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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Radii problems for sections of convex functions


Author: Herb Silverman
Journal: Proc. Amer. Math. Soc. 104 (1988), 1191-1196
MSC: Primary 30C45; Secondary 30C50
DOI: https://doi.org/10.1090/S0002-9939-1988-0942638-3
MathSciNet review: 942638
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Abstract: A classical theorem of Szegö states that the sections $ {f_n}(z) = z + \sum\nolimits_{k = 2}^n {{a_k}{z^k}} $ of a convex function $ f(z) = z + \sum\nolimits_{k = 2}^\infty {{a_k}{z^k}} $ must be convex for $ \left\vert z \right\vert < \frac{1}{4}$. We determine disks $ \left\vert z \right\vert < {r_n}$ in which $ {f_n}$ is starlike and starlike of a positve order. Our proofs rely on some properties of convolutions.


References [Enhancements On Off] (What's this?)

  • [1] A. W. Goodman and I. J. Schoenberg, On a theorem of Szegö on univalent convex maps of the unit circle, J. Analyse Math. 44 (1984/85), 200-204. MR 801293 (87b:30013)
  • [2] A. Marx, Untersuchungen über Schlichte Abbildungen, Math. Ann. 107 (1932/33), 40-67.
  • [3] G. Pólya and I. J. Schoenberg, Remarks on de la Vallée Poussin means and convex conformal maps of the circle, Pacific J. Math. 8 (1958), 295-334. MR 0100753 (20:7181)
  • [4] M. S. Robertson, The partial sums of multivalently star-like functions, Ann. of Math. 42 (1941), 829-838. MR 0004905 (3:79b)
  • [5] St. Ruscheweyh and T. Sheil-Small, Hadamard products of schlicht functions and the Pólya-Schoenberg conjecture, Comment Math. Helv. 48 (1973), 119-135. MR 0328051 (48:6393)
  • [6] E. Strohhäcker, Beiträge zur Theorie der schlichten Funktionen, Math. Z. 37 (1933), 356-380. MR 1545400
  • [7] G. Szegö, Zur Theorie der schlichten Abbildungen, Math. Ann. 100 (1928), 188-211. MR 1512482

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0942638-3
Keywords: Univalent, starlike, convex, convolution
Article copyright: © Copyright 1988 American Mathematical Society

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