DOI QR코드

DOI QR Code

THE TILTED CARATHÉODORY CLASS AND ITS APPLICATIONS

  • Wang, Li-Mei (Division of Mathematics Graduate School of Information Sciences Tohoku University)
  • Received : 2010.07.22
  • Published : 2012.07.01

Abstract

This paper mainly deals with the tilted Carath$\acute{e}$odory class by angle ${\lambda}$ ${\in}$ ($-{\pi}/2$, ${\pi}/2$), denoted by $P{\lambda}$) an element of which maps the unit disc into the tilted right half-plane {<${\omega}$ : Re $e^{i{\lambda}}{\omega}$ > 0}. Firstly we will characterize $P{\lambda}$ from different aspects, for example by subordination and convolution. Then various estimates of functionals over $P{\lambda}$ are deduced by considering these over the extreme points of $P{\lambda}$ or the knowledge of functional analysis. Finally some subsets of analytic functions related to $P{\lambda}$ including close-to-convex functions with argument ${\lambda}$, ${\lambda}$-spirallike functions and analytic functions whose derivative is in $P{\lambda}$ are also considered as applications.

Keywords

References

  1. O. P. Ahuja and H. Silverman, A survey on spiral-like and related function classes, Math. Chronicle 20 (1991), 39-66.
  2. S. D. Bernardi, New distortion theorems for functions of positive real part and applications to the partial sums of univalent convex functions, Proc. Amer. Math. Soc. 45 (1974), 113-118. https://doi.org/10.1090/S0002-9939-1974-0357755-9
  3. P. L. Duren, Univalent Functions, Grundlehren Math. Wiss. 259, Springer-Verlag, New York, 1983.
  4. S. Gelfer, On the class of regular functions which do not take on any pair of values ${\omega}$ and $-{\omega}$, Mat. Sb. 19 (1946), 33-46.
  5. A. W. Goodman, Univalent Functions. Vol. II, Mariner Publishing Co. Inc., 1983.
  6. I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker, Inc., New York, 2003.
  7. F. Gray and St. Ruscheweyh, Functions whose derivatives take values in a half-plane, Proc. Amer. Math. Soc. 104 (1988), no. 1, 215-218. https://doi.org/10.1090/S0002-9939-1988-0958069-6
  8. D. J. Hallenbeck and T. H. MacGregor, Linear Problems and Convexity Techniques in Geometric Function Theory, Monographs and Studies in Mathematics, 22. Pitman (Advanced Publishing Program), Boston, MA, 1984.
  9. G. Herglotz, Uber Potenzreihen mit positivem, reellen Teil in Einheitskreis, Ber. Verh. Sachs. Akad. Wiss. Leipzig (1911), 501-511.
  10. F. Holland, The extreme points of a class of functions with positive real part, Math. Ann. 202 (1973), 85-87. https://doi.org/10.1007/BF01351208
  11. Y. C. Kim and T. Sugawa, Growth and coefficient estimates for uniformly locally univalent functions on the unit disk, Rocky Mountain J. Math. 32 (2002), no. 1, 179-200. https://doi.org/10.1216/rmjm/1030539616
  12. Y. C. Kim and T. Sugawa, A conformal invariant for nonvanishing analytic functions and its applications, Michigan Math. J. 54 (2006), no. 2, 393-410. https://doi.org/10.1307/mmj/1156345602
  13. Y. C. Kim and T. Sugawa, Norm estimates of the pre-Schwarzian derivatives for certain classes of univalent functions, Proc. Edinb. Math. Soc. (2) 49 (2006), no. 1, 131-143. https://doi.org/10.1017/S0013091504000306
  14. Y. C. Kim and T. Sugawa, A note on Bazilevic functions, Taiwanese J. Math. 13 (2009), no. 5, 1489-1495. https://doi.org/10.11650/twjm/1500405555
  15. R. A. Kortram, The extreme points of a class of functions with positive real part, Bull. Belg. Math. Soc. Simon Stevin 4 (1997), no. 4, 449-459.
  16. R. J. Libera, Some radius of convexity problems, Duke Math. J. 31 (1964), 143-158. https://doi.org/10.1215/S0012-7094-64-03114-X
  17. T. H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc. 104 (1962), 532-537. https://doi.org/10.1090/S0002-9947-1962-0140674-7
  18. M. Nunokawa, On the univalency and multivalency of certain analysis functions, Math. Z. 104 (1968), 394-404. https://doi.org/10.1007/BF01110431
  19. M. S. Robertson, Variational methods for functions with positive real part, Trans. Amer. Math. Soc. 102 (1962), 82-93. https://doi.org/10.1090/S0002-9947-1962-0133454-X
  20. M. S. Robertson, Extremal problems for analytic functions with positive real part and applications, Trans. Amer. Math. Soc. 106 (1963), 236-253. https://doi.org/10.1090/S0002-9947-1963-0142756-3
  21. M. S. Robertson, Radii of star-likeness and close-to-covexity, Proc. Amer. Math. Soc. 16 (1965), 847-852.
  22. St. Ruscheweyh, Convolution in geometric function theory, Sem. Math. Sup. 83, University of Montreal, Montreal, Quebec, Canada 1982.
  23. St. Ruscheweyh and V. Singh, On certain extremal problems for functions with positive real part, Proc. Amer. Math. Soc. 61 (1976), no. 2, 329-334. https://doi.org/10.1090/S0002-9939-1976-0425102-1
  24. K. Sakaguchi, A variational method for functions with positive real part, J. Math. Soc. Japan 16 (1964), 287-297. https://doi.org/10.2969/jmsj/01630287
  25. I. Schur, Uber Potenzreihen, die im Innern des Einheitskreises beschrankt sind, J. Reine Angew. Math. 147 (1917), 205-232.
  26. I. Schur, Uber Potenzreihen, die im Innern des Einheitskreises beschrankt sind, J. Reine Angew. Math. 148 (1918), 122-145.
  27. L. Spacek, Contribution a la theorie des fonctions univalentes, Casopis Pest. Mat.-Fys. 62 (1932), 12-19.
  28. L.-M. Wang, Coefficient estimates for close-to-convex functions with argument ${\lambda}$, Bull. Berg. Math. Soc. (to appear).
  29. S. Yamashita, Gel'fer functions, integral means, bounded mean oscillation, and univalency, Trans. Amer. Math. Soc. 321 (1990), no. 1, 245-259.
  30. V. A. Zmorovic, On bounds of convexity for starlike function of order ${\alpha}$ in the circle |z| < 1 and in the circular region 0 < |z| < 1, Mat. Sb. 68 (110) (1965), 518-526; English transl., Amer. Math. Soc. Transl. (2) 80 (1969), 203-213.
  31. V. A. Zmorovic, On the bounds of starlikeness and univalence in certain classes of functions regular in the circle |z| < 1, Ukrain. Mat. Z. 18 (1966), 28-39; English transl., Amer. Math. Soc. Transl. (2) 80 (1969), 227-242.