomials with geometric properties and coefficient bounds, European Journal of Pure and Applied Mathematics, 2025, 18(3), 6294. doi: 10.29020/nybg.ejpam.v18i3.6294 CrossRef Google Scholar
[44]	M. Thirucheran and T. Stalin, On a new subclass of multivalent functions defined by Al-Oboudi differential operator, Global J. Pure Appl. Math., 2018, 14(5), 733–741. Google Scholar
[45]	B. A. Uralegaddi and C. Somanatha, Certain classes of univalent functions, Current Topics in Analytic Function Theory, 1992, 371–374. Google Scholar