Results on bi-univalent functions
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- by D. Styer and D. J. Wright PDF
- Proc. Amer. Math. Soc. 82 (1981), 243-248 Request permission
Abstract:
When the class $\sigma$ of bi-univalent functions was first defined, it was known that functions of the form $\phi \circ {\psi ^{ - 1}} \in \sigma$ when $\phi$ and $\psi$ are univalent, map the unit disc ${\mathbf {B}}$ onto a set containing ${\mathbf {B}}$, and satisfy $\phi (0) = \psi (0) = 0$, $\phi ’(0) = \psi ’(0)$. It is shown here that such functions form a proper subset of $\sigma$, and that $\sigma$ is a proper subset of the set of functions of the form $\phi \circ {\psi ^{ - 1}}$, where $\phi$ and $\psi$ are locally univalent, at most $2$-valent, each maps a subregion of ${\mathbf {B}}$ univalently onto ${\mathbf {B}}$, and $\phi (0) = \psi (0) = 0$, $\phi ’(0) = \psi ’(0)$, ${\psi ^{ - 1}}(0) = 0$. It is also shown that there are $f(z) = z + {a_2}{z^2} + \cdots$ in $\sigma$ with $\left | {{a_2}} \right | > 4/3$. However, doubt is cast that $\left | {{a_2}} \right |$ can be as large as $3/2$.References
- D. Bshouty, W. Hengartner, and G. Schober, Estimates for the Koebe constant and the second coefficient for some classes of univalent functions, Canadian J. Math. 32 (1980), no. 6, 1311–1324. MR 604686, DOI 10.4153/CJM-1980-101-9
- M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63–68. MR 206255, DOI 10.1090/S0002-9939-1967-0206255-1
- E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in $z<\,1$, Arch. Rational Mech. Anal. 32 (1969), 100–112. MR 235110, DOI 10.1007/BF00247676
- T. J. Suffridge, A coefficient problem for a class of univalent functions, Michigan Math. J. 16 (1969), 33–42. MR 240297
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 82 (1981), 243-248
- MSC: Primary 30C45; Secondary 30C75
- DOI: https://doi.org/10.1090/S0002-9939-1981-0609659-5
- MathSciNet review: 609659