Skip to main content
Log in

Starlikeness of analytic functions using special functions and subordination

  • Original Article
  • Published:
Boletín de la Sociedad Matemática Mexicana Aims and scope Submit manuscript

Abstract

Let \(\mathcal {P}\) be the class of analytic functions having positive real part in the complex plane. The association of subordination and special functions is used to find sharp estimates on the parameter \(\beta \) such that the analytic function \(p\in \mathcal {P}\) is subordinate to certain functions having positive real part whenever \(p(z)+\beta z p'(z)\) is subordinate to the Janowski function. Further, the concept of admissibility is employed to establish certain second and third order differential subordination relations between the analytic function p and the functions associated with right half plane. As a sequel, we demonstrate the starlikeness of various well-known analytic functions as well.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Ahuja, O.P., Kumar, S., Ravichandran, V.: Applications of first order differential subordination for functions with positive real part. Stud. Univ. Babeş-Bolyai Math. 63(3), 303–311 (2018)

    Article  MathSciNet  Google Scholar 

  2. Ali, R.M., Ravichandran, V., Seenivasagan, N.: Sufficient conditions for Janowski starlikeness. Int. J. Math. Math. Sci. 2007, 62925 (2007)

    Article  MathSciNet  Google Scholar 

  3. Antonino, J.A., Miller, S.S.: Third-order differential inequalities and subordinations in the complex plane. Complex Var. Elliptic Equ. 56(5), 439–454 (2011)

    Article  MathSciNet  Google Scholar 

  4. Bohra, N., Kumar, S., Ravichandran, V.: Some special differential subordinations. Hacet. J. Math. Stat. 48(4), 1017–1034 (2019)

    MathSciNet  Google Scholar 

  5. Cho, N.E., Lecko, A.: Differential subordination of harmonic mean. Comput. Methods Funct. Theory 15(3), 427–437 (2015)

    Article  MathSciNet  Google Scholar 

  6. Cho, N.E., Srivastava, H.M.: Subordinations by \(\eta \)-convex functions for a class of nonlinear integral operators. Bull. Sci. Math. 187, 103304 (2023)

    Article  MathSciNet  Google Scholar 

  7. Cho, N.E., Kumar, V., Kumar, S.S., Ravichandran, V.: Radius problems for starlike functions associated with the sine function. Bull. Iran. Math. Soc. 45(1), 213–232 (2019)

    Article  MathSciNet  Google Scholar 

  8. Goel, P., Kumar, S.S.: Certain class of starlike functions associated with modified sigmoid function. Bull. Malays. Math. Sci. Soc. 43(1), 957–991 (2020)

    Article  MathSciNet  Google Scholar 

  9. Janowski, W.: Extremal problems for a family of functions with positive real part and for some related families. Ann. Polon. Math. 23, 159–177 (1970)

    Article  MathSciNet  Google Scholar 

  10. Kumar, S., Ravichandran, V.: A subclass of starlike functions associated with a rational function. Southeast Asian Bull. Math. 40(2), 199–212 (2016)

    MathSciNet  Google Scholar 

  11. Kumar, S., Ravichandran, V.: Subordinations for functions with positive real part. Complex Anal. Oper. Theory 12(5), 1179–1191 (2018)

    Article  MathSciNet  Google Scholar 

  12. Kumar, V., Cho, N.E., Ravichandran, V., Srivastava, H.M.: Sharp coefficient bounds for starlike functions associated with the Bell numbers. Math. Slovaca 69(5), 1053–1064 (2019)

    Article  MathSciNet  Google Scholar 

  13. Küstner, R.: On the order of starlikeness of the shifted Gauss hypergeometric function. J. Math. Anal. Appl. 334(2), 1363–1385 (2007)

    Article  MathSciNet  Google Scholar 

  14. Ma, W. C., Minda, D.: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis (Tianjin, 1992), pp. 157–169, I, Int. Press, Cambridge

