\( \alpha\)-monotone sequences and the Lorentz theorem (Q6115275)

In this note, the authors focus their attention on the properties of \(\alpha\)-monotone sequences when \(0<\alpha<1\). More precisely, the authors obtain a relationship between \(\alpha\)-monotonicity and the limiting rate of change of the corresponding Cesaro numbers (cf. Theorem 1). Also, the authors show that the multiplication of these sequences could not belong to the class \(M_\alpha\) (cf. Theorem 2). Finally, the authors show an analog of the Lorentz theorem on the Fourier series with coefficients from the classes \(M_\alpha\) for \(0<\alpha<1\) of the functions from the classes \(\operatorname{Lip} \beta\), with \(0<\beta<1\) and show that their result is the best possible (see Theorems 3 and 4, respectively).