A method for investigating geometric properties of support points and applications

Brown, Johnny E.

doi:10.1090/S0002-9947-1985-0766220-8

A method for investigating geometric properties of support points and applications
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Trans. Amer. Math. Soc. 287 (1985), 285-291 Request permission

Abstract:

A normalized univalent function $f$ is a support point of $S$ if there exists a continuous linear functional $L$ (which is nonconstant on $S$) for which $f$ maximizes $\operatorname {Re} L(g),g \in S$. For such functions it is known that $\Gamma = {\text {C}} - f(U)$ is a single analytic arc that is part of a trajectory of a certain quadratic differential $Q(w)\;d{w^2}$. A method is developed which is used to study geometric properties of support points. This method depends on consideration of $\operatorname {Im} \{ {w^2}Q(w)\}$ rather than the usual $\operatorname {Re} \{ {w^2}Q(w)\}$. Qualitative, as well as quantitative, applications are obtained. Results related to the Bieberbach conjecture when the extremal functions have initial real coefficients are also obtained.

References

Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln

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