Geodesic
Estimating the integral of the derivative of the sum of a trigonometric series with quasi-convex coefficients
Sbornik. Mathematics, Tome 186 (1995) no. 11, pp. 1659-1669 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {S. A. Telyakovskii},
     title = {Estimating the integral of the~derivative of the~sum of a~trigonometric series with quasi-convex coefficients},
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S. A. Telyakovskii. Estimating the integral of the derivative of the sum of a trigonometric series with quasi-convex coefficients. Sbornik. Mathematics, Tome 186 (1995) no. 11, pp. 1659-1669. http://geodesic.mathdoc.fr/item/SM_1995_186_11_a4/

[1] Zigmund A., Trigonometricheskie ryady, GONTI, M.–L., 1939

[2] Telyakovskii S. A., “Nekotorye otsenki dlya trigonometricheskikh ryadov s kvazivypuklymi koeffitsientami”, Matem. sb., 63 (105) (1964), 426–444 | MR

[3] Telyakovskii S. A., “Lokalizatsiya uslovii integriruemosti trigonometricheskikh ryadov”, Tr. MIAN, 210, Nauka, M., 1995, 264–273 | MR | Zbl