Article ID: MTJPAM-D-20-00042

Title: Fekete-Szegö Inequalities for Certain Subclasses of Analytic Functions Related with Leaf-Like Domain


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00042; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 305-316

Document Type: Research Paper

Author(s): Gangadharan Murugusundaramoorthy a

aSchool of Advanced Science, Vellore Institute of Technology, Deemed to be University, Vellore – 632014, India

Received: 27 November 2020, Accepted: 29 March 2021, Published: 25 April 2021.

Corresponding Author: Gangadharan Murugusundaramoorthy (Email address: gmsmoorthy@yahoo.com)

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Abstract

The purpose of this paper is to consider coefficient estimates in a class of functions  \mathscr{M}_{\alpha,\,\lambda}(q) consisting of analytic functions  f normalized by   f(0)=f'(0)-1=0  in the open unit disk Δ = {z : z ∈ ℂ  and  |z| < 1} subordinating with leaf like domain, to derive certain coefficient estimates a2, a3 and Fekete-Szegö inequality for f\in\mathscr{M}_{\alpha,\,\lambda}(q). A similar results have been done for the function f^{-1}. Further application of our results to certain functions defined by convolution products with a normalized functions analytic is given, and in particular we obtain Fekete-Szegö inequalities for certain subclasses of functions defined through Poisson distribution series.

Keywords: Analytic functions, Starlike functions, Convex functions, Subordination, Fekete-Szegö inequality, Poisson distribution series, Hadamard product

References:
  1. S. M. El-Deeb and T. Bulboacă, Fekete-Szegö inequalities for certain class of analytic functions connected with q-anlogue of Bessel function, J. Egyptian. Math. Soc. 27 (42), 1-11, 2019, https://doi.org/10.1186/s42787-019-0049-2.
  2. D. Guo and M-S. Liu, On certain subclass of Bazilevič functions, J. Inequal. Pure Appl. Math. 8 (1), Article 12, 1-11, 2007.
  3. U. Grenander and G.Szegö, Toeplitz Forms and Their Applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley, 1958.
  4. F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20, 8-12, 1969.
  5. B. Kowalczyk, A. Lecko and H. M. Srivastava, A note on the Fekete-Szegö problem for close-to-convex functions with respect to convex functions, Publ. Inst. Math. (Beograd) (Nouvelle Sér.) 101 (115), 143-149, 2017.
  6. R. J. Libera and E. J. Zlotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (2), 225-230, 1982.
  7. W. Ma and D. Minda, A Unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis, Z. Li, F. Ren, L. Yang, and S. Zhang(Eds.), Int. Press, 157-169, 1994.
  8. G. Murugusundaramoorthy, Subclasses of starlike and convex functions involving Poisson distribution series, Afr. Mat. 28, 1357-1366, 2017.
  9. G. Murugusundaramoorthy and T. Bulboacă, Hankel determinants for new subclasses of analytic functions related to a shell shaped region, Mathematics 8, 1041, 2020.
  10. G. Murugusundaramoorthy, K. Vijaya and S. Porwal, Some inclusion results of certain subclass of analytic functions associated with Poisson distribution series, Hacettepe J. Math. Stat. 45 (4), 1101-1107, 2016.
  11. S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39, 1057-1077, 1987.
  12. S. Porwal, An application of a Poisson distribution series on certain analytic functions, J. Complex Anal. Art. ID 984135, 1-3, 2014.
  13. S. Porwal and M. Kumar, A unified study on starlike and convex functions associated with Poisson distribution series, Afr. Mat. 27, 1021-1027, 2016.
  14. R. K. Raina and J. Sokol, On coeffcient estimates for a certain class of starlike functions, Hacettepe Journal of Math. and Stat. 44 (6), 1427-1433, 2015.
  15. R. B. Sharma and M. Haripriya, On a class of a-convex functions subordinate to a shell shaped region, J. Anal. 25, 93-105, 2016, DOI10.1007/s41478-017-0031-z.
  16. J. Sokol and D. K. Thomas, Further results on a class starlike functions related to the Bernoulli lemniscate, Houston J. Math. 44, 83-95, 2018.
  17. 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, 327-344, 2020.
  18. H. M. Srivastava, S. Hussain, A. Raziq and M. Raza, The Fekete-Szegö functional for a subclass of analytic functions associated with quasi-subordination, Carpathian J. Math. 34, 103-113, 2018.
  19. H. M. Srivastava, N. Khan, M. Darus, S. Khan, Q. Z. Ahmad and S. Hussain, Fekete-Szegö type problems and their applications for a subclass of q-starlike functions with respect to symmetrical points, Mathematics 8, 842, 1-18, 2020.
  20. H. M. Srivastava, A. K. Mishra and M. K. Das, The Fekete-Szegö problem for a subclass of close-to-convex functions, Complex Variables Theory Appl. 44 (2), 145-163, 2001.
  21. H. M. Srivastava, A. O. Mostafa, M. K. Aouf and H. M. Zayed, Basic and fractional q-calculus and associated Fekete-Szegö problem for p-valently q-starlike functions and p-valently q-convex functions of complex order, Miskolc Math. Notes 20, 489-509, 2019.
  22. H. M. Srivastava, N. Raza, E. S. A. AbuJarad, G. Srivastava and M. H. AbuJarad, Fekete-Szegö inequality for classes of (p, q)-starlike and (p, q)-convex functions, Rev. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat. (RACSAM) 113, 3563-3584, 2019.