Intrinsic metrics defined with arithmetic and logarithmic mean values

We introduce several new functions that measure the distance between two points $x$ and $y$ in a domain $G\subsetneq\mathbb{R}^n$ by using the arithmetic or the logarithmic mean of the Euclidean distances from the points $x$ and $y$ to the boundary of $G$. We study in which domains these functions are metrics and find sharp inequalities between them and the hyperbolic metric. We also present one result about their distortion under quasiregular mappings.Comment: 15 page

Notes on the norm of pre-Schwarzian derivatives of certain analytic functions

In this paper, we obtain sharp bounds for the norm of pre-Schwarzian derivatives of certain analytic functions. Initially this problem was handled by H. Rahmatan, Sh. Najafzadeh and A. Ebadian [Stud. Univ. Babes-Bolyai Math. 61(2016), no. 2, 155-162]. We pointed out that their proofs are incorrect and present correct proofs

Harmonic approximations of analytic functions

This paper aims to introduce a measure of the non-univalency of a harmonic mapping. By using it, we find the best approximation of an analytic function by a univalent harmonic mapping.</p

Formulas for the visual angle metric

We prove several new formulas for the visual angle metric of the unit disk in terms of the hyperbolic metric and apply these to prove a sharp Schwarz lemma for the visual angle metric under quasiconformal mapping.Comment: 14 pages, 9 Figure

Landen transformations applied to approximation

We study computational methods for the approximation of special functions recurrent in geometric function theory and quasiconformal mapping theory. The functions studied can be expressed as quotients of complete elliptic integrals and as inverses of such quotients. In particular, we consider the distortion function $\varphi_K(r)$ which gives a majorant for $|f(x)|$ when $f: \mathbb{B}^2 \to \mathbb{B}^2, f(0)=0,$ is a quasiconformal mapping of the unit disk $\mathbb{B}^2.$ It turns out that the approximation method is very simple: five steps of Landen iteration is enough to achieve machine precision.Comment: 14 pages, 4 figure