## A note on a problem of Robinson

HTML articles powered by AMS MathViewer

- by Kent Pearce PDF
- Proc. Amer. Math. Soc.
**89**(1983), 623-627 Request permission

## Abstract:

Let $\mathcal {S}$ be the usual class of univalent analytic functions on $\left | z \right | < 1$ normalized by $f(0) = 0$ and $f’(0) = 1$. Let $\mathfrak {L}$ be the linear operator on $\mathcal {S}$ given by $\mathfrak {L}f = \tfrac {1}{2}(zf)’$ and let ${r_{{\mathcal {S}_t}}}$ be the minimum radius of starlikeness of $\mathfrak {L}f$ for $f$ in $\mathcal {S}$. In 1947 R. M. Robinson initiated the study of properties of $\mathfrak {L}$ acting on $\mathcal {S}$ when he showed that ${r_{{\mathcal {S}_t}}} > .38$. Later, in 1975, R. W. Barnard gave an example which showed ${r_{{\mathcal {S}_t}}} < .445$. It is shown here, using a distortion theorem and Jenkin’s region of variability for $zf’(z)/f(z)$, $f$ in $\mathcal {S}$, that ${r_{{\mathcal {S}_t}}} > .435$. Also, a simple example, a close-to-convex half-line mapping, is given which again shows ${r_{{\mathcal {S}_t}}} < .445$.## References

- Hassoon S. Al-Amiri,
*On the radius of univalence of certain analytic functions*, Colloq. Math.**28**(1973), 133–139. MR**328041**, DOI 10.4064/cm-28-1-133-139 - Hassoon S. Al-Amiri,
*On the radius of starlikeness of certain analytic functions*, Proc. Amer. Math. Soc.**42**(1974), 466–474. MR**330431**, DOI 10.1090/S0002-9939-1974-0330431-4 - P. L. Bajpai and Prem Singh,
*The radius of starlikeness of certain analytic functions*, Proc. Amer. Math. Soc.**44**(1974), 395–402. MR**340578**, DOI 10.1090/S0002-9939-1974-0340578-4 - 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
—, - Roger W. Barnard and Charles Kellogg,
*Applications of convolution operators to problems in univalent function theory*, Michigan Math. J.**27**(1980), no. 1, 81–94. MR**555840** - Susheel Chandra and Prem Singh,
*Certain subclasses of the class of functions regular and univalent in the unit disc*, Arch. Math. (Basel)**26**(1975), 60–63. MR**367171**, DOI 10.1007/BF01229704 - James A. Jenkins,
*Univalent functions and conformal mapping*, Reihe: Moderne Funktionentheorie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958. MR**0096806** - 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 - Raphael M. Robinson,
*Univalent majorants*, Trans. Amer. Math. Soc.**61**(1947), 1–35. MR**19114**, DOI 10.1090/S0002-9947-1947-0019114-6

*On Robinson’s*$\tfrac {1}{2}$

*conjecture*, Proc. Amer. Math. Soc.

**72**(1978), 135-139.

## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**89**(1983), 623-627 - MSC: Primary 30C45
- DOI: https://doi.org/10.1090/S0002-9939-1983-0718985-2
- MathSciNet review: 718985