On Robinson’s 1\over2 conjecture

Barnard, Roger W.

doi:10.1090/S0002-9939-1978-0503547-0

On Robinson’s ${1\over 2}$ conjecture
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Proc. Amer. Math. Soc. 72 (1978), 135-139 Request permission

Abstract:

In 1947, R. Robinson conjectured that if f is in S, i.e. a normalized univalent function on the unit disk, then the radius of univalence of $[zf(z)]’/2$ is at least $\tfrac {1}{2}$. He proved in that paper that it was at least .38. The conjecture has been shown to be true for most of the known subclasses of S. This author shows through use of the Grunski inequalities, that the minimum lower bound over the class S lies between .49 and .5.

References

Roger W. Barnard, On the radius of starlikeness of $(zf)^{\prime }$ for $f$ univalent, Proc. Amer. Math. Soc. 53 (1975), no. 2, 385–390. MR 382615, DOI 10.1090/S0002-9939-1975-0382615-8

Applications of convolution operator techniques to problems in univalent function theory

S. D. Bernardi, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 24 (1970), 312–318. MR 251202, DOI 10.1090/S0002-9939-1970-0251202-X
James A. Jenkins, Univalent functions and conformal mapping, Reihe: Moderne Funktionentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR 0096806
J. Krzyż, The radius of close-to-convexivity within the family of univalent functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 201–204. MR 148887
R. J. Libera and A. E. Livingston, On the univalence of some classes of regular functions, Proc. Amer. Math. Soc. 30 (1971), 327–336. MR 288244, DOI 10.1090/S0002-9939-1971-0288244-5
A. E. Livingston, On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 17 (1966), 352–357. MR 188423, DOI 10.1090/S0002-9939-1966-0188423-X
K. S. Padmanabhan, On the radius of univalence of certain classes of analytic functions, J. London Math. Soc. (2) 1 (1969), 225–231. MR 247062, DOI 10.1112/jlms/s2-1.1.225
Christian Pommerenke, Univalent functions, Studia Mathematica/Mathematische Lehrbücher, Band XXV, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen. MR 0507768
Raphael M. Robinson, Univalent majorants, Trans. Amer. Math. Soc. 61 (1947), 1–35. MR 19114, DOI 10.1090/S0002-9947-1947-0019114-6
St. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119–135. MR 328051, DOI 10.1007/BF02566116