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Abstract:
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This work consists of three chapters. The first one contains some well known
facts about Hardy classes of harmonic, analytic, and logarithmically subharmonic functions
in the unit disk, as well as their applications. Then we briefly talk about the harmonic
and minimal surfaces, the classical isoperimetric inequality, and the more recent
results related to this inequality. One of the most elegant way to establish the isoperimetric
inequality is via Carleman’s inequality for analytic functions in disks. In the second
chapter we present the results from our recent work [29] for harmonic mappings of a disc
onto a Jordan surface. In this chapter we establish the versions of classical theorems of
Carath´eodory and Smirnov for mappings of the previous type. At the end of the head
we apply these results to prove the isoperimetric inequality for Jordan harmonic surfaces
bounded by rectifiable curves. In the third chapter, according to the author paper [35], we
prove an inequality of the isoperimetric type, similar to Carleman’s, for functions of several
variables. The first version of this inequality is for analytic functions in a Reinhardt
domain. The second one concerns the functions that belong to Hardy spaces in polydiscs. |