Paweł Zaprawa

Estimates of Initial Coefficients for Bi-Univalent Functions

We consider the Fekete-Szego inequalities for classes which were defined by Murugusundaramoorthy et al. (2013). These inequalities will result in bounds of the third coefficient which are better than these obtained by Murugusundaramoorthy et al. (2013). Moreover, we discuss two other classes of bi-univalent functions. The estimates of initial coefficients in these classes are obtained.

Domains of Univalence for Typically-real Odd Functions

The univalence problems in the class of typically-real functions were considered by Golusin [G. Golusin (1950). On typically-real functions, (Russian). Mat. Sb. , 27 (69), 201-218]. He proved that the domain of local and global univalence for typically-real functions coincide. In this note we have proved that this is not the case for typically-real odd functions. Moreover, we have observed that for this class there can be infinitely many domains of univalence. We have determined some of them.

Second Hankel Determinants for the Class of Typically Real Functions

We discuss the Hankel determinants for typically real functions, that is, analytic functions which satisfy the condition in the unit disk Δ. Main results are concerned with and . The sharp upper and lower bounds are given. In general case, for , the results are not sharp. Moreover, we present some remarks connected with typically real odd functions.

Covering domains for classes of functions with real coefficients

The integral formulae for the classes of univalent functions having real coefficients, namely for CVR, CVR(i), STR(1/2) consisting of convex functions, convex in the direction of the imaginary axis functions, and starlike of order 1/2 functions, are known. However, studying those classes with the use of appropriate formulae is rather difficult. The main idea of this article is to obtain some results for the special class TR(1/2), which is a superclass of CVR, CVR(i), STR(1/2). It occurs that the results remain true also for these subclasses of TR(1/2). We will be concerned with the generalized Koebe domains and generalized covering domains over some subsets of .

Covering problems for functions n-fold symmetric and convex in the direction of the real axis

Abstract Let F denote the class of all functions univalent in the unit disk and convex in the direction of the real axis. In the paper we discuss the functions of the class F which are n -fold symmetric, where n is positive even integer. For the class of such functions we find the Koebe set as well as the covering set, i.e. ⋂ f ∈ F f ( Δ ) and ⋃ f ∈ F f ( Δ ) . Moreover, the Koebe constant and the covering constant are obtained.

The Concept of Robotic Station for Measurement of the Friction Force between Fibre-Optic Cable and Duct

The interaction of an optotelecommunication cable and casing depends on numerous factors e.g. the type of cable, its structure, dimensions, as well as applied lubricants and sliding layers [8]. In addition, the course of the cable, velocity of installation and its maximum force also play a role in the interaction of the tribological pair. The paper outlines the idea and operation of a robotized testing station for evaluation of sliding friction. The station enables cables’ friction to be determined in relation to the length of contact between the cable and the casing.

The Influence of the Shape of a Polymer Wall on Selected Mechanical Properties

This paper presents results of personal research concerning designing a new construction in the form of axially symmetric ducts, characterized by a complex construction of the carrying wall. These ducts were designed so that in the carrying wall they have a specified number of micro canals, in which additional cables can be placed. This paper presents research results of selected mechanical properties of the products, the values of force required to achieve a deformation of 3%, useful to calculating the peripheral stiffness of the construction. The research results allowed to determine the range of change of thinning possibility of the wall of the product in relation to the volume of the material and the mathematical function depicting the outline of the duct in cross-section.

The Analysis of Selected Rheological Models of the Cement Dispersion in the Pipe Flow

The paper presents the new flow equations of the cement dispersion, using different rheological models. The derived solutions are exact ones, obtained without previous approximations. The results of calculations carried out on the basis of the presented equations are compared to available literature data, obtained their good agreement.