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

On removability properties of ψ\psi-uniform domains in Banach spaces

Suppose that $E$ denotes a real Banach space with the dimension at least 2. The main aim of this paper is to show that a domain $D$ in $E$ is a $\psi$-uniform domain if and only if $D\backslash P$ is a $\psi_1$-uniform domain, and $D$ is a uniform domain if and only if $D\backslash P$ also is a uniform domain, whenever $P$ is a closed countable subset of $D$ satisfying a quasihyperbolic separation condition. This condition requires that the quasihyperbolic distance (w.r.t. $D$) between each pair of distinct points in $P$ has a lower bound greater than or equal to $\frac{1}{2}$.Comment: arXiv admin note: text overlap with arXiv:1303.3335 by other author

On conformal moduli of polygonal quadrilaterals

The change of conformal moduli of polygonal quadrilaterals under some geometric transformations is studied. We consider the motion of one vertex when the other vertices remain fixed, the rotation of sides, polarization, symmetrization, and averaging transformation of the quadrilaterals. Some open problems are formulated.Comment: 7 figure

Reflections on Ramanujan's Mathematical Gems

The authors provide a survey of certain aspects of their joint work with the late M. K. Vamanamurthy. Most of the results are simple to state and deal with special functions, a topic of research where S. Ramanujan's contributions are well-known landmarks. The comprehensive bibliography includes references to the latest contributions to this field

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

Region of variability for exponentially convex univalent functions

For \alpha\in\IC\setminus \{0\} let $\mathcal{E}(\alpha)$ denote the class of all univalent functions $f$ in the unit disk $\mathbb{D}$ and is given by $f(z)=z+a_2z^2+a_3z^3+\cdots$, satisfying {\rm Re\,} \left (1+ \frac{zf''(z)}{f'(z)}+\alpha zf'(z)\right)>0 \quad {in ${\mathbb D}$}. For any fixed $z_0$ in the unit disk $\mathbb{D}$ and $\lambda\in\overline{\mathbb{D}}$, we determine the region of variability $V(z_0,\lambda)$ for $\log f'(z_0)+\alpha f(z_0)$ when $f$ ranges over the class \mathcal{F}_{\alpha}(\lambda)=\left\{f\in\mathcal{E}(\alpha) \colon f''(0)=2\lambda-\alpha %\quad{and} f'''(0)=2[(1-|\lambda|^2)a+ %(\lambda-\alpha)^2 -\lambda\alpha] \right\}. We geometrically illustrate the region of variability $V(z_0,\lambda)$ for several sets of parameters using Mathematica. In the final section of this article we propose some open problems.Comment: 11 pages and 8 figure

Topics in Special Functions

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