On presentation of Gelfond-Leontiev operators of generalized differentiation in simply connected region

Some new properties of the multiplier are determined. A class of simply connected regions whose multiplier is a connected set is described. This class is characterized by the availability of spirals in a multiplier. Let the Gelfond - Leontiev generalized differentiation operator be continuous in the space of the analytic functions in simply connected region G of a complex plane. It is known to be presented as an operator of general complex convolution. The convolution kernel is generated by the many-valued function of one variable. The set M(G) with the property M(G)·G⊆G is called multiplier G. Let the region multiplier be connected, and it does not align with identity. It is proved in the paper that the functions under consideration will be univalent under these conditions. If multiplier G is unconnected, then there is always a generalized differentiation Gelfond-Leontiev operator with a many-valued generating function.