On a radius problem concerning a class of close-to-convex functions

Richard Fournier

Banach Center Publications (1995)

  • Volume: 31, Issue: 1, page 187-195
  • ISSN: 0137-6934

Abstract

top
The problem of estimating the radius of starlikeness of various classes of close-to-convex functions has attracted a certain number of mathematicians involved in geometric function theory ([7], volume 2, chapter 13). Lewandowski [11] has shown that normalized close-to-convex functions are starlike in the disc | z | < 4 √ 2 - 5 . Krzyż [10] gave an example of a function f ( z ) = z + ∑ n = 2 ∞ a n z n , non-starlike in the unit disc , and belonging to the class H = f | f’() lies in the right half-plane. More generally let H* = f | f’() lies in some half-plane not containing 0. To the best of our knowledge, the radii of starlikeness of both H and H* are still unknown, in spite of the fact that corresponding extremal functions can be described in a relatively simple way (by using, for example, Ruscheweyh’s duality theory [15]). This paper is a survey of recent results concerning the radius of starlikeness of K = f ∈ H | |f’(z)-1| < 1, z ∈ .

How to cite

top

Fournier, Richard. "On a radius problem concerning a class of close-to-convex functions." Banach Center Publications 31.1 (1995): 187-195. <http://eudml.org/doc/262651>.

@article{Fournier1995,
abstract = {The problem of estimating the radius of starlikeness of various classes of close-to-convex functions has attracted a certain number of mathematicians involved in geometric function theory ([7], volume 2, chapter 13). Lewandowski [11] has shown that normalized close-to-convex functions are starlike in the disc $|z| < 4√2 - 5$. Krzyż [10] gave an example of a function $f(z) = z + ∑_\{n=2\}^∞ a_n z^n$, non-starlike in the unit disc , and belonging to the class H = f | f’() lies in the right half-plane. More generally let H* = f | f’() lies in some half-plane not containing 0. To the best of our knowledge, the radii of starlikeness of both H and H* are still unknown, in spite of the fact that corresponding extremal functions can be described in a relatively simple way (by using, for example, Ruscheweyh’s duality theory [15]). This paper is a survey of recent results concerning the radius of starlikeness of K = f ∈ H | |f’(z)-1| < 1, z ∈ .},
author = {Fournier, Richard},
journal = {Banach Center Publications},
language = {eng},
number = {1},
pages = {187-195},
title = {On a radius problem concerning a class of close-to-convex functions},
url = {http://eudml.org/doc/262651},
volume = {31},
year = {1995},
}

TY - JOUR
AU - Fournier, Richard
TI - On a radius problem concerning a class of close-to-convex functions
JO - Banach Center Publications
PY - 1995
VL - 31
IS - 1
SP - 187
EP - 195
AB - The problem of estimating the radius of starlikeness of various classes of close-to-convex functions has attracted a certain number of mathematicians involved in geometric function theory ([7], volume 2, chapter 13). Lewandowski [11] has shown that normalized close-to-convex functions are starlike in the disc $|z| < 4√2 - 5$. Krzyż [10] gave an example of a function $f(z) = z + ∑_{n=2}^∞ a_n z^n$, non-starlike in the unit disc , and belonging to the class H = f | f’() lies in the right half-plane. More generally let H* = f | f’() lies in some half-plane not containing 0. To the best of our knowledge, the radii of starlikeness of both H and H* are still unknown, in spite of the fact that corresponding extremal functions can be described in a relatively simple way (by using, for example, Ruscheweyh’s duality theory [15]). This paper is a survey of recent results concerning the radius of starlikeness of K = f ∈ H | |f’(z)-1| < 1, z ∈ .
LA - eng
UR - http://eudml.org/doc/262651
ER -

References

top
  1. [1] R. P. Boas, Entire Functions, Academic Press, New York, 1954. Zbl0058.30201
  2. [2] P. C. Cochrane and T. H. MacGregor, Fréchet differentiable functionals and support points for families of analytic functions, Trans. Amer. Math. Soc. 236 (1978), 75-92. Zbl0377.30010
  3. [3] K. de Leeuw and W. Rudin, Extreme points and extremum problems in H 1 , Pacific J. Math. 8 (1958), 467-485. Zbl0084.27503
  4. [4] P. L. Duren, Univalent Functions, Springer, New York, 1983. 
  5. [5] R. Fournier, On integrals of bounded analytic functions in the unit disc, Complex Variables 11 (1989), 125-133. Zbl0639.30016
  6. [6] R. Fournier, The range of a continuous linear functional over a class of functions defined by subordination, Glasgow Math. J. 32 (1990), 381-387. Zbl0715.30010
  7. [7] A. W. Goodman, Univalent Functions, Mariner Publishing Company, Tampa, 1983. 
  8. [8] D. J. Hallenbeck and T. H. MacGregor, Support points of families of analytic functions defined by subordination, Trans. Amer. Math. Soc. 278 (1983), 523-546. Zbl0521.30018
  9. [9] D. J. Hallenbeck and T. H. MacGregor, Linear Problems and Convexity Techniques in Geometric Function Theory, Pitman, Boston, 1984. Zbl0581.30001
  10. [10] J. Krzyż, A counterexample concerning univalent functions, Folia Soc. Scient. Lubliniensis 2 (1962), 57-58. 
  11. [11] Z. Lewandowski, Sur l'identité de certaines classes de fonctions univalentes, Ann. Univ. M. Curie-Skłodowska 14 (1960), 19-46. Zbl0108.07502
  12. [12] T. H. MacGregor, A class of univalent functions, Proc. Amer. Math. Soc. 15 (1964), 311-317. Zbl0137.05205
  13. [13] R. M. McLeod, The Generalized Riemann Integral, Mathematical Association of America, 1980. Zbl0486.26005
  14. [14] P. T. Mocanu, Some starlikeness conditions for analytic functions, Rev. Roumaine Math. Pures Appl. 33 (1988), 117-124. Zbl0652.30004
  15. [15] St. Ruscheweyh, Convolutions in Geometric Function Theory, Les Presses de l'Université de Montréal, Montréal, 1982. 
  16. [16] St. Ruscheweyh, Duality for Hadamard products with applications to extremal problems for functions regular in the unit disc, Trans. Amer. Math. Soc. 210 (1975), 63-74. Zbl0311.30011
  17. [17] O. Toeplitz, Die linearen volkommenen Räume der Funktionentheorie, Comment. Math. Helv. 23 (1949), 222-242. Zbl0035.07301
  18. [18] V. Singh, Univalent functions with bounded derivative in the unit disc, Indian J. Pure Appl. Math. 5 (1974), 733-754. Zbl0346.30011

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.