Exponential Starlikeness and Convexity of Confluent Hypergeometric, Lommel, and Struve Functions

Abstract

Sufficient conditions are obtained on the parameters of Lommel function of the first kind, generalized Struve function of the first kind, and the confluent hypergeometric function under which these special functions become exponential convex and exponential starlike in the open unit disk. The method of differential subordination is employed in proving the results. Few examples are also provided to illustrate the results obtained.

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

Fig. 1
Fig. 2
Fig. 3

References

  1. 1.

    Ali, R.M., Mondal, S.R., Ravichandran, V.: On the Janowski convexity and starlikeness of the confluent hypergeometric function. Bull. Belg. Math. Soc. Simon Stevin 22(2), 227–250 (2015)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Baricz, A.: Geometric properties of generalized Bessel functions. Publ. Math. Debrecen 73(1–2), 155–178 (2008)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Baricz, A.: Generalized Bessel Functions of the First Kind. Lecture Notes in Mathematics (1994). Springer, Berlin (2010)

    Google Scholar 

  4. 4.

    Baricz, A., Ponnusamy, S.: Starlikeness and convexity of generalized Bessel functions. Integral Transforms Spec. Funct. 21(9–10), 641–653 (2010)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Baricz, A., Yağmur, N.: Geometric properties of some Lommel and Struve functions. Ramanujan J. 42(2), 325–346 (2017)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bohra, N., Ravichandran, V.: On confluent hypergeometric functions and generalized Bessel functions. Anal. Math. 43(4), 533–545 (2017)

    MathSciNet  Article  Google Scholar 

  7. 7.

    de Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154(1–2), 137–152 (1985)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Farzana Habibullah, A., Bhaskaran, A.S., Muthusamy Palani, J.: Univalent and starlike properties for generalized Struve function. Int. J. Math. Math. Sci. 20, 7 (2016). (Art. ID 3987231)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Kanas, S., Mondal, S.R., Mohammed, A.D.: Relations between the generalized Bessel functions and the Janowski class. Math. Inequal. Appl. 21(1), 165–178 (2018)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    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), 157–169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA (1992)

  11. 11.

    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)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Miller, S.S., Mocanu, P.T.: Univalence of Gaussian and confluent hypergeometric functions. Proc. Am. Math. Soc. 110(2), 333–342 (1990)

    MathSciNet  Article  Google Scholar 

  13. 13.

    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 

  14. 14.

    Mondal, S.R., Al Dhuain, M.: Inclusion of the generalized Bessel functions in the Janowski class. Int. J. Anal. 2016, 8 (2016). (Art. ID 4740819)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Naz, A., Nagpal, S., Ravichandran, V.: Geometric properties of generalized Bessel function associated with the exponential function. (2019) (Preprint)

  16. 16.

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

    MathSciNet  Article  Google Scholar 

  17. 17.

    Noreen, S., Raza, M., Deniz, E., Kazımoğlu, S.: On the Janowski class of generalized Struve functions. Afr. Mat. 30(1–2), 23–35 (2019)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Orhan, H., Yagmur, N.: Geometric properties of generalized Struve functions. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 63(2), 229–244 (2017)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Owa, S., Srivastava, H.M.: Univalent and starlike generalized hypergeometric functions. Can. J. Math. 39(5), 1057–1077 (1987)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Ponnusamy, S., Vuorinen, M.: Univalence and convexity properties for confluent hypergeometric functions. Complex Var. Theory Appl. 36(1), 73–97 (1998)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Prajapat, J.K.: Certain geometric properties of normalized Bessel functions. Appl. Math. Lett. 24(12), 2133–2139 (2011)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Radhika, V., Sivasubramanian, S., Cho, N.E., Murugusundaramoorthy, G.: Geometric properties of Bessel functions for the classes of Janowski starlike and convex functions. J. Comput. Anal. Appl. 25(3), 452–466 (2018)

    MathSciNet  Google Scholar 

  23. 23.

    Ruscheweyh, S., Singh, V.: On the order of starlikeness of hypergeometric functions. J. Math. Anal. Appl 113(1), 1–11 (1986)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Sim, Y., Kwon, O., Cho, N.E.: Geometric properties of Lommel functions of the first kind. Symmetry 10, 455 (2018)

    Article  Google Scholar 

  25. 25.

    Szász, R., Kupán, P.A.: About the univalence of the Bessel functions. Stud. Univ. Babeş-Bolyai Math. 54(1), 127–132 (2009)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Yağmur, N.: Hardy space of Lommel functions. Bull. Korean Math. Soc. 52(3), 1035–1046 (2015)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Yagmur, N., Orhan, H.: Starlikeness and convexity of generalized Struve functions. Abstr. Appl. Anal. 2013, 6 (2013). (Art. ID 954513)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sumit Nagpal.

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

Verify currency and authenticity via CrossMark

Cite this article

Naz, A., Nagpal, S. & Ravichandran, V. Exponential Starlikeness and Convexity of Confluent Hypergeometric, Lommel, and Struve Functions. Mediterr. J. Math. 17, 204 (2020). https://doi.org/10.1007/s00009-020-01621-4

Download citation

Keywords

  • Differential subordination
  • exponential function
  • Struve function
  • Lommel function
  • confluent hypergeometric function

Mathematics Subject Classification

  • Primary 30C45
  • Secondary 30C80