Results on bi-univalent functions

Authors:
D. Styer and D. J. Wright

Journal:
Proc. Amer. Math. Soc. **82** (1981), 243-248

MSC:
Primary 30C45; Secondary 30C75

MathSciNet review:
609659

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Abstract: When the class of bi-univalent functions was first defined, it was known that functions of the form when and are univalent, map the unit disc onto a set containing , and satisfy , . It is shown here that such functions form a proper subset of , and that is a proper subset of the set of functions of the form , where and are locally univalent, at most -valent, each maps a subregion of univalently onto , and , , . It is also shown that there are in with . However, doubt is cast that can be as large as .

**[1]**D. Bshouty, W. Hengartner, and G. Schober,*Estimates for the Koebe constant and the second coefficient for some classes of univalent functions*, Canad. J. Math.**32**(1980), no. 6, 1311–1324. MR**604686**, 10.4153/CJM-1980-101-9**[2]**M. Lewin,*On a coefficient problem for bi-univalent functions*, Proc. Amer. Math. Soc.**18**(1967), 63–68. MR**0206255**, 10.1090/S0002-9939-1967-0206255-1**[3]**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**0235110****[4]**T. J. Suffridge,*A coefficient problem for a class of univalent functions*, Michigan Math. J.**16**(1969), 33–42. MR**0240297**

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DOI:
https://doi.org/10.1090/S0002-9939-1981-0609659-5

Article copyright:
© Copyright 1981
American Mathematical Society