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Estimation of the curvature of level curves in the class of all functions regular and univalent in the circle. (Russian) Zbl 0585.30020

Let S denote the usual class of analytic, univalent functions \(g(z)=z+a_ 2z^ 2+...\) in the unit disk. It is known that the level curve \(g(| z| =r)\) has curvature \[ K_ r=| z_ 0g'(z_ 0)|^{-1} Re[1+z_ 0g''(z_ 0)/g'(z_ 0)] \] at the point \(w=g(z_ 0)\), where \(| z_ 0| =r\). In an earlier paper, the author found the value of \(\sup_{g\in S_ r}K_ r\) for \(z_ 0=x_ 0\) \((0<x_ 0<1)\), where \(S_ r\) consists of the functions in S with real coefficients [Mat. Zametki 19, 381-388 (1976; Zbl 0334.30010)].
In this paper, the author improves the previously known upper bound of \(K_ r\) for functions in S. In particular, it is shown that for \(0<r\leq 0.279624\), \[ K_ r\leq (1/r)\frac{(1+x)^ 4}{(1+x^ 2)(1-x)^ 2}\exp (\frac{4x}{(1-x^ 2)}l\quad n(r/x)), \] where x \((0<x<r)\) is the root of a certain transcendental equation, and for \(0.279624<r<1\), \[ K_ r\leq \frac{1+r^ 2}{r}(\frac{1+x}{1-x})^ 4 \exp (\frac{4x}{1-x^ 2}l\quad n(r,x)), \] where x \((0<x<r)\) is again the root of a certain equation. The proof uses the logarithmic coefficients of functions in S.
Reviewer: Renate McLaughlin

MSC:

30C55 General theory of univalent and multivalent functions of one complex variable
30C75 Extremal problems for conformal and quasiconformal mappings, other methods

Citations:

Zbl 0334.30010
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