Extremal problems in the class of close-to-convex functions

Pinchuk, Bernard

doi:10.1090/S0002-9947-1967-0217279-7

Extremal problems in the class of close-to-convex functions
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by Bernard Pinchuk

Trans. Amer. Math. Soc. 129 (1967), 466-478

DOI: https://doi.org/10.1090/S0002-9947-1967-0217279-7

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Abstract:

The class K of normalized close-to-convex functions in $D = \{ z:|z| < 1\}$ has a parametric representation involving two Stieltjes integrals. Using a variational method due to G. M. Goluzin [2] for classes of analytic functions defined by a Stieltjes integral, variational formulas are developed for K. With these variational formulas, two general extremal problems within K are solved. The first problem is to maximize the functional $\operatorname {Re} F[\log f’(z)]$ over K where $F(w)$ is a given entire function and z a given point in D. A special case of this is the rotation theorem for K. The second problem solved is a general coefficient problem. Both problems are solved by characterizing the measures which appear in the integral representation for the extremal functions. The classes of convex univalent functions in D and functions whose derivative has a positive real part in D are proper subclasses of K. The methods used to solve the extremal problems in K can be used for these subclasses as well. Some of the results for the subclasses are known and are not presented here, even though the methods differ from those used previously. It should be mentioned that Goluzin originally used these methods to solve extremal problems of the first type mentioned above within the classes of starlike and typically real functions.

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