On the existence of analytical functions with nonvanishing finite differences of several orders in the half-plane
Santrauka
In the present paper multistage nite di erences of di erent orders for functions, analytical in the half-plane, are investigated. The main result is assertion on the absence of functions analytic in the half-plane, which have two nonvanishing nite di erences and the di erence between their orders exceeds a certain positive number. The proof is based on the properties of univalent functions. In particular, estimations for the derivatives of univalent functions, obtained by I.A. Aleksandrov as the corollary of Louis de Branges theorem about the coe cients of the univalent and normalized in the unit disk functions, are used.