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UNIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS INVOLVING PASCAL DISTRIBUTION SERIES

  • Received : 2019.11.30
  • Accepted : 2020.02.05
  • Published : 2020.07.31

Abstract

The aim of this article is to make a connection between the Pascal distribution series and some subclasses of normalized analytic functions whose coefficients are probabilities of the Pascal distribution. To be more precise, we investigate such connections with the classes of analytic univalent functions with positive coefficients in the open unit disk 𝕌.

Keywords

Acknowledgement

The authors are grateful to the referees of this article, that gave valuable comments and advices, in order to revise and improve the results of the paper in the present form.

References

  1. T. R. Caplinger and W. M. Causey, A class of univalent functions, Proc. Amer. Math. Soc. 39 (1973), 357-361. https://doi.org/10.2307/2039647
  2. N. E. Cho, S. Y. Woo, and S. Owa, Uniform convexity properties for hypergeometric functions, Fract. Calc. Appl. Anal. 5 (2002), no. 3, 303-313.
  3. L. de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), no. 1-2, 137-152. https://doi.org/10.1007/BF02392821
  4. K. K. Dixit and V. Chandra, On subclass of univalent functions with positive coefficients, Aligarh Bull. Math. 27 (2008), no. 2, 87-93.
  5. K. K. Dixit and S. K. Pal, On a class of univalent functions related to complex order, Indian J. Pure Appl. Math. 26 (1995), no. 9, 889-896.
  6. K. K. Dixit and A. L. Pathak, A new class of analytic functions with positive coefficients, Indian J. Pure Appl. Math. 34 (2003), no. 2, 209-218.
  7. S. M. El-Deeb, T. Bulboaca, and J. Dziok, Pascal distribution series connected with certain subclasses of univalent functions, Kyungpook Math. J. 59 (2019), no. 2, 301-314. https://doi.org/10.5666/KMJ.2019.59.2.301
  8. E. P. Merkes and W. T. Scott, Starlike hypergeometric functions, Proc. Amer. Math. Soc. 12 (1961), 885-888. https://doi.org/10.2307/2034382
  9. A. O. Mostafa, A study on starlike and convex properties for hypergeometric functions, JIPAM. J. Inequal. Pure Appl. Math. 10 (2009), no. 3, Article 87, 8 pp.
  10. G. Murugusundaramoorthy, Subclasses of starlike and convex functions involving Poisson distribution series, Afr. Mat. 28 (2017), no. 7-8, 1357-1366. https://doi.org/10.1007/s13370-017-0520-x
  11. G. Murugusundaramoorthy, Univalent functions with positive coefficients involving Poisson distribution series, Honam Math. J. 40 (2018), no. 3, 529-538. https://doi.org/10.5831/HMJ.2018.40.3.529
  12. G. Murugusundaramoorthy, K. Vijaya, and S. Porwal, Some inclusion results of certain subclass of analytic functions associated with Poisson distribution series, Hacet. J. Math. Stat. 45 (2016), no. 4, 1101-1107. https://doi.org/10.15672/HJMS.20164513110
  13. K. S. Padmanabhan, On a certain class of functions whose derivatives have a positive real part in the unit disc, Ann. Polon. Math. 23 (1970/71), 73-81. https://doi.org/10.4064/ap-23-1-73-81
  14. S. Porwal, An application of a Poisson distribution series on certain analytic functions, J. Complex Anal. 2014 (2014), Art. ID 984135, 3 pp. https://doi.org/10.1155/2014/984135
  15. H. Silverman, Starlike and convexity properties for hypergeometric functions, J. Math. Anal. Appl. 172 (1993), no. 2, 574-581. https://doi.org/10.1006/jmaa.1993.1044
  16. H. M. Srivastava, G. Murugusundaramoorthy, and S. Sivasubramanian, Hypergeometric functions in the parabolic starlike and uniformly convex domains, Integral Transforms Spec. Funct. 18 (2007), no. 7-8, 511-520. https://doi.org/10.1080/10652460701391324
  17. B. A. Uralegaddi, M. D. Ganigi, and S. M. Sarangi, Univalent functions with positive coefficients, Tamkang J. Math. 25 (1994), no. 3, 225-230. https://doi.org/10.5556/j.tkjm.25.1994.4448