$\alpha$-monotone sequences and the Lorentz theorem

Properties of $\alpha$-monotone sequences are studied. A relationship between $\alpha$-monotonicity and the limiting rate of change of coefficients is revealed. Operations on sequences that do not lead out of the class $M_\alpha$ are discussed. An analogue of the Lorentz theorem for trigonometric series with coefficients from the classes $M_\alpha$ for $0 <\alpha <1$ is proved.