Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 20 (2025), 235 -- 250

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

NOTE ON λ- PSEUDO BI-STARLIKE AND BI-CONVEX FUNCTIONS RELATED WITH SINE FUNCTIONS

Kaliyappan Vijaya, Gangadharan Murugusundaramoorthy and Hatun Özlem Güney

Abstract. In this paper we introduce two new classes ℬΣλ(μ) of λ-pseudo bi-starlike functions and 𝒦ℬΣλ(μ) of λ-pseudo bi-convex functions. Also, we determine the bounds for |a2| and |a3| where a2, a3 are the initial Taylor coefficients of f∈ℬΣλ(μ) and f∈𝒦ℬΣλ(μ), respectively. Furthermore, we estimate the Fekete-Szegö functionals for the functions in these classes.

2020 Mathematics Subject Classification: 30C45, 30C50.
Keywords: Analytic Function, Starlike Function, Convex Function, Bi-Univalent Function, Sine Function, Subordination.

Full text

References

  1. Ş. Alt\i nkaya, S.Y. Özkan, On Salagean type pseudo-starlike functions. Acta et Commentationes Universitatis Tartuensis de Mathematica (21), No.2, 2017. https://doi.org/10.12697/ACUTM.2017.21.19.

  2. A. Mohammadi, N.E. Cho, E.A. Adegani, A. Motamednezhad, Argument and coefficient estimates for certain analytic functions, Mathematics (2020) 8 (1), 88. https://doi.org/10.3390/math8010088.

  3. A. Alotaibi, M. Arif, M.A. Alghamdi, S. Hussain, Starlikeness associated with cosine hyperbolic function, Mathematics, 8(7):1118, 2020. https://doi.org/10.3390/math8071118.

  4. M. Arif, M. Raza, Huo Tang, S. Hussain, H. Khan, Hankel determinant of order three for familiar subsets of analytic functions related with sine function, Open Math., (17)2019, 1615--1630. MR4048485. Zbl 1441.30019.

  5. K.O. Babalola, On λ -pseudo-starlike functions, J. Class. Anal. 3(2)(2013), 137--147. MR3322264. Zbl 1412.30026.

  6. D. Bansal, J. Sokó\l , Coefficient bound for a new class of analytic and bi-univalent functions, J. Fract. Calc. Appl. 5(1)(2014) 122--128. MR3234102. Zbl 1488.30040.

  7. D.M. Campbell, J.G. Clunie, W.K. Hayman, Research problems in complex analysis. Aspects of contemporary complex analysis, (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979), pp. 527-572, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980 ISBN:0-12-125950-1. MR0623497. Zbl 0497.30001.

  8. B. A. Frasin, M. K. Aouf, New subclasses of bi-univalent functions, Appl. Matt. Lett. 24(9)(2011), 1569--1573. MR2803711. Zbl 1218.30024.

  9. N.E.  Cho, V. Kumar, S.S. Kumar, V. Ravichandran, Radius problems for starlike functions associated with the Sine function, Bull. Iran. Math. Soc. 45, No.1(2019), 213-232. MR3913990. Zbl 1409.30009.

  10. S. Joshi, S. Joshi, H. Pawar, On some subclasses of bi-univalent functions associated with pseudo-starlike function, J. Egyptian Math. Soc. 24(2016) 522--525. MR3553869. Zbl 1352.30009.

  11. S. Joshi, Ş. Alt\i nkaya, S. Yalc\i n, Coefficient Estimates for Salagean Type λ --bi-pseudo-starlike Functions, Kyungpook Mathematical Journal 57, No. 4(2017), 613-621 . MR3745142. Zbl 1400.30023.

  12. M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Am. Math. Soc. 18(1967), 63--68. MR0206255. Zbl 0158.07802.

  13. W. Ma, D. Minda, A Unified treatment of some special cases of univalent functions, Li, Zhong (ed.) et al., Proceedings of the conference on complex analysis, held June 19-23, 1992 at the Nankai Institute of Mathematics, Tianjin, China. Cambridge, MA: International Press. Conf. Proc. Lect. Notes Anal. 1(1994), 157-169. Zbl 0823.30007.

  14. R. Mendiratta, S. Nagpal, V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc., (2) 38, (2015) No. 1, 365-386. MR3394060. Zbl 1312.30019.

  15. G. Murugusundaramoorthy, J. Sokó\l , On λ - Pseudo Bi-Starlike Functions related to some conic domains, Bulletin of the Transilvania University of Braşov Series III: Mathematics, Informatics, Physics, 12(61), no.2(2019), 381-392. https://doi.org/10.31926/but.mif.2019.12.61.2.15, MR4059168. Zbl 1488.30106.

