Albert Baernstein

On the harmonic measure of slit domains

Let be distinct real numbers mod 2π and K be a closed subset of [0,1]. Define Ω to be the unit disk Δ from which the slits e i,a K. j = 1,…n have been removed, and let u be the harmonic measure of ∂Δ relative to Ω. Let v be the corresponding harmonic measure when the a j. are changed to In case Ω is simply connected Dubinin has proved that v(0) u(0). It is not known whether this holds in general when Ω is multiply connected. In this paper, we prove it is true when n n  3. In fact, using ∗-function arguments, we obtain a much stronger theorem involving integral means.

Coefficients of univalent functions with restricted maximum modulus

Suppose that is univalent in Classical results due essentially to Littlewood and Paley assert that an inequality implies provided a α > ½ In this paper we show that this implication remains true for α ≧.497. In particular, this confirms Szegos conjecture that coefficients of fourfold symmetric functions satisfy . Tools include Haymans theorem which asserts that a univalent function cannot be too big at too many different places, and a localized version of an inequality of Clunie and Pommerenke which those authors had used to prove an=(n−.503) for bounded univalent f

Some conjectures aboutL p norms of k-plane transforms

For 1≤k≤n-1, the k-plane transformTk,n carries functionsf defined onRn to functionsTk,nf defined on the set of affine k-planes inRn. It is known thatTk,n mapsLp intoLq for certain values ofp andq. In this article we formulate conjectures for the exact values of the norm of theTk,n, and state also a conjecture asserting that theLq norm ofTk,nf changes monotonically whenf is replaced by its symmetric decreasing rearrangement.

Estimates for inverse coefficients of univalent functions from integral means

The purpose of this note is to point out that sharp coefficient bounds for the inverses of univalent functions from certain families are fairly direct corollaries of results on integral means. As an example, in §1 the method is applied to the familiar schlicht classS. The resulting coefficient estimates for the inverses of functions inS were first obtained by K. Löwner. Following this prototype, in §2 we obtain corresponding results, which are new, for a classS(p) of meromorphic schlicht functions in |z|<1.

An extremal property of meromorphic functions with n–fold symmetry

Let Ω be a simply connected domain on the Riemann sphere, and F: D → Ω be a conformal mapping of the unit disk onto Ω. We prove integral mean inequalities of the form for certain functions f where F n(z)= F(z n)and u is subharmonic on the range of f. These reverse the usual subordination inequalities. We also prove results of this type when f and F n are polynomials, with the zeros of F nuniformly spaced on a circle. These results may bear on some well known open extremal problems for which the extremals are expected to have n–fold symmetry.

Convolution and rearrangement on circle

Let f g h be real functions on the unit circle, and let f 0 g 0 h 0 denote their symmetric decreasing rearrangements. We prove, under appropriate integrability assumptions, the inequality .