DOI QR코드

DOI QR Code

COEFFICIENT DISCS AND GENERALIZED CENTRAL FUNCTIONS FOR THE CLASS OF CONCAVE SCHLICHT FUNCTIONS

  • Received : 2013.09.04
  • Published : 2014.09.30

Abstract

We consider functions that map the open unit disc conformally onto the complement of an unbounded convex set with opening angle ${\pi}{\alpha}$, ${\alpha}{\in}(1,2]$, at infinity. We derive the exact interval for the variability of the real Taylor coefficients of these functions and we prove that the corresponding complex Taylor coefficients of such functions are contained in certain discs lying in the right half plane. In addition, we also determine generalized central functions for the aforesaid class of functions.

References

  1. F. G. Avkhadiev, Ch. Pommerenke, and K.-J. Wirths, Sharp inequalities for the coefficients of concave schlicht functions, Comment. Math. Helv. 81 (2006), no. 4, 801-807.
  2. F. G. Avkhadiev and K.-J. Wirths, Concave schlicht functions with bounded opening angle at infinity, Lobachevskii J. Math. 17 (2005), 3-10.
  3. U. Bednarz and J. Sokol, On T-neighborhoods of analytic functions, J. Math. Appl. 32 (2010), 25-32.
  4. B. Bhowmik, On concave univalent functions, Math. Nachr. 285 (2012), no. 5-6, 606-612. https://doi.org/10.1002/mana.201000063
  5. B. Bhowmik, S. Ponnusamy, and K.-J.Wirths, Characterization and the pre-Schwarzian norm estimate for concave univalent functions, Monatsh. Math. 161 (2010), no. 1, 59-75. https://doi.org/10.1007/s00605-009-0146-7
  6. B. Bhowmik and K.-J. Wirths, Central functions for classes of concave univalent functions , Math. Slovaca, To Appear.
  7. W. Rogosinski, Uber positive harmonische Entwicklungen und typisch reelle Potenzreihen, Math. Z. 35 (1932), no. 1, 93-121. https://doi.org/10.1007/BF01186552
  8. St. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81 (1981), no. 4, 521-527. https://doi.org/10.1090/S0002-9939-1981-0601721-6
  9. T. Sheil-Small, On the convolution of analytic functions, J. Reine Angew. Math. 258 (1973), 137-152.
  10. T. Sheil-Small and E. M. Silvia, Neighborhoods of analytic functions, J. Analyse Math. 52 (1989), 210-240.
  11. J. Sokol, On neighborhoods of analytic functions with positive real part, Math. Nachr. 284 (2011), no. 11-12, 1547-1553. https://doi.org/10.1002/mana.200910074