Coefficient inequality for transforms of parabolic starlike and uniformly convex functions

D. Vamshee Krishna[1]; B. Venkateswarlu[1]; T. RamReddy[2]

  • [1] Department of Mathematics GIT, GITAM University Visakhapatnam- 530 045, A.P., India.
  • [2] Department of Mathematics, Kakatiya University, Warangal- 506 009, A.P., India.

Annales mathématiques Blaise Pascal (2014)

  • Volume: 21, Issue: 2, page 39-56
  • ISSN: 1259-1734

Abstract

top
The objective of this paper is to obtain sharp upper bound to the second Hankel functional associated with the k t h root transform f ( z k ) 1 k of normalized analytic function f ( z ) belonging to parabolic starlike and uniformly convex functions, defined on the open unit disc in the complex plane, using Toeplitz determinants.

How to cite

top

Vamshee Krishna, D., Venkateswarlu, B., and RamReddy, T.. "Coefficient inequality for transforms of parabolic starlike and uniformly convex functions." Annales mathématiques Blaise Pascal 21.2 (2014): 39-56. <http://eudml.org/doc/275600>.

@article{VamsheeKrishna2014,
abstract = {The objective of this paper is to obtain sharp upper bound to the second Hankel functional associated with the $k^\{th\}$ root transform $\left[ f(z ^k ) \right] ^\{\frac\{1\}\{k\}\}$ of normalized analytic function $f(z)$ belonging to parabolic starlike and uniformly convex functions, defined on the open unit disc in the complex plane, using Toeplitz determinants.},
affiliation = {Department of Mathematics GIT, GITAM University Visakhapatnam- 530 045, A.P., India.; Department of Mathematics GIT, GITAM University Visakhapatnam- 530 045, A.P., India.; Department of Mathematics, Kakatiya University, Warangal- 506 009, A.P., India.},
author = {Vamshee Krishna, D., Venkateswarlu, B., RamReddy, T.},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Analytic function; parabolic starlike and uniformly convex functions; upper bound; second Hankel functional; positive real function; Toeplitz determinants; parabolic star-like functions; uniformly convex functions},
language = {eng},
month = {7},
number = {2},
pages = {39-56},
publisher = {Annales mathématiques Blaise Pascal},
title = {Coefficient inequality for transforms of parabolic starlike and uniformly convex functions},
url = {http://eudml.org/doc/275600},
volume = {21},
year = {2014},
}

TY - JOUR
AU - Vamshee Krishna, D.
AU - Venkateswarlu, B.
AU - RamReddy, T.
TI - Coefficient inequality for transforms of parabolic starlike and uniformly convex functions
JO - Annales mathématiques Blaise Pascal
DA - 2014/7//
PB - Annales mathématiques Blaise Pascal
VL - 21
IS - 2
SP - 39
EP - 56
AB - The objective of this paper is to obtain sharp upper bound to the second Hankel functional associated with the $k^{th}$ root transform $\left[ f(z ^k ) \right] ^{\frac{1}{k}}$ of normalized analytic function $f(z)$ belonging to parabolic starlike and uniformly convex functions, defined on the open unit disc in the complex plane, using Toeplitz determinants.
LA - eng
KW - Analytic function; parabolic starlike and uniformly convex functions; upper bound; second Hankel functional; positive real function; Toeplitz determinants; parabolic star-like functions; uniformly convex functions
UR - http://eudml.org/doc/275600
ER -

References

top
  1. A. Abubaker, M. Darus, Hankel Determinant for a class of analytic functions involving a generalized linear differential operators, Int. J. Pure Appl. Math. 69(4) (2011), 429-435 Zbl1220.30011MR2847841
  2. J. W. Alexnader, Functions which map the interior of the unit circle upon simple regions, Annal. of. Math. (2)17 (1915), 12-22 MR1503516
  3. R. M. Ali, Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc.(second series) 26(1) (2003), 63-71 Zbl1185.30010MR2055766
  4. R. M. Ali, Starlikeness associated with parabolic regions, Int. J. Math. Math. Sci. 4 (2005), 561-570 Zbl1077.30011MR2172395
  5. R. M. Ali, S. K. Lee, V. Ravichandran, S. Supramaniam, The Fekete-Szeg o ¨ coefficient functional for transforms of analytic functions, Bull. Iran. Math. Soc. 35(2) (2009), 119-142 Zbl1193.30006MR2642930
  6. R. M. Ali, V. Singh, Coefficients of parabolic starlike functions of order ρ , (1995), 23-36, World Sci. Publ. River Edge, New Jersey Zbl0874.30007MR1415158
  7. P. L. Duren, Univalent functions, (1983), 259, Grundlehren der Mathematischen Wissenschaften, New York, Springer-verlag XIV, 328 Zbl0514.30001MR708494
  8. R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly 107(6) (2000), 557-560 Zbl0985.15006MR1767065
  9. A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1) (1991), 87-92 Zbl0744.30010MR1145573
  10. U. Grenander, G. Szegő, Toeplitz forms and their applications, (1984), Second edition. Chelsea Publishing Co., New York Zbl0611.47018MR890515
  11. A. Janteng, S. A. Halim, M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Inequl. Pure Appl. Math. 7(2) (2006), 1-5 Zbl1134.30310MR2221331
  12. A. Janteng, S. A. Halim, M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal., (Ruse) 4 (no. 13-16) (2007), 619-625 Zbl1137.30308MR2370200
  13. J. W. Layman, The Hankel transform and some of its properties, J. Integer Seq. 4 (1) (2001), 1-11 Zbl0978.15022MR1848942
  14. R. J. Libera, E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in ℙ , Proc. Amer. Math. Soc. 87 (1983), 251-257 Zbl0488.30010MR681830
  15. W. C. Ma, D. Minda, Uniformly convex functions, Ann. Polon. Math. 57(2) (1992), 165-175 Zbl0760.30004MR1182182
  16. J. W. Noonan, D. K. Thomas, On the second Hankel determinant of areally mean p - Valent functions, Trans. Amer. Math. Soc. 223(2) (1992), 337-346 Zbl0346.30012MR422607
  17. K. I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Et Appl. 28(8) (1983), 731-739 Zbl0524.30008MR725316
  18. Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc. 41 (1966), 111-122 Zbl0138.29801MR185105
  19. Ch. Pommerenke, Univalent functions, (1975), Vandenhoeck and Ruprecht, Gottingen Zbl0298.30014MR507768
  20. F. Ronning, A survey on uniformly convex and uniformly starlike functions, Ann. Univ. Mariae Curie - Sklodowska Sect. A. 47 (1993), 123-134 Zbl0879.30004MR1344982
  21. B. Simon, Orthogonal polynomials on the unit circle, Part 1. Classical theory, (2005), AMS Colloquium Publ. 54, Part 1, American Mathematical Society, Providence, RI Zbl1082.42021MR2105088
  22. D. VamsheeKrishna, T. RamReddy, Coefficient inequality for uniformly convex functions of order α , J. Adv. Res. Pure Math. 5(1) (2013), 25-41 MR3020966

NotesEmbed ?

top

You must be logged in to post comments.