An extremal decomposition problem for harmonic measure

Let $E$ be a continuum in the closed unit disk $|z|\le 1$ of the complex $z$-plane which divides the open disk $|z| < 1$ into $n\ge 2$ pairwise non-intersecting simply connected domains $D_k,$ such that each of the domains $D_k$ contains some point $a_k$ on a prescribed circle $|z| = \rho, 0 <\rho <1, k=1,...,n\,.$ It is shown that for some increasing function $\Psi\,$ independent of $E$ and the choice of the points $a_k,$ the mean value of the harmonic measures \Psi^{-1}\[ \frac{1}{n} \sum_{k=1}^{k} \Psi(\omega(a_k,E, D_k))] is greater than or equal to the harmonic measure $\omega(\rho, E^*, D^*)\,,$ where $E^* = \{z: z^n \in [-1,0] \}$ and $D^* =\{z: |z|<1, |{\rm arg} z| < \pi/n\} \,.$ This implies, for instance, a solution to a problem of R.W. Barnard, L. Cole, and A. Yu. Solynin concerning a lower estimate of the quantity $\inf_{E} \max_{k=1,...,n} \omega(a_k,E, D_k)\,$ for arbitrary points of the circle $|z| = \rho \,.$ These authors stated this hypothesis in the particular case when the points are equally distributed on the circle $|z| = \rho \,.$Comment: 6 pages, 2 figure

Region of variability for certain classes of univalent functions satisfying differential inequalities

In this paper we determine the region of variability for certain subclasses of univalent functions satisfying differential inequalities. In the final section we graphically illustrate the region of variability for several sets of parameters.Comment: 24 pages, 5 figure

On John domains in Banach spaces

We study the stability of John domains in Banach spaces under removal of a countable set of points. In particular, we prove that the class of John domains is stable in the sense that removing a certain type of closed countable set from the domain yields a new domain which also is a John domain. We apply this result to prove the stability of the inner uniform domains. Finally, we consider a wider class of domains, so called $\psi$-John domains and prove a similar result for this class.Comment: 22page

The minimal surfaces over the slanted half-planes, vertical strips and single slit

In this paper, we discuss the minimal surfaces over the slanted half-planes, vertical strips, and single slit whose slit lies on the negative real axis. The representation of these minimal surfaces and the corresponding harmonic mappings are obtained explicitly. Finally, we illustrate the harmonic mappings of each of these cases together with their minimal surfaces pictorially with the help of mathematica.Comment: 18 pages (including 26 figures), with a journa

Freely quasiconformal maps and distance ratio metric

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least $2$ and that $D\subset E$ and $D'\subset E'$ are domains. In this paper, we establish, in terms of the $j_D$ metric, a necessary and sufficient condition for the homeomorphism $f: E \to E'$ to be FQC. Moreover, we give, in terms of the $j_D$ metric, a sufficient condition for the homeomorphism $f: D\to D'$ to be FQC. On the other hand, we show that this condition is not necessary.Comment: 10 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1110.626

Topics in Special Functions

The authors survey recent results in special functions, particularly the gamma function and the Gaussian hypergeometric function.Comment: 22 page

Teichm\"uller's problem in space

Quasiconformal homeomorphisms of the whole space Rn, onto itself normalized at one or two points are studied. In particular, the stability theory, the case when the maximal dilatation tends to 1, is in the focus. Our main result provides a spatial analogue of a classical result due to Teichm\"uller. Unlike Teichm\"uller's result, our bounds are explicit. Explicit bounds are based on two sharp well-known distortion results: the quasiconformal Schwarz lemma and the bound for linear dilatation. Moreover, Bernoulli type inequalities and asymptotically sharp bounds for special functions involving complete elliptic integrals are applied to simplify the computations. Finally, we discuss the behavior of the quasihyperbolic metric under quasiconformal maps and prove a sharp result for quasiconformal maps of R^n \ {0} onto itself.Comment: 25 pages, 2 figure

On quasiplanes in Euclidean spaces

A variational inequality for the images of $k$-dimensional hyperplanes under quasiconformal maps of the $n$-dimensional Euclidean space is proved when $1\le k\le n-2 .$Comment: 12 page

Ahlfors theorems for differential forms

Some counterparts of theorems of Phragm\'en-Lindel\"of and of Ahlfors are proved for differential forms of ${\cal WT}$--classes.

Hypergeometric functions and hyperbolic metric

We obtain new inequalities for certain hypergeometric functions. Using these inequalities, we deduce estimates for the hyperbolic metric and the induced distance function on a certain canonical hyperbolic plane domain.Comment: 13 pages, 1 figur