Bernard Pinchuk

A variational method for functions of bounded boundary rotation

Introduction. In this paper we present a variational method for functions of bounded boundary rotation and solve certain general extremal problems for these functions. The variational method is based upon a general method of G. M. Goluzin [1], and has previously been used by the present author to solve extremal problems for several classes of univalent functions ([7] and [8]). Once the variation formulas are derived, the details of solving the extremal problems are very similar to those in [7] and [8]. Theorem 1 here has been obtained independently by Professor J. Pfaltzgraff, who used essentially the same methods. I wish to thank Professor Pfaltzgraff for providing me with his unpublished work. Those functions of bounded boundary rotation which are also univalent form a subclass which has recently been studied by M. M. Schiffer and 0. Tammi [9]. They use a different (though closely related) variational method. We shall compare their results with ours.

Harmonic measure on an annulus

Gaiers problem is to find a continuum in the closed unit disc which contains two given points a and b so as to minimize ω(0) where ω(z) is the harmonic measure of Ω in the unit disc. A variational method is used m derive the differential equation for the extremal case of this problem. A special transformation simplifies the analysis of the problem in the symmetric case

The Hardy class of functions of bounded boundary rotation

The Hardy class for functions of bounded boundary rotation and their derivatives is determined. In the univalent case, a more precise description of the Hardy class in terms of the behavior of the representing measure is obtained.

Constrained extremal problems for classes of meromorphic functions

where m(t) is a nondecreasing function on 0^/^27r, Jl dm(t) = 1 and g(z, t) is given for the class. We shall assume that g(z, t) and gt(z, t) are analytic functions of z in D and Lipschitz continuous with respect to t (uniformly for z in a compact subset of D). G. M. Goluzin has developed a variational method for §>(g) [5] which has proved to be useful in the study of extremal problems within various classes of analytic functions [4], [5], [9], [lO]. In this note we present a generalization of the Goluzin variational method. The generalization enables one to preserve side conditions imposed on the functions m{t) in (1). Our work was motivated by the fact that various classes of meromorphic functions have structural formulas based on (1) where m(t) must satisfy the additional condition /%* e-dm{t) = 0. Complete proofs and applications of our results will be published elsewhere [8].

Extremal problems for functions of bounded boundary rotation

9. R. Palais, Ljusternik-Schnirelmann theory on Banach manifolds, Topology 15 (1966), 115-132. 10. F. Rellich, Störungstheorie der Spektralzerlegung. I, Math. Annalen 113 (1936), 600-619. 11. E. Schmidt, Zur Theorie der linear en und nichtlinearen Integralgleichungen. I l l , Teil, Math. Ann. 65 (1910), 370-399. 12. J. Schwartz, Generalizing the Ljusternik-Schnirelrnann theory of critical points, Comm. Pure Appl. Math. 27 (1964), 307-315. 13. C. L. Siegel, Vorlesungen Uber Himmelsmechanik, Springer-Verlag, Berlin, 1965.

Integral means of analytic functions

Sharp bounds for general integral means of analytic functions in the unit disc are determined. These bounds depend only on the moduli of the points on the boundary of the image domain nearest to and farthest from the origin. The proof is shown to be a simple application of a deep theorem of A. Beurling in potential theory.