Skip to main content

On Applications of Nunokawa and Sokół Theorem for p-Valency

Abstract

We present a generalization of a result for p-valent functions obtained by Nunokawa and Sokół (Appl Math Comput 22(219): 10768–10776, 2013) and give a few corollaries and examples yielding some classical results on univalent functions.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  1. 1.

    Ali, R.A., Ravichandran, V.: Integral operators on Ma-Minda type starlike and convex functions. Math. Comput. Model. 53(5–6), 581–586 (2011)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Duren, P.L.: Univalent Functions. Grundlehren der Mathematischen Wissenschaften, vol. 259. Springer, New York (1983)

    Google Scholar 

  3. 3.

    Ebadian, A., Sokół, J.: On the subordination and superordination of strongly starlike functions. Math. Slovaca 66(4), 815–822 (2016)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Goodman, A.W.: On the Schwarz–Christoffel transformation and \(p\)-valent functions. Trans. Am. Math. Soc. 68(2), 204–223 (1950)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Liu, J.-L.: Some argument inequalities for certain analytic functions. Math. Slovaca 62(1), 25–28 (2012)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Liu, J.-L.: The Noor integral and strongly starlike functions. J. Math. Anal. Appl. 261(2), 441–447 (2001)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Miller, S.S., Mocanu, P.T.: Differential subordinations: theory and applications. CRC Press, Boca Raton (2000)

    Book  Google Scholar 

  8. 8.

    Mocanu, P.T.: Some starlikeness conditions for analytic functions. Rev. Roum. Math. Pures Appl. 33, 117–124 (1988)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Nunokawa, M.: On the order of strongly starlikeness of strongly convex functions. Proc. Jpn. Acad. Ser. A Math. Sci. 69(7), 234–237 (1993)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Nunokawa, M.: On properties of non-Carathéodory functions. Proc. Jpn. Acad. Ser. A 6(68), 152–153 (1992)

    Article  Google Scholar 

  11. 11.

    Nunokawa, M., Kwon, O.S., Sim, Y.J., Cho, N.E.: Sufficient conditions for Carathéodory functions. Filomat 32(3), 1097–1106 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Nunokawa, M., Owa, S., Polatoğlu, Y., Çağlar, M., Duman, E.M.: Some sufficient conditions for starlikeness and convexity. Turk. J. Math. 3(34), 333–338 (2010)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Nunokawa, M., Sokół, J.: An improvement of Ozaki’s condition. Appl. Math. Comput. 22(219), 10768–10776 (2013)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Şan, M., Irmak, H.: Ordinary differential operator and some of its applications to certain meromorphically \(p\)-valent functions. Appl. Math. Comput. 218(3), 817–821 (2011)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Sokół, J., Nunokawa, M.: On the order of strongly close-to-convexity of strongly convex functions (On Schwarzian Derivatives and Its Applications). RIMS Kôkyûroku No. 1824, 100–106 (2013)

    Google Scholar 

  16. 16.

    Vinod, K., Shukla, S.L.: Certain integrals for classes of \(p\)-valent meromorphic functions. Bull. Aust. Math. Soc. 25(1), 85–97 (1982)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to express sincere thanks to the referees for careful reading and suggestions which help us to improve the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Vali Soltani Masih.

Additional information

Publisher's Note

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

Communicated by Ali Abkar.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Masih, V.S., Ebadian, A. & Najafzadeh, S. On Applications of Nunokawa and Sokół Theorem for p-Valency. Bull. Iran. Math. Soc. 46, 471–486 (2020). https://doi.org/10.1007/s41980-019-00270-y

Download citation

Keywords

  • Nunokawa and Sokół theorem
  • Subordination
  • p-valent strongly starlike and convex function of order \(\upalpha \)
  • p-valent meromorphic function

Mathematics Subject Classification

  • 30A10
  • 30C45