# Miljan KneževićUniversity of Belgrade · Faculty of Mathematics

Miljan Knežević

PhD

## About

8

Publications

3,342

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113

Citations

## Publications

Publications (8)

Let A denotes the absolute plane and da the distance function on it. Using a constructive approach which leads to the functional equations, we study which conditions on a “measure” of segments on a given half-line l in the absolute plane are essential to be the restriction of da on l.

We give a new glance to the theorem of Wan (Theorem 1.1) which is related to the hyperbolic bi-Lipschicity of the K-quasiconformal, K 1, hyperbolic harmonic mappings of the unit disk D onto itself. Especially, if f is such a mapping and f (0) = 0, we obtained that the following double inequality is valid 2|z|/(K + 1) |f (z)| √ K|z|, whenever z ∈ D.

We analyze the properties of harmonic quasiconformal mappings and by comparing some suitably chosen conformal metrics defined in the unit disc we obtain some geometrically motivated inequalities for those mappings (see for instance [15, 17, 20]). In particular, we obtain the answers to many questions concerning these classes of functions which are...

We are analyzing the properties of holomorphic functions and the hyperbolic metric to obtain some geometrically motivated inequalities for quasi-conformal and generalized harmonic mappings. Also, we are interested in which properties of hyperbolic harmonic mappings and the hyperbolic metric are essential for validity of some versions of the Ahlfors...

We provide some theoretical background for infinite descent principle and its relation to the principle of mathematical induction and the well ordering property. Also, we provide some interesting examples by applying the infinite descent principle as an extremal principle in several situations. At the end we prove several assertions which confirm t...

In this paper we consider some facts related to the models of economic growth. A generalization of the golden rule of capital accumulation and dynamic inefficiency is proposed. We analyze the production change and the evolution of the physical capital and give a new approach.

Suppose that h is a harmonic mapping of the unit disc onto a C 1, α domain D. We give sufficient and necessary conditions in terms of boundary function that h is q.c. We announce some new results and also outline application to existence problem of mean distortion minimizers in the Universal Teichmüller space.

We prove versions of the Ahlfors–Schwarz lemma for quasiconformal euclidean harmonic functions and harmonic mappings with respect to the Poincaré metric.

## Projects

Projects (3)

We plane to investigate inequalities related to Geometric Analysis
including isoperimetric type, harmonic and analytic function, Sobolev spaces, etc.

We study the growth of gradients of solutions of elliptic equations, including
the Dirichlet eigenfunction solutions on bounded plane convex domain.
In particular if those mappings are quasiconformal we study Bi-Lipschicity properties with respect to quasi-hyperbolic, euclidean and the others metrics.

The Schwarz lemma as one of the most influential results in complex analysis and it has
a great impact to the development of several research fields, such as geometric function
theory, hyperbolic geometry, complex dynamical systems,
and theory of quasi-conformal mappings.
We plan to study Schwarz lemma at the boundary of strongly
pseudoconvex domains, and versions of now called the Caratheodory-Cartan-Kaup-Wu theorem, which
generalizes the classical Schwarz lemma for holomorphic functions to higher dimensions, and iterations of holomorphic mappings.
PS. Let $G$ be a nonempty domain in a complex
Banach space and let a holomorphic function h maps G strictly inside a subset $G$, then h is a contraction.
We proved this probably around 1980 (we found a hand written manuscript 1990 and did not pay much attention to it at that time).
But we realized these days that it is a version of the Earle-Hamilton (1968) fixed point theorem, which may
be viewed as a holomorphic formulation of Banach's contraction mapping
theorem. The result was proved in 1968 (when I enroled Math Faculty in Belgrade) by Clifford Earle and Richard Hamilton by showing that, with respect to the Carath\'{e}odory metric on the domain, the holomorphic mapping becomes a contraction mapping to which the Banach fixed-point theorem can be applied.