On the radius of starlikeness of $(zf)^{β}$ for $f$ univalent
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- by Roger W. Barnard PDF
- Proc. Amer. Math. Soc. 53 (1975), 385-390 Request permission
Abstract:
Let $S$ be the standard class of normalized univalent functions. For a given function $f,\;f(z) = z + {a_2}{z^2} + \ldots$, regular for $|z| < 1$, let $r(f)$ be the radius of starlikeness of $f$. In 1947, R. M. Robinson considered the combination ${g_f}(z) = (zf)β/2$ for $f\epsilon S$. He found a lower bound of .38 for $r({g_f})$ for all $f\epsilon S$. He noted that the standard Koebe function $k,\;k(z) = z{(1 - z)^2}$, has its $r({g_k})$ equal to $1/2$. A question that has been asked since Robinsonβs paper is whether $1/2$ is the minimum $r({g_f})$ for all $f$ in $S$. It is shown here that this is not the case by giving examples of functions $f$ whose $r({g_{f}})$ is less than $1/2$.References
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Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 53 (1975), 385-390
- MSC: Primary 30A32
- DOI: https://doi.org/10.1090/S0002-9939-1975-0382615-8
- MathSciNet review: 0382615