45607022
Faber polynomials in the theory of univalent functions
Introduction. The Faber polynomials play an important role in the theory of univalent functions. Grunsky [ l] succeeded in establishing a set of conditions for a given function which are necessary and in their totality sufficient for the uni valency of this function, and in these conditions the coefficients of the Faber polynomials play an important role. Schiffer [2] gave a differential equation for univalent functions solving certain extremum problems with respect to the coefficients of such functions ; in this differential equation appears again a polynomial which is just the derivative of a Faber polynomial (cf. Schiffer [3]; see also Schaeffer-Spencer [4]). It seems, therefore, of interest to study these Faber polynomials more closely, in particular their dependence on the given function with respect to which they are defined, their variation with the latter and certain characteristic inequalities for them and their coefficients. This investigation is carried out in the present paper. In §1 we establish a generating function for all Faber polynomials with respect to a given function. In §2 we establish variation formulas for the Faber polynomials and their coefficients. In §3 we solve certain extremum problems with respect to the coefficients of these polynomials and find again all the inequalities which have been established by Grunsky. In §4 we use our method in order to generalize our results and to find inequalities for the Faber polynomials themselves.
118096935
Stefan Bergman (1895--1977) in Memoriam
115850274
Fredholm Eigenvalues and Conformal Mapping
This paper is mainly of expository nature. Its aim is to give a connected account of various results regarding the Fredholm eigenfunctions and the Fredholm eigenvalues of plane domains. The Fredholm eigenvalues of a curve system are a set of functionals of these curves whose study leads to may useful applications. They are closely related to the boundary value problem for harmonic functions. They are important in the theory of conformal mapping, in the theory of kernel functions and orthonormal series. They play a role in the theory of the Hilbert transform and in the theory of univalent functions. Their dependence on the curve system is displayed by a very elegant and convenient variational formula. Some applications of these results are given to show the significance of the identities and formulas obtained. For a more detailed development of many points the reader is referred to [10], [11] and [12].
126639698
Potential-theoretic methods in the theory of functions of two complex variables
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45663311
Some Linear Operators in the Theory of Partial Differential Equations.
124400876
THE COEFFICIENT PROBLEM FOR MULTIPLY-CONNECTED DOMAINS
There is sometimes considerable advantage to be gained from generalizing a problem which has been intensively studied in particular cases. At the very least, relations may be brought to light which were disregarded or covered up in the particular cases studied. These relations may not only turn out to be important in themselves, but they may also throw light on the original problems. With these thoughts in mind, we have generalized the coefficient problem for schlicht functions, a problem which has heretofore been considered mainly in the case when the function is defined over a simply-connected domain. The word "schlicht" has little meaning unless the surface is specified on which the values are carried. For example, a function may map the interior of the unit circle onto a subdomain of a Riemann surface and the subdomain may be schlicht relative to the Riemann surface but not relative to the plane. In the case of functions which are schlicht in a simply-connected domain, the meaning is that the regions of values form schlicht subdomains of the sphere. A certain symmetry is inherent in this definition because the region of definition of the function and the surface on which the values lie (namely the sphere) are about equal from the function-theory point of view. As soon as we go over to more complicated domains of definition of the function, the importance of the surface which carries the values of the function becomes evident. In this paper we have attempted to restore the symmetry by supposing that the values of the function on the multiply-connected domain lie schlicht on another multiply-connected domain. But we could go much further and choose either the domain of definition or the carrying surface or both to be domains of higher structure. By choosing the carrying surface to be of higher structure, we are brought close to the theory of subordination in the sense of Lindelof. Problems relating to domains of higher connectivity cannot be approached without first discussing the invariants of the domains including differentials. In the case of the sphere the associated invariants are of rational character, but in the case of a domain of higher structure they are transcendental. As was pointed out by Schottky [11], and later by Picard [7] and Klein [3],1 the invariants of a multiply-connected domain are best understood in terms of the so-called "double" of that domain. The double is a closed Riemann surface of genus m 1 if the domain itself is of genus zero and has m boundary components. It is the symmetry of the double which lies behind many of the relations connecting the invariants of a multiply-connected domain.
121602047
Naked singularities cannot be observed (An information-theoretic approach to the cosmic censorship conjecture)
In this article we review some recent results obtained in the interplay between field theory in curved space-time and information theory, and explore some consequences. We shall explain how the Weyl tensor becomes responsible for the creation of quantum noise which, in turn, degrades a considerable fraction of the information which is optically transferred in a curved space-time. We conclude that causal connection between spacetime regions, although being a necessary condition for transmitting information between them, is by no means sufficient. This is because this quantum noise might preclude the observation of very distant objects in the Universe and even closer ones if it becomes very intense. We argue that naked singularities are very powerful sources of noise and, therefore, cannot be observed optically. This leads to the intriguing possibility that the ‘no-hair’ theorem might be more general than was thought before.