Miodrag Mateljevic

Inner estimate and quasiconformal harmonic maps between smooth domains

We prove a type of “inner estimate” for quasi-conformal diffeomorphisms, which satisfies a certain estimate concerning their Laplacian. This, in turn, implies that quasiconformal harmonic mappings between smooth domains (with respect to an approximately analytic metric), have bounded partial derivatives; in particular, these mappings are Lipschitz. We discuss harmonic mappings with respect to (a) spherical and Euclidean metrics (which are approximately analytic) (b) the metric induced by a holomorphic quadratic differential.

On harmonic quasiconformal quasi-isometries

The purpose of this paper is to explore conditions which guarantee Lipschitz-continuity of harmonic maps with respect to quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz with respect to quasihyperbolic metrics.

A new version of the main inequality and the uniqueness of harmonic maps

In this paper, we state a new version of an inequality of Reich and Strebel, namely their so-called Main Inequality, and use it to study the uniqueness property of harmonic mappings. To state the main inequality we need the following notation.

Distortion of Quasiregular Mappings and Equivalent Norms on Lipschitz-Type Spaces

We prove a quasiconformal analogue of Koebe’s theorem related to the average Jacobian and use a normal family argument here to prove a quasiregular analogue of this result in certain domains in -dimensional space. As an application, we establish that Lipschitz-type properties are inherited by a quasiregular function from its modulo. We also prove some results of Hardy-Littlewood type for Lipschitz-type spaces in several dimensions, give the characterization of Lipschitz-type spaces for quasiregular mappings by the average Jacobian, and give a short review of the subject.

Uniformly Bounded Maximal ϕ-Disks, Bers Space and Harmonic Maps

We characterize Bers space by means of maximal ϕ-disks. As an application we show that the Hopf differential of a quasiregular harmonic map with respect to strongly negatively curved metric belongs to Bers space. Also we give further sufficient or necessary conditions for a holomorphic function to belong to Bers space.

The Lower Bound for the Modulus of the Derivatives and Jacobian of Harmonic Injective Mappings

We give the lower bound for the modulus of the radial derivatives and Jacobian of harmonic injective mappings from the unit ball onto convex domain in plane and space. As an application we show co-Lipschitz property of some classes of qch mappings. We also review related results in planar case using some novelty.

An extension of the area theorem

We prove an inequality between the area of the omitted set of an analytic function f and its coefficients. We show that equality holds if and only if f is a univalent function.

Energy of Graphs and Orthogonal Matrices

In this paper, we characterize graphs of maximal energy by means of orthogonal matrices. The result makes it possible to estimate energy of graphs without direct computation of eigenvalues. As an illustration, we compute the maximum energy among all graphs with n = 4 k vertices, which corresponds to strongly regular graphs found by Koolen and Moulton, and apply our result to conference graphs, computing the asymptotic formula for maximal graph energy.

Extremal metrics and modulus

AbstractWe give a new proof of Beurlings result related to the equality of the extremal length and the Dirichlet integral of solution of a mixed Dirichlet-Neuman problem.Our approach is influenced by Gehrings work in