Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

HARDY SPACE OF LOMMEL FUNCTIONS

  • Yagmur, Nihat (Department of Mathematics Faculty of Science and Art Erzincan University)
  • Received : 2014.07.21
  • Published : 2015.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this work we present some geometric properties (like star-likeness and convexity of order ${\alpha}$ and also close-to-convexity of order ($1+{\alpha}$)/2) for normalized of Lommel functions of the first kind. In order to prove our main results, we use the technique of differential subordinations and some inequalities. Furthermore, we present some applications of convexity involving Lommel functions associated with the Hardy space of analytic functions, i.e., we obtain conditions for the function $h_{{\mu},{\upsilon}}(z)$ to belong to the Hardy space $H^p$.

Keywords

References

  1. A. Baricz, Bessel transforms and Hardy space of generalized Bessel functions, Mathematica 48(71) (2006), no. 2, 127-136.
  2. A. Baricz, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen 73 (2008), no. 1-2, 155-178.
  3. A. Baricz, Generalized Bessel functions of the first kind, Lecture Notes in Mathematics, vol. 1994, Springer-Verlag, Berlin, 2010.
  4. A. Baricz and S. Koumandos, Turan type inequalities for some lommel functions of the first kind, arxiv:1308.6477.
  5. A. Baricz and S. Ponnusamy, Starlikeness and convexity of generalized Bessel functions, Integral Transforms Spec. Funct. 21 (2010), no. 9-10, 641-653. https://doi.org/10.1080/10652460903516736
  6. A. Baricz and R. Szasz, Close-to-convexity of some special functions and their derivatives, arxiv:1402.0692.
  7. P. L. Duren, Theory of $H^p$ Spaces, Academic Press, New York/London, 1970.
  8. P. J. Eenigenburg and F. R. Keogh, The Hardy class of some univalent functions and their derivatives, Michigan Math. J. 17 (1970), 335-346. https://doi.org/10.1307/mmj/1029000519
  9. S. S.Miller and P. T. Mocanu, Differential subordinations and inequalities in the complex plane, J. Differential Equations 67 (1987), no. 2, 199-211. https://doi.org/10.1016/0022-0396(87)90146-X
  10. H. Orhan and N. Yagmur, Geometric properties of generalized struve functions, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) (in press).
  11. S. Owa, M. Nunokawa, H. Saitoh, and H. M. Srivastava, Close-to-convexity, starlikeness, and convexity of certain analytic functions, Appl. Math. Lett. 15 (2002), no. 1, 63-69. https://doi.org/10.1016/S0893-9659(01)00094-5
  12. S. Ponnusamy, The Hardy space of hypergeometric functions, Complex Variables Theory Appl. 29 (1996), no. 1, 83-96. https://doi.org/10.1080/17476939608814876
  13. S. Ponnusamy, Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane, Rocky Mountain J. Math. 28 (1998), no. 2, 695-733. https://doi.org/10.1216/rmjm/1181071795
  14. J. Stankiewicz and Z. Stankiewicz, Some applications of the Hadamard convolutions in the theory of functions, Ann. Univ. Mariae Curie-Sklodowska 40 (1986), 251-265.
  15. G. N.Watson, A Treatise on the Theory of Bessel Functions, Second edition, Cambridge University Press, Cambridge, London and New York, 1944.
  16. N. Yagmur and H. Orhan, Starlikeness and convexity of generalized Struve functions, Abstr. Appl. Anal. 2013 (2013), article ID 954513, 6 pages; doi:10.1155/2013/954513.
  17. N. Yagmur and H. Orhan, Hardy space of generalized Struve functions, Complex Var. Elliptic Equ. 59 (2014), no. 7, 929-936. https://doi.org/10.1080/17476933.2013.799148

Cited by

  1. Certain Geometric Properties of Normalized Wright Functions vol.2016, 2016, https://doi.org/10.1155/2016/1896154
  2. Geometric Properties of Lommel Functions of the First Kind vol.10, pp.10, 2018, https://doi.org/10.3390/sym10100455