Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

STARLIKENESS AND SCHWARZIAN DERIVATIVES OF HIGHER ORDER OF ANALYTIC FUNCTIONS

  • Received : 2016.03.21
  • Published : 2017.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we apply the third-order differential subordination results to normalized analytic functions in the open unit disk. We obtain appropriate classes of admissible functions and find some sufficient conditions of functions to be starlike associated with Tamanoi's Schwarzian derivative of third order. Several interesting examples are also discussed.

Keywords

References

  1. D. Aharonov, A necessary and sufficient condition for univalence of a meromorphic function, Duke Math. J. 36 (1969), 599-604. https://doi.org/10.1215/S0012-7094-69-03671-0
  2. R. M. Ali, V. Ravichandran, and N. Seenivasagan, Subordination and superordination on Schwarzian derivatives, J. Inequal. Appl. 2008 (2008), Article ID 712328, 18 pp.
  3. A. A. Antonino and S. S. Miller, Third-order differential inequalities and subordinations in the complex plane, Complex Var. Elliptic Equ. 56 (2011), no. 5, 439-454. https://doi.org/10.1080/17476931003728404
  4. R. Harmelin, Aharonov invariants and univalent functions, Israel J. Math. 43 (1982), no. 3, 244-254. https://doi.org/10.1007/BF02761945
  5. M. P. Jeyaraman and T. K. Suresh, Third-order differential subordination of analytic functions, Acta Univ. Apulensis Math. Inform. 35 (2013), 187-202.
  6. S. A. Kim and T. Sugawa, Invariant Schwarzian derivatives of higher order, Complex Anal. Oper. Theory 5 (2011), no. 3, 659-670. https://doi.org/10.1007/s11785-010-0081-6
  7. S. S. Miller and P. T. Mocanu, Second order differential inequalities in the complex plane, J. Math. Anal. Appl. 65 (1978), no. 2, 289-305. https://doi.org/10.1016/0022-247X(78)90181-6
  8. S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Marcel Dekker, New York, 2000.
  9. Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545-551. https://doi.org/10.1090/S0002-9904-1949-09241-8
  10. Z. Nehari, Some criteria of univalence, Proc. Amer. Math. Soc. 5 (1954), 700-704. https://doi.org/10.1090/S0002-9939-1954-0064145-2
  11. S. Owa and M. Obradovic, An application of differential subordinations and some criteria for univalency, Bull. Aust. Math. Soc. 41 (1990), no. 3, 487-494. https://doi.org/10.1017/S0004972700018360
  12. E. Schippers, Distortion theorems for higher order Schwarzian derivatives of univalent functions, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3241-3249. https://doi.org/10.1090/S0002-9939-00-05623-9
  13. H. Tamanoi, Higher Schwarzian operators and combinatorics of the Schwarzian derivative, Math. Ann. 305 (1996), no. 1, 127-151. https://doi.org/10.1007/BF01444214

Cited by

  1. Sharp Bounds on the Higher Order Schwarzian Derivatives for Janowski Classes vol.10, pp.8, 2018, https://doi.org/10.3390/sym10080348