  16. M. Raza, S.N. Malik, Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli. J Inequal Appl 2013, 412 (2013). MR3339521. Zbl 1291.30106.

  17. H. Özlem Güney, G. Murugusundaramoorthy, New classes of pseudo-type bi-univalent functions, Rev. R. Acad. Cienc. Exactas F\is. Nat., Ser. A Mat., RACSAM 114, No. 2, Paper No. 65, 8 p. (2020). MR4054920. Zbl 1435.30047.

  18. S.O. Olatunji, P.T. Ajai, On subclasses of bi-univalent functions of Bazilevi\vc type involving linear and S\val\vagean Operator, Inter. J. Pure Appl. Math. 92(5)(2015) 645--656.

  19. C. Pommerenke, Univalent functions. With a chapter on quadratic differentials by Gerd Jensen Studia, Studia Math./Math. Lehrbücher, Band XXV[Studia Mathematica/Mathematical Textbooks], Vandenhoeck & Ruprecht, Göttingen, 1975. 376 pp. MR0507768. Zbl 0298.30014.

  20. R. K. Raina, J. Sokó\l , On coefficient estimates for a class of starlike functions, Hacet. J. Math. Stat. 44(2015), No. 6, 1427-1433. MR3469339. Zbl 1350.30029.

  21. F. Rø nning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), no.1, 189-196. MR1128729. Zbl 0805.30012.

  22. K. Sharma, N. K. Jain, V. Ravichandran, Starlike functions associated with a cardioid, Afr. Mat. 27 (2016), no. 5-6, 923-939. MR3536076. Zbl 1352.30015.

  23. S. Salehian, A. Zireh, Coefficient estimates for certain subclass of meromorphic and bi-univalent functions, Commun. Korean Math. Soc. 32 (2017), no. 2, 389--397. MR3641173. Zbl 1375.30022.

  24. J. Sokó\l , Radius problems in the class \mathcalYL, Appl. Math. Comput. 214 (2009), no. 2, 569-573. MR2541692. Zbl 1170.30005.

  25. H.M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23(2010) 1188--1192. MR2665593. Zbl 1201.30020.

  26. H.M. Srivastava, G. Murugusundaramoorthy, N. Magesh, Certain subclasses of bi-univalent functions associated with Hoholov operator, Global J. Math. Anal. 1(2013), no. ~2, 67--73.

  27. H.M. Srivastava, S. Bulut, M. Ca\vglar and N. Ya\vgmur, Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat. 27(2013), no. 5, 831--842. MR3186102. Zbl 1432.30014.

  28. H.M. Srivastava, Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. Iran. J. Sci. Technol. Trans. A Sci. 44 (2020), no. 1, 327-344. MR4064730.

  29. H.M. Srivastava, Per W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood Ser. Math. Appl., Ellis Horwood Ltd., Chichester; Halsted Press: a division of John Wiley & Sons. 425 p. (1985). Zbl 0552.33001.

  30. H.M. Srivastava, D. Raducanu, P.A. Zaprawa, A certain subclass of analytic functions defined by means of differential subordination, Filomat 30 (2016), no. 14, 3743-3757. MR3593746. Zbl 1458.30038.

  31. H.M. Srivastava, Certain q-polynomial expansions for functions of several variables, IMA J. Appl. Math. 30 (1983), no. 3, 315-323. MR0719983. Zbl 0504.33003.

  32. P. Zaprawa, On the Fekete-Szeg\Ho problem for classes of bi-univalent functions, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 1, 169-178. MR3178538. Zbl 1300.30035.

  33. A. Zireh, E. Analouei Audegani, Coefficient estimates for a subclass of analytic and bi-univalent functions, Bull. Iranian Math. Soc., 42 (2016), no. 4, 881-889. MR3545119. Zbl 1373.30033.




Kaliyappan Vijaya
Department of Mathematics, School of Advanced Sciences,
Vellore Institute of Technology (Deemed to be University),
Vellore-632014. India.
email: kvijaya@vit.ac.in


Gangadharan Murugusundaramoorthy
Department of Mathematics, School of Advanced Sciences,
Vellore Institute of Technology (Deemed to be University),
Vellore-632014. India.
email: gmsmoorthy@yahoo.com


Hatun Özlem Güney - Corresponding author
Dicle University,
Faculty of Science, Department of Mathematics,
Diyarbak\i r-Türk\.iye.
email: ozlemg@dicle.edu.tr

https://www.utgjiu.ro/math/sma