Mappings With Direct And Inverse Poletsky Inequalities 1st Edition Evgeny Sevostyanov by Evgeny Sevost'yanov 9783031454172, 3031454170 instant download after payment.

functions. In itself, the definition of analytic functions is simple and quite accessible

to the average student. However, this definition gives little in terms of the methodology of their research. How, for example, to investigate the question of the boundary

extension of these mappings? It is unlikely that the Cauchy-Riemann conditions

clarify anything in this sense. We believe that, without the Cauchy formula, we

would have a very poor theory of analytic functions without a complete description

of their properties.

Cauchy’s integral formula changes everything. It allows, first of all, to establish

the possibility of expanding an analytic function into a series, from which the

classical theorems of complex analysis follow. Thus, Cauchy’s integral formula

is one of the most important tools for the study of analytic functions, while their

definition itself is not yet such a tool.

Let us suppose that we are now investigating a certain class of mappings, not

necessarily defined on the complex plane. By analogy with analytic functions, we

need to find a research tool for such a class. As a rule, Cauchy’s integral formula is

no longer valid for it, as well as any other integral representation. As in the case of

analytic functions, logically, such a tool should not be completely trivial.

Just such a tool is the modulus of families of paths, and the Poletsky inequalities

are the bridge that connects it with mappings. (The modulus of families of paths is

some outher mesure on families of paths. We give its definitions and main properties

in the text of the monograph.) Since analytic functions can be interpreted as flat

mappings, one can compare the application of the Cauchy integral formula and

the Poletsky inequalities. It is worth noting that Poletsky’s inequalities are more

powerful in