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1
Content available remote The real and complex convexity
EN
We prove that the holomorphic differential equation ϕ’’(ϕ+c) = γ(ϕ’)² (ϕ:C→C be a holomorphic function and (γ, c) ϵ C²) plays a classical role on many problems of real and complex convexity. The condition exactly γ ϵ [wzór] (independently of the constant c) is of great importance in this paper. On the other hand, let n ≥ 1, (A₁, A₂) ϵ C² and g₁, g₂ : Cᵑ → C be two analytic functions. Put u(z, w) = │A ₁w - g₁(z) │² + │A₂w - g₂(z) │²v(z,w) = │A₁w - g₁(z) │² + │ A₂w - g₂(z) │², for (z,w) ϵ Cᵑ x C. We prove that u is strictly plurisubharmonic and convex on Cᵑ x C if and only if n = 1, (A₁, A₂) ϵ C² \{0} and the functions g₁ and g₂ have a classical representation form described in the present paper. Now v is convex and strictly psh on Cᵑ x C if and only if (A₁, A₂) ϵ C² \{0}, n ϵ {1,2} and and g₁, g₂ have several representations investigated in this paper.
EN
The regularity theorem is a result stating that functions which have extremal growth or decrease in the given class display a regular behaviour. Such theorems for linearly invariant families of analytic functions are well known. We prove regularity theorems for some classes of harmonic functions. Many presented statements are new even in the analytic case.
EN
In this paper we define classes of harmonic functions related to the Janowski functions and we give some necessary and sufficient conditions for these classes. Some topological properties and extreme points of the classes are also considered. By using extreme points theory we obtain coefficients estimates, distortion theorems, integral mean inequalities for the classes of functions.
PL
Praca podejmuje problematykę planowania ruchu i sterowania robota mobilnego w środowisku dwuwymiarowym z eliptyczną przeszkodą statyczną w oparciu o metodę przepływu płynu. Opisano w niej algorytm projektowania strumienia dla przeszkody eliptycznej w przypadku celu statycznego i dynamicznego (trajektorii). Rozważania teoretyczne poparto wynikami symulacji numerycznych i wynikami badań eksperymentalnych, w których wykorzystano robota dwukołowego i metodę linearyzacji modelu kinematyki.
EN
The paper presents planning motion problem and control of mobile robot in two-dimensional environment with elliptical static obstacle based on hydrodynamics description. The design algorithm of the flow with respect to elliptical obstacle for static and dynamic goal is discussed. Theoretical considerations are supported by numerical simulations as well as experimental results using two-wheeled mobile robot and linearization technique.
5
EN
In this article we investigate some classes of meromorphic or complex harmonie functions with a pole, which are generated either by analytic conditions or by "coefficient inequalities". There are given theorems, which combine the geometrical properties of functions of the introduced classes. Some results broaden knowledge about the classes of functions, which were investigated in [15]. The main inspiration for the reaserch were the papers [4] and [11]. The part of results were presented in the XII-th International Mathematically-Informatical Conference in Chełm (2nd-5th July, 2006) [12].
6
Content available remote On weighted harmonic Bergman spaces
EN
This paper is devoted to the investigation of the weighted Bergman harmonic spaces bp/alpha(B] in the unit ball in Rn. The reproducing kernel Ralpha for the ball is constructed and the integral representation for functions in bp/alpha(B) by means of this kernel is obtained. Besides an linear mapping between the bp/alpha(B) spaces and the ordinary L2-space on the unit sphere, which has an explicit form of integral operator along with its inversion, is established.
EN
In 1984 J. Clunie and T. Sheil-Small initiated studies of complex functions harmonic in the unit disc. In 1987 W. Hergartner and G. Schober considered mappings of this type, defined in the domain U = {z is an element of C : \z\ > 1}. Several mathematicians examine classes of complex harmonic functions with some coefficient conditions, defined in the unit disc (e.g. [2], [5], [10], [1] [9]) or in U (e.g. [8], [7]). We investigate the classes of mappings harmonic in U with coefficient conditions more general than the considered in paper [8].
8
Content available remote On some generalization of coefficient conditions for complex harmonic mappings
EN
Let h = u + iv, where u, v are real harmonic functions in the unit disc A. Such functions are called complex mappings harmonic in A. The function h may be written in the form h = f + g, where f,g are functions holomorphic in the unit disc, of course. Studies of complex harmonic functions were initiated in 1984 by J. Clunie and T. Sheil-Small ([2]) and were continued by many others mathematicians. We can find some papers on functions harmonic in A, satisfying certain coefficient conditions, e.g. [1], [4], [6], [7], [8]. We investigate some more general problem, i. e. a coefficient inequality with any fixed sequence of real positive numbers.
EN
In the present paper the second order differential subordination (1 - alpha)f(z/z + alphaf'{z) + betazf(z) [...] 1 + Mz \z is an element of U) is investigated. The best dominant of subordination (1) is founded. Connections between subordination (1) and subordination of f(z)/z, f'{z) are given. Further the convexity of the function / satisfying the subordination (1) for special choice of parameters a, alpha, beta and M are derived.
EN
In this paper we present some properties of normalized (fixed in points x1, x2, infinity k-quasihomographies of the real line.
EN
Let h = u + w, where u,v are real harmonic functions in the unit disc delta. Such functions are called complex mappings harmonic in delta. The function h may be written in the form h = f + g, where f, g are functions holomorphic in the unit disc, of course. Studies of complex harmonic functions were initiated in 1984 by J. Clunie and T. Sheil-Small ([CS-S]) and were continued by many others mathematicians. We can find some papers on functions harmonic in delta, satisfying certain coefficient conditions, e.g. [AZ], [S], [G]. We investigate some more general problems, which appeared during the seminar conducted by Professor Z. Jakubowski.
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