Universal holomorphic and harmonic functions with additional properties

Based on the concept of so-called (total) omnipresence of operators, several results on the generity of (translation-dilation) universal functions are proved. Mainly to have a unified approach to holomorphic and harmonic functions, in the first part operators on spaces of P-holomorphic functions are considered. The second part is devoted to holomorphic functions having lacunary power series structure and to holomorphic functions which are univalent in certain prescribed sets.