  15. Mendiratta, R., Nagpal, S., Ravichandran, V.: On a subclass of strongly starlike functions associated with exponential function. Bull. Malays. Math. Sci. Soc. 38(1), 365–386 (2015)

    Article  MathSciNet  Google Scholar 

  16. Mushtaq, S., Raza, M., Sokół, J.: Differential subordination related with exponential functions. Quaest. Math. 45(6), 889–899 (2022)

    Article  MathSciNet  Google Scholar 

  17. Miller, S.S., Mocanu, P.T.: Second-order differential inequalities in the complex plane. J. Math. Anal. Appl. 65(2), 289–305 (1978)

    Article  MathSciNet  Google Scholar 

  18. Miller, S.S., Mocanu, P.T.: Differential Subordinations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 225. Marcel Dekker Inc, New York (2000)

    Google Scholar 

  19. Mushtaq, S., Raza, M., Sokół, J.: Differential subordination related with exponential functions. Quaest. Math. 45(6), 889–899 (2021)

    Article  MathSciNet  Google Scholar 

  20. Naz, A., Nagpal, S., Ravichandran, V.: Star-likeness associated with the exponential function. Turk. J. Math. 43(3), 1353–1371 (2019)

    Article  MathSciNet  Google Scholar 

  21. Nunokawa, M., Obradović, M., Owa, S.: One criterion for univalency. Proc. Am. Math. Soc. 106(4), 1035–1037 (1989)

    Article  MathSciNet  Google Scholar 

  22. Palani, J., Lavanya, V.A.S.J., Farzana, H.A.: Third order differential subordination associated with Janowski functions. Math. Appl. 11(1), 45–55 (2022)

    Article  MathSciNet  Google Scholar 

  23. Ponnusamy S., Juneja, O. P.: Third-order differential inequalities in the complex plane. In: Current Topics in Analytic Function Theory, pp. 274–290. World Scientific Publication, River Edge

  24. Raina, R.K., Sokół, J.: On coefficient estimates for a certain class of starlike functions. Hacet. J. Math. Stat. 44(6), 1427–1433 (2015)

    MathSciNet  Google Scholar 

  25. Sharma, M., Kumar, S., Jain, N.K.: Differential subordination implications for Certain Carathéodory functions. Stud. Univ. Babeş-Bolyai Math. 68(4), 775–787 (2023)

    Article  MathSciNet  Google Scholar 

  26. Sharma, M., Kumar, S., Jain, N.K.: Differential subordinations for functions with positive real part using admissibility conditions. Asian-Eur. J. Math. 15(4), 2250066 (2022)

    Article  MathSciNet  Google Scholar 

  27. Sokół, J., Stankiewicz, J.: Radius of convexity of some subclasses of strongly starlike functions. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19, 101–105 (1996)

    MathSciNet  Google Scholar 

  28. Srivastava, H.M., Prajapati, A., Gochhayat, P.: Third-order differential subordination and differential superordination results for analytic functions involving the Srivastava-Attiya operator. Appl. Math. Inf. Sci. 12(3), 469–481 (2018)

    Article  MathSciNet  Google Scholar 

  29. Ullah, K., Srivastava, H.M., Rafiq, A., Arif, M., Arjika, S.: A study of sharp coefficient bounds for a new subfamily of starlike functions. J. Inequal. Appl. 194, 20 (2021)

    MathSciNet  Google Scholar 

  30. Wani, L.A., Swaminathan, A.: Starlike and convex functions associated with a nephroid domain. Bull. Malays. Math. Sci. Soc. 44(1), 79–104 (2021)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sushil Kumar.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sharma, M., Jain, N.K. & Kumar, S. Starlikeness of analytic functions using special functions and subordination. Bol. Soc. Mat. Mex. 30, 55 (2024). https://doi.org/10.1007/s40590-024-00630-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40590-024-00630-7

Keywords

Mathematics Subject Classification

Navigation