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On confluent hypergeometric functions and generalized Bessel functions

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Abstract

Conditions are determined on the parameters a and c so that the confluent hypergeometric function Φ(a, c; z) = 1 F 1(a, c; z) is strongly convex of order 1/2 and the function zΦ(a, c; z) is strongly starlike of order 1/2 in D. Also sufficient conditions are obtained on a and c which ensure that the subordination (c/a)Φ′(a, c; z) ≺ ((1 + z)/(1 − z))1/2 holds. We also derive conditions on the parameters associated with the generalized and normalized Bessel function u p (z) of order p so that u p is strongly convex of order 1/2 and zu p is strongly starlike of order 1/2. As an application interesting examples are given including some mapping properties of Alexander and Libera operators.

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References

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

    MathSciNet  MATH  Google Scholar 

  2. H. A. Al-Kharsani, Á. Baricz and K. S. Nisar, Differential subordinations and superordinations for generalized Bessel functions, Bull. Korean Math. Soc., 53 (2016), 127–138.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Balasubramanian, S. Ponnusamy and M. Vuorinen, On hypergeometric functions and function spaces, J. Comput. Appl. Math., 139 (2002), 299–322.

    Article  MathSciNet  MATH  Google Scholar 

  4. Á. Baricz, Applications of the admissible functions method for some differential equations, Pure Math. Appl., 13 (2002), 433–440.

    MathSciNet  MATH  Google Scholar 

  5. Á. Baricz, Geometric properties of generalized Bessel functions of complex order, Mathematica, 48(71) (2006), 13–18.

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  7. Á. Baricz and B. A. Frasin, Univalence of integral operators involving Bessel functions, Appl. Math. Lett., 23 (2010), 371–376.

    Article  MathSciNet  MATH  Google Scholar 

  8. Á. Baricz and S. Ponnusamy, Differential inequalities and Bessel functions, J. Math. Anal. Appl., 400 (2013), 558–567.

    Article  MathSciNet  MATH  Google Scholar 

  9. Á. Baricz and R. Szász, The radius of convexity of normalized Bessel functions of the first kind, Anal. Appl. (Singap.), 12 (2014), 485–509.

    Article  MathSciNet  MATH  Google Scholar 

  10. Á. Baricz, P. A. Kupán and R. Szász, The radius of starlikeness of normalized Bessel functions of the first kind, Proc. Amer. Math. Soc., 142 (2014), 2019–2025.

    Article  MathSciNet  MATH  Google Scholar 

  11. Á. Baricz, E. Deniz, M. Çăglar and H. Orhan, Differential subordinations involving generalized Bessel functions, Bull. Malays. Math. Sci. Soc., 38 (2015), 1255–1280.

    Article  MathSciNet  MATH  Google Scholar 

  12. Á. Baricz, D. K. Dimitrov and I. Mező, Radii of starlikeness and convexity of some q-Bessel functions, J. Math. Anal. Appl., 435 (2016), 968–985.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Ponnusamy and M. Vuorinen, Univalence and convexity properties for Gaussian hypergeometric functions, Rocky Mountain J. Math., 31 (2001), 327–353.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. T. N. Shanmugam, Convolution and differential subordination, Internat. J. Math. Math. Sci., 12 (1989), 333–340.

    Article  MathSciNet  MATH  Google Scholar 

  21. T. N. Shanmugam, Hypergeometric functions in the geometric function theory, Appl. Math. Comput., 187 (2007), 433–444.

    MathSciNet  MATH  Google Scholar 

  22. A. Swaminathan, Certain sufficiency conditions on Gaussian hypergeometric functions, J. Inequal. Pure Appl. Math., 5 (2004), Article 83, 10 pp.

  23. R. Szász, About the radius of starlikeness of Bessel functions of the first kind, Monatsh. Math., 176 (2015), 323–330.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Delhi, Delhi, 110 007, India

    N. Bohra & V. Ravichandran

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  1. N. Bohra
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  2. V. Ravichandran
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Correspondence to N. Bohra.

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Bohra, N., Ravichandran, V. On confluent hypergeometric functions and generalized Bessel functions. Anal Math 43, 533–545 (2017). https://doi.org/10.1007/s10476-017-0203-8

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  • DOI: https://doi.org/10.1007/s10476-017-0203-8

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