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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A variational method for functions of bounded boundary rotation


Author: H. B. Coonce
Journal: Trans. Amer. Math. Soc. 157 (1971), 39-51
MSC: Primary 30.43
MathSciNet review: 0274737
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Abstract: Let $ f$ be a function analytic in the unit disc, properly normalized, with bounded boundary rotation. There exists a Stieltjes integral representation for $ 1 + zf''(z)/f'(z)$. From this representation, and in view of a known variational formula for functions of positive real part, a variational formula is derived for functions of the form $ q(z) = 1 + zf''(z)/f'(z)$. This formula is for functions of arbitrary boundary rotation and does not assume the functions to be univalent.

A new proof for the radius of convexity for functions of bounded boundary rotation is given. The extremal function for Re$ \{ F(f'(z))\} $ is derived. Examples of univalent functions with arbitrary boundary rotation are given and estimates for the radius in which Re$ \{ f'(z)\} > 0$ are computed.

The coefficient problem is solved for $ {a_4}$ for all values of the boundary rotation and without the assumption of univalency.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0274737-8
PII: S 0002-9947(1971)0274737-8
Keywords: Bounded boundary rotation, coefficient problems, extremal problems, radius of convexity, univalent functions, variational methods
Article copyright: © Copyright 1971 American Mathematical Society