Peter Duren

Extension of a theorem of Carleson

Carlesons proof of this theorem involves a difficult covering argument and the consideration of a certain quadratic form (see also [ l ] ) . L. Hörmander later found a proof which appeals to the Marcinkiewicz interpolation theorem and avoids any discussion of quadratic forms. The main difficulty in this approach is to show that a certain sublinear operator is of weak type (1, 1). Here a covering argument reappears which is similar to Carlesons but apparently easier (see [4]). We wish to point out that Hörmanders argument, with appropriate modifications, actually proves the theorem in the following extended form.

A variational method for functions schlicht in an annulus

Abstract : Consideration is given to the family of function f(z) regular analytic and schlicht in the annulus R and satisfying the following conditions: (1) f(z) maps R onto the unit disk minus some continuum G, and (2) G contains the origin. Extremal problems (maximum modulus on the inner boundary and maximum displacement on the outer boundary of R) are solved using a specific method of variation within the family. The variation leads from every given f(z) to a large set of comparison functions within the family. The use of the variational method is further illustrated in finding the maximum diameter of the continuum G for all functions in the family F.

Zeros of the hypergeometric polynomials F(−n, b; 2b; z)

Abstract Our purpose is to study the zeros of hypergeometric polynomials, especially those of F(−n,b; 2b; z), where b > − 1 2 . Although some properties are implicit in known connections with classical orthogonal polynomials, whose zeros are well understood, the implications for zeros of hypergeometric polynomials appear not to have been generally recognized. The results for polynomials F(−n,b; 2b; z) are applied to give information about the zeros of other systems of hypergeometric functions, including Jacobi polynomials and Legendre functions. In a subsequent paper [5], we will discuss the behaviour of the zeros as b descends below the critical value − 1 2 .

Curvature Properties of Planar Harmonic Mappings

A geometric interpretation of the Schwarzian of a harmonic mapping is given in terms of geodesic curvature on the associated minimal surface, generalizing a classical formula for analytic functions. A formula for curvature of image arcs under harmonic mappings is applied to derive a known result on concavity of the boundary. It is also used to characterize fully convex mappings, which are related to fully starlike mappings through a harmonic analogue of Alexander’s theorem.

Univalence criteria for lifts of harmonic mappings to minimal surfaces.

A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass-Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian defined for conformai metrics and on a Schwarzian introduced by Ahlfors for curves. Convexity plays a central role.

Coefficients of univalent functions

The interplay of geometry and analysis is perhaps the most fascinating aspect of complex function theory. The theory of univalent functions is concerned primarily with such relations between analytic structure and geometric behavior. A function is said to be univalent (or schlichi) if it never takes the same value twice: f(z{) # f(z2) if zx #= z2. The present survey will focus upon the class S of functions

Robin functions and distortion of capacity under conformal mapping

Under normalized conformal mappings of a multiply connected domain ω it is found that the sharp lower bound for the capacity of the image of a given set is the Robin capacity of A with respect to ω. The result is generalized to a sharp inequality for quadratic forms involving Greens function and the Robin function associated with A and ω. AMS No. 30C85, 30C35, 30C70, 31A15

THE THEORY OF THE SECOND VARIATION IN EXTREMUM PROBLEMS FOR UNIVALENT FUNCTIONS

Abstract : An attempt is made to improve the method of variations by considering further necessary conditions for the extremum function which arises from a study of the second variation. The formulas for the second variation of coefficients of univalent functions are, in general, so involved as to be impractical for a finer study of the extremum function. However, using the fact that this function satisfies a differential equation due to the first variational condition, success was reached in simplifying the expressions considerably. It was shown that a whole new set of necessary extremum conditions can be obtained to test every competing solution of the first variational condition. The new extremum conditions have the form of quadratic inequalities which are similar in type to those occurring through the method of contour integration. The characteristic difference lies in the fact that the quadratic inequalities have only to hold in the case of the extremum function, while in the other case the inequalities are asserted for all univalent functions. Nevertheless, it seems that the theory of the second variation is more closely connected with the method of contour integration than is that of the first variation. It might be possible to combine both methods for a unified approach to the general coefficient problem. (Author)

Linear extremal problems for harmonic mappings of the disk

Sharp bounds for Fourier coefficients and distortion are established for harmonic mappings of the unit disk onto itself.

A variational method for harmonic mappings onto convex regions

Harmonic mappings arise in several areas of analysis and geometry. Our purpose is to present a general variational method for treating extremal problems over families of univalent harmonic functions which map the unit disk onto a given convex region. In the special case where the target region is again the unit disk, the method leads to sharp estimates for coefficients and other functionals.