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Banach Center Publications

1995 | 31 | 1 | 187-195
Tytuł artykułu

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

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EN
Abstrakty
EN
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 𝔻, and belonging to the class H = {f | f'(𝔻) lies in the right half-plane.} More generally let H* = {f | f'(𝔻) 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 ∈ 𝔻}.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
187-195
Opis fizyczny
Daty
wydano
1995
Twórcy
autor
• Département de Mathématiques et de Statistique, Université de Montréal, Montréal (QC) H3C 3J7, Canada
Bibliografia
• [1] R. P. Boas, Entire Functions, Academic Press, New York, 1954.
• [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.
• [3] K. de Leeuw and W. Rudin, Extreme points and extremum problems in $H_1$, Pacific J. Math. 8 (1958), 467-485.
• [4] P. L. Duren, Univalent Functions, Springer, New York, 1983.
• [5] R. Fournier, On integrals of bounded analytic functions in the unit disc, Complex Variables 11 (1989), 125-133.
• [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.
• [7] A. W. Goodman, Univalent Functions, Mariner Publishing Company, Tampa, 1983.
• [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.
• [9] D. J. Hallenbeck and T. H. MacGregor, Linear Problems and Convexity Techniques in Geometric Function Theory, Pitman, Boston, 1984.
• [10] J. Krzyż, A counterexample concerning univalent functions, Folia Soc. Scient. Lubliniensis 2 (1962), 57-58.
• [11] Z. Lewandowski, Sur l'identité de certaines classes de fonctions univalentes, Ann. Univ. M. Curie-Skłodowska 14 (1960), 19-46.
• [12] T. H. MacGregor, A class of univalent functions, Proc. Amer. Math. Soc. 15 (1964), 311-317.
• [13] R. M. McLeod, The Generalized Riemann Integral, Mathematical Association of America, 1980.
• [14] P. T. Mocanu, Some starlikeness conditions for analytic functions, Rev. Roumaine Math. Pures Appl. 33 (1988), 117-124.
• [15] St. Ruscheweyh, Convolutions in Geometric Function Theory, Les Presses de l'Université de Montréal, Montréal, 1982.
• [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.
• [17] O. Toeplitz, Die linearen volkommenen Räume der Funktionentheorie, Comment. Math. Helv. 23 (1949), 222-242.
• [18] V. Singh, Univalent functions with bounded derivative in the unit disc, Indian J. Pure Appl. Math. 5 (1974), 733-754.
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