Alexander Yu. Solynin

A minimal area problem in conformal mapping

LetS denote the usual class of functionsf holomorphic and univalent in the unit diskU such thatf(0)=f′(0)−1=0. The main result of the paper is that area (f(U) ≥27π/7)(2-α)−2 for allf∈S such that |f″(0)|=2α, 1/2<α<2. This solves a long-standing extremal problem for the class of functions considered.

Quadratic Differentials and Weighted Graphs on Compact Surfaces

We prove that for every simply connected graph Γ embedded in a compact surface R of genus g≥o, whose edges e i kj carry positive weights w i kj , there exist a complex structure on R and a Jenkins-Strebel quadratic differential Q(z) dz 2 , whose critical graph ΦQ complemented, if necessary, by second degree vertices on its edges, is homeomorphic to Γ on R and carries the same set of weights. In other words, every positive simply connected graph on R can be analytically embedded in R. We also discuss a problem on the extremal partition of R relative to such analytical embedding. As a consequence, we establish the existence of systems of disjoint simply connected domains on R with a prescribed combinatorics of their boundaries, which carry proportional harmonic measures on their boundary arcs.

A SCHWARZ LEMMA FOR MEROMORPHIC FUNCTIONS AND ESTIMATES FOR THE HYPERBOLIC METRIC

We prove a generalization of the Schwarz lemma for meromorphic functions f mapping the unit disk D onto Riemann surfaces R with bounded in mean radial distances from f(0) to the boundary of R. A new variant of the Schwarz lemma is also proved for the Caratheodory class of analytic functions having positive real part in D. Our results lead to several improved estimates for the hyperbolic metric.

Hyperbolic convexity and the analytic fixed point function

We answer a question raised by D. Mejia and Ch. Pommerenke by showing that the analytic fixed point function is hyperbolically convex in the unit disc.

Concentration of area in half-planes

For the standard class S of normalized univalent functions f analytic in the unit disk U, we consider a problem on the minimal area of the image f(U) concentrated in any given half-plane. This question is related to a well-known problem posed by A. W. Goodman in 1949 that regards minimizing area covered by analytic univalent functions under certain geometric constraints. An interesting aspect of this problem is the unexpected behavior of the candidates for extremal functions constructed via geometric considerations.

ON THE DISTRIBUTION OF HARMONIC MEASURE ON SIMPLY CONNECTED PLANAR DOMAINS

For a simply connected planar domain D with 0 2 D and dist.0;@D/ D 1, let h D.r/ be the harmonic measure of@ D \f jz jrg evaluated at 0. The function h D.r/ is the distribution of harmonic measure. It has been studied by B. L. Walden and L. A. Ward. We continue their study and answer some questions raised by them by constructing domains with pre-specified distribution.

Root-counting Measures of Jacobi Polynomials and Topological Types and Critical Geodesics of Related Quadratic Differentials

Two main topics of this paper are asymptotic distributions of zeros of Jacobi polynomials and topology of critical trajectories of related quadratic differentials. First, we will discuss recent developments and some new results concerning the limit of the root-counting measures of these polynomials. In particular, we will show that the support of the limit measure sits on the critical trajectories of a quadratic differential of the form \( Q(z) \, dz^{2} = \frac{az^{2}+bz+c} {(z^{2}-1)^{2}} \, dz^{2} \) Then we will give a complete classification, in terms of complex parameters a, b, and c, of possible topological types of critical geodesics for the quadratic differential of this type.

The Poincaré metric and isoperimetric inequalities for hyperbolic polygons

We prove several isoperimetric inequalities for the conformal radius (or equivalently for the Poincare density) of polygons on the hyperbolic plane. Our results include, as limit cases, the isoperimetric inequality for the conformal radius of Euclidean n-gons conjectured by G. Polya and G. Szego in 1951 and a similar inequality for the hyperbolic n-gons of the maximal hyperbolic area conjectured by J. Hersch. Both conjectures have been proved in previous papers by the third author. Our approach uses the method based on a special triangulation of polygons and weighted inequalities for the reduced modules of trilaterals developed by A. Yu. Solynin. We also employ the dissymmetrization transformation of V. N. Dubinin. As an important part of our proofs, we obtain monotonicity and convexity results for special combinations of the Euler gamma functions, which appear to have a significant interest in their own right.

A note on equilibrium points of Green’s function

We answer a question raised by Ahmet Sebbar and Therese Falliero (2007) by showing that for every finitely connected planar domain Ω there exists a compact subset K C Ω, independent of w, containing all critical points of Greens function G(z,w) of Ω with pole at w ∈ Ω.

Overdetermined Boundary Value Problems, Quadrature Domains and Applications

We discuss an overdetermined problem in planar multiply connected domains Ω. This problem is solvable in Ω if and only if Ω is a quadrature domain carrying a solid-contour quadrature identity for analytic functions. At the same time the existence of such quadrature identity is equivalent to the solvability of a special boundary value problem for analytic functions. We give a complete solution of the problem in some special cases and discuss some applications concerning the shape of electrified droplets and small air bubbles in a fluid flow.