A variational method for functions of bounded boundary rotation

Coonce, H. B.

doi:10.1090/S0002-9947-1971-0274737-8

A variational method for functions of bounded boundary rotation
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by H. B. Coonce PDF

Trans. Amer. Math. Soc. 157 (1971), 39-51 Request permission

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 $\text {Re} \{ F(f’(z))\}$ is derived. Examples of univalent functions with arbitrary boundary rotation are given and estimates for the radius in which $\text {Re} \{ f’(z)\} > 0$ 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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