Uniform Convergence of Sine Series with Fractional-Monotone Coefficients

We study how the well-known criterion for the uniform convergence of a sine series with monotone coefficients changes if, instead of monotonicity, one imposes the condition of $\alpha$-monotonicity with $0<\alpha <1$. Moreover, we obtain an addition to the well-known Kolmogorov theorem on the integrability of the sum of a cosine series with convex coefficients tending to zero.