Estimating the integral of the derivative of the sum of a trigonometric series with quasi-convex coefficients

For the sum of series in sines or in cosines with quasi-convex coefficients, representation of their first derivatives are found in terms of the second differences of the sequence of coefficients of the series obtained by formal differentiation. Estimates of the integrals of the absolute values of these derivatives over a part of the period are obtained in terms of coefficients of the